monotone operator theory for unsteady problems on non-cylindrical

MONOTONE OPERATOR THEORY FOR UNSTEADY PROBLEMS ON NON-CYLINDRICAL DOMAINS

Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau

vorgelegt von Philipp Nägele

April 2015

Dekan: 1. Gutachter: 2. Gutachter: Datum der mündlichen Prüfung:

Prof. Dr. Dietmar Kröner Prof. Dr. Michael Růžička Prof. Dr. Céline Grandmont 24.07.2015

Erklärung Ich erkläre hiermit, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt übernommenen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet. Insbesondere habe ich hierfür nicht die entgeltliche Hilfe von Vermittlungs- beziehungsweise Beratungsdiensten (Promotionsberater/-beraterinnen oder anderer Personen) in Anspruch genommen. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer Prüfungsbehörde vorgelegt.

Ort, Datum

Unterschrift

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Contents Notation

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0. Introduction

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1. Tensor Analysis on Manifolds and Selected Applications 1.1. Manifolds and Tangent Spaces . . . . . . . . . . . . . 1.2. Vector Bundles . . . . . . . . . . . . . . . . . . . . . 1.3. Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. The Lie Derivative . . . . . . . . . . . . . . . . . . . 1.5. k-Forms and the Bundle Λk (M ) . . . . . . . . . . . . 1.6. Extension of the Lie Derivative . . . . . . . . . . . . 1.7. The Exterior Derivative and Cartan’s Magic Formula 1.8. The Piola Transform . . . . . . . . . . . . . . . . . . 1.9. The Reynolds Transport Theorem . . . . . . . . . .

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7 7 9 11 13 17 19 20 22 25

2. Function Spaces on Non-cylindrical Domains 2.1. Non-cylindrical Domains . . . . . . . . . . . . . . . . . . 2.2. Isomorphisms Induced by Diffeomorphisms . . . . . . . . 2.3. Generalized Bochner Spaces . . . . . . . . . . . . . . . . 2.3.1. Time Derivatives in Bochner Spaces . . . . . . . 2.3.2. Time Derivatives in Generalized Bochner Spaces

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3. Unsteady Monotone Problems on Non-cylindrical Domains 3.1. Informal Discussion of Relevant Examples and Results . 3.2. The p-Stokes System as a Model Problem . . . . . . . . 3.3. Galerkin Approximation . . . . . . . . . . . . . . . . . . 3.4. Density Result . . . . . . . . . . . . . . . . . . . . . . . 3.5. Integration by Parts Formula . . . . . . . . . . . . . . . 3.6. Minty’s Trick . . . . . . . . . . . . . . . . . . . . . . . .

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4. Unsteady Problems with Compact Perturbations 4.1. The Compactness Principle of Landes and Mustonen . . . . . . 4.2. Existence of Solutions for the Truncated Problems . . . . . . . 4.3. Fixed Point Argument . . . . . . . . . . . . . . . . . . . . . . . 4.4. Passage to the Limit in the Truncated Nonlinearity . . . . . . . 4.5. The Navier–Stokes Equations on Time-Dependent Domains . . 4.5.1. Functional Setting and Weak Formulation . . . . . . . . 4.5.2. Existence of Approximate Solutions and Fixed Points . 4.5.3. Passage to the Limit in the Truncated Convective Term

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5. The Aubin–Lions Lemma in Generalized Bochner Spaces 5.1. Basic Classical Version of the Aubin–Lions Lemma . . . . . . 5.2. Towards a Non-cylindrical Version of the Aubin–Lions Lemma 5.2.1. Arzelà–Ascoli-type Argument . . . . . . . . . . . . . . 5.2.2. Ehrling-type Estimates . . . . . . . . . . . . . . . . . . 5.3. The Non-cylindrical Aubin–Lions Compactness Lemma . . . . 5.4. Uniform Ehrling Property of the Basic Function Spaces . . .

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6. Shear Thickening Fluids in Non-cylindrical Domains 6.1. Motivation and Model . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Nonhomogeneous Boundary Conditions and a Pitfall in the Penalty 6.2.1. Lacking Energy Identity . . . . . . . . . . . . . . . . . . . . 6.2.2. Non-Applicability of the Classical Aubin–Lions Lemma . . . 6.3. Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Compactness of the Solution Operator . . . . . . . . . . . . 6.3.2. Passage to the Limit . . . . . . . . . . . . . . . . . . . . . . 6.4. Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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109 . 109 . 111 . 111 . 114 . 115 . 116 . . . . . . . .

121 121 125 125 128 136 139 145 147

7. Shear Thinning Fluids in Non-cylindrical Domains 151 7.1. Existence of Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2. Identification via Solenoidal Lipschitz Truncation . . . . . . . . . . . . . . . . 161 8. Outlook A. Appendix A.1. Lebesgue Spaces . . A.2. Sobolev Spaces . . . A.3. Bochner Spaces . . . A.4. Monotone Operators A.5. Auxiliary Results . . Bibliography

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Notation c, C, c(k) x·y A:B u⊗v ∇ Du Tϕ div ∆ ∂t d dt

dt ϕ∗ , ϕ∗ Jϕ Pϕ LX LX = ∂t + LX Z

·, ·Z ·, · H ∗ →, *, * i

,→, ,→, ,→,→ Ω(t) Q Γ Lr (I, Z) W 1,p,q (Z, Y ) Lr (I, X(t)) 0 W 1,p,p (Vσ (t), Vσ (t)0 ) MB1 (t)0 (un , η, Ft ) NB1 (t)0 (un , Ft )

Generic constants that may change from line to line. The constant c(k) especially depends on k Euclidean scalar product of two vectors Matrix scalar product Tensor product of two vector fields (Weak) gradient with respect to the spatial variables Symmetric part of the gradient of u Tangent mapping of a mapping between (local) manifolds, see (1.2). In the Euclidean case, T ϕ is identified with ∇ϕ Divergence with respect to the spatial variables Laplacian with respect to the spatial variables (Weak) time derivative Time derivative in the sense of vector-valued distributions, see (2.13) Generalized time derivative, see Definition 2.12 Pull-back and push-forward with respect to ϕ Jacobian determinant of a diffeomorphism ϕ, see (1.20) Piola transform of a vector field with respect to ϕ, see Definition 1.4 Lie derivative with respect to a vector field X, see (1.6) Dynamic Lie derivative with respect to X, see (1.7) Generic Banach space with dual space Z 0 Duality pairing in Z Scalar product in the Hilbert space H Strong, weak and weak-∗ convergence in Banach spaces or dual spaces Continuous, injective and compact embedding Deformed initial domain at time t, see (2.1). Noncylindrical space-time domain, see (2.3) Lateral boundary of a noncylindrical domain Q, see (2.4) Generic Bochner space Sobolev-Bochner space, see (2.14) Generic generalized Bochner space, see Definition 2.6 Generalized Sobolev-Bochner space, see Theorem 3.20 Functional defined in Theorem 5.4 Functional defined in Theorem 5.4

1

2

Notation

0. Introduction In the 50 years that have passed since Minty introduced the concept of monotonicity for mappings between Banach spaces and their dual spaces, the theory of monotone operators has become an indispensable tool in many areas of nonlinear analysis and nonlinear partial differential equations (PDEs). While Minty first applied his theory to the abstract study of electrical networks, it was Felix Browder who extended the method to study fixed point problems, the Brouwer degree, the calculus of variations and especially nonlinear elliptic and parabolic PDEs. This progress was accompanied by further seminal contributions of Brezis, Vishik, Lions and Landes to name just a few. It is hardly possible to give a complete list of the most important publications in retrospect, even more so as such a list would inevitably reflect the personal liking of the author. Nevertheless, Lions’ monograph [Lio69] is certainly one of the landmark publications for the field of nonlinear differential equations. The sheer extent of the topics treated in that book is an evidence for the relevance of the combination of monotonicity and compactness methods as abstract tools for tackling a vast variety of problems arising in applied mathematics. One particular application we are interested in is the mathematical theory of fluid dynamics and especially the theory of incompressible generalized Newtonian fluids. These fluids are characterized by their ability to change their mechanical properties due to applied or internal shear stresses. In the constitutive theory for generalized Newtonian fluids, this behaviour manifests itself in a strongly nonlinear dependence of the Cauchy stress tensor S on the symmetric part of the deformation tensor of the fluid Du. One of the typical examples for a nonlinear Cauchy stress tensor is given by the relation p−2 S(Du) := Du Du for some exponent 1 < p < ∞. The latter relation then leads to the p-Laplacian operator p−2 −div Du Du which in turn appears in the so-called p-Navier-Stokes equations for the velocity u and the pressure π of an incompressible generalized Newtonian fluid. These equations read as p−2 ∂t u + u · ∇ u = f + div Du Du − π idR3 , div u = 0, and have to be complemented by suitable boundary and initial conditions. Since the p-Laplacian operator is one of the most prominent examples in the theory of monotone operators, it will come as no surprise that the p-Navier–Stokes equations had already been treated in Lions’ book to further illustrate this theory. However, the mathematical theory of generalized Newtonian fluids still is subject to ongoing research activity. Besides numerous applications in engineering sciences, a further current field of application is the

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4

Chapter 0. Introduction

modelling of the human cardiovascular system and especially the blood flow in human arteries. This is simply due to the fact that the mechanical properties of red blood cells suspended in liquid plasma give rise to a non-Newtonian behaviour of blood. But still, there is another substantial peculiarity one has to take into account: The elastic tissue surrounding the blood may itself change its shape due to the dynamics caused by the mechanical interaction between the blood and the tissue. On the mathematical side, the equations governing the dynamics of blood flow therefore have to be formulated on domains Ω(t) that vary in time. The aim of the present work is to extend some of the cornerstones of the classical theory of monotone operators in order to develop an existence theory for a certain class of nonlinear initial-boundary value problems that are formulated on non-cylindrical domains. Although we are interested in a rather general theory, we shall elaborate our approach using variants of the incompressible p-Navier-Stokes equations as guiding examples, since these already contain typical mathematical challenges for a fruitful abstract theory. To this end, we shall start by recalling basic constructions from tensor analysis in the first chapter. We use them to define generalized Bochner spaces as a functional framework for evolution equations on time-dependent domains in Chapter 2. Chapter 3 is devoted to the well-posedness of the unsteady p-Stokes equations in non-cylindrical domains. There, our results rely on an extension of the classical integration by parts formula to the framework of generalized Bochner spaces. This tool enables us to apply Minty’s monotonicity trick in the existence proof. In Chapter 4 we turn our attention to abstract unsteady nonlinear problems on non-cylindrical domains, containing also lower order compact perturbations. Here, we first elaborate on a seemingly less-known compactness result of Landes and Mustonen and illustrate its applicability by showing existence of weak solutions of the unsteady Navier-Stokes equations on a time-dependent domain. The generalization of the well-known Aubin-Lions compactness lemma to the framework of generalized Bochner spaces is the content of Chapter 5. This result is subsequently used in Chapter 6 to prove existence of weak solutions of an unsteady p-Navier-Stokes system governing the dynamics of shear-thickening fluids in non-cylindrical domains. In this chapter, we shall also discuss problems connected to lacking energy identities on the one hand and to the penalty method in the context of unsteady incompressible fluid flow on the other hand. Chapter 7 contains an existence result for shear-thinning fluids in non-cylindrical domains. This result is based on the implications of the so-called solenoidal Lipschitz truncation method. Eventually, in Chapter 8 we briefly sketch further fields of application for the current methods.

5

Acknowledgement I would like to express my gratitude to my advisor Prof. Dr. Michael Růžička for his guidance, his continual patience and also for constantly providing a very nice atmosphere. I am deeply indepted to my friends and colleagues Sarah Eckstein, Daniel Lengeler, Hannes Eberlein, Erik Bäumle, Susanne Knies, Johannes Daube, Swen Kiesel and Yann Bernard for hours of discussion as well as for the huge fun we had during our time together at the department. There are of course many other people who, wittingly or unwittingly, contributed to the present work through encouragement or support, so that it is hardly possible to mention every single one. Instead, I want to dedicate this work to my mother Marija in return for her infinite affection, for completely backing me up and if nothing else, for helping me and my brother in literally every possible way!

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Chapter 0. Introduction

1. Tensor Analysis on Manifolds and Selected Applications 1.1. Manifolds and Tangent Spaces In this introductory chapter we present the basic definitions and constructions from differential geometry and tensor analysis on manifolds. The reason for this is the following: Although we are interested in equations on subsets of the Euclidean space Rd we will have to use (geometric) transformations of vector fields that will become much clearer if they are formulated in the language of differential geometry. Our presentation will closely follow the one given in [AMKK67], [AMR88] and [MH94]. We start with the definition of d-dimensional C k -manifolds as these are the basic objects of differential geometry. Let M be a set. We call a mapping ϕ : U → ϕ(U ) from a subset U ⊂ M to an open subset of Rd a local chart. For k ∈ N ∪ {∞}, a C k -atlas A := { (Ui , ϕi ) | i ∈ I } is a family of charts with the property that the sets Ui cover M and that the overlap maps d ϕji := ϕj ◦ ϕ−1 i : ϕi (Ui ∩ Uj ) → R , i, j ∈ I,

are C k -diffeomorphisms for any choice of two local charts (Ui , ϕi ), (Uj , ϕj ) ∈ A with Ui ∩Uj 6= ∅. Here, we also require the sets ϕi (Ui ∩ Uj ) to be open in Rd . Two C k -atlases A1 and A2 are called equivalent if the union A1 ∪ A2 is a C k -atlas. This notion of equivalence induces an equivalence relation on C k -atlases. A differentiable structure D on M is an equivalence class of C k -atlases. A C k -manifold is a pair (M, D) consisting of a set M and a differentiable structure D on M . A local chart (Ui , ϕi ) ∈ D is called an admissible chart. A differentiable manifold M is a d-dimensional manifold if every admissible chart takes its values in a subset of Rd . Any differentiable manifold is a topological space by declaring A ⊂ M to be open if for every a ∈ A there is an admissible chart (U, ϕ) such that a ∈ U and U ⊂ A. Any open subset Ω ⊂ Rd is a d-dimensional smooth manifold with the single chart (Ω, idRd ). Open subsets in Euclidean space will be called local manifolds. If (M1 , D1 ) and (M2 , D2 ) are two differentiable manifolds we can form the product manifold M1 ×M2 with the differentiable structure D1 ×D2 generated by { (U1 ×U2 , ϕ1 ×ϕ2 ) | (Ui , ϕi ) ∈ Di , i = 1, 2 }. As a consequence, any cylindrical domain I×Ω, where I ⊂ R is an open interval and Ω ⊂ Rd is open, can be seen as a (d + 1)-dimensional manifold in R × Rd ∼ = Rd+1 . The next step is to introduce the notion of smoothness for mappings between C k -manifolds and especially the tangent mapping which generalizes the classical derivative. Let M and N be two C k -manifolds. A mapping f : M → N is said to be of class C r , 0 ≤ r ≤ k, if for every m ∈ M and every admissible chart (V, ψ) on N with f (m) ∈ V there is an admissible chart (U, ϕ) on M with m ∈ U such that the local representative of f fϕψ := ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ),

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Chapter 1. Tensor Analysis on Manifolds and Selected Applications

is C r in the Euclidean sense. In the case r = 0, this definition of continuity is consistent with the continuity of f regarded as a mapping between topological spaces. Also, if f : M → N is continuous then it is a C r -mapping if the local representatives of f are C r . Using local representatives, it is readily checked that the concatenation of two C r mappings between manifolds yields a C r mapping. A map f : M → N between manifolds is called a C r diffeomorphism if f is C r , bijective and if the inverse f −1 : N → M is C r , too. Before we come to define the tangent space and the tangent bundle of a manifold M , we first define the notion of tangency of curves in M , which may be seen as the geometric approach of defining these objects. A curve c at m ∈ M is a C 1 -mapping from an open interval containing 0 to M such that c(0) = m. Two curves c1 and c2 at m ∈ M are tangent at m with respect to an admissible chart (U, ϕ) with m ∈ U if (ϕ ◦ c1 )0 (0) = (ϕ ◦ c2 )0 (0). Note that ϕ ◦ ci : I → Rd , i = 1, 2, is a curve in Euclidean space Rd and the prime denotes the usual differentiation on I ⊆ R. It is easily checked that this notion of tangency is independent of the chart and therefore induces an equivalence relation on the set of curves at m ∈ M . If we let [c]m denote a representative of such a class of curves then the tangent space to m ∈ M is defined by n o Tm M := [c]m c is a curve at m ∈ M . The tangent bundle T M of M is then defined by [ T M := Tm M. m∈M

The mapping τM : T M → M given by τM ([c]m ) := m is called the tangent bundle projection. If Ω ⊂ Rd is an open subset then T Ω can be identified with Ω × Rd . In fact, if the curve c is tangent at x ∈ Ω, Taylor’s theorem shows that the curve defined by cx,c0 (0) (t) := x + tc0 (0) is tangent at x, too. This in turn implies that the mapping i : Ω × Rd → T Ω, i(x, e) := [cx,e ]x

(1.1)

yields a bijection. Now, if f : M → N is a C 1 -mapping between manifolds and c1 , c2 are tangent at m ∈ M , it follows that the curves f ◦ c1 and f ◦ c2 are tangent at f (m) ∈ N . Therefore, the tangent T f : T M → T N is defined by T f ([c]m ) := [f ◦ c]f (m) . With this definition, the following diagram commutes: TM τM

Tf

M

/ TN

f

τN

/ N.

(1.2)

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1.2 Vector Bundles

For a C 1 -mapping f : Ω1 ⊂ Rd → Ω2 ⊂ Rn between local manifolds, the derivative f 0 is given by f 0 : Ω1 × Rd → Ω2 × Rn , f 0 (x, e) := f (x), ∇f (x)e ,

(1.3)

where ∇f : Ω1 → Rd×n denotes the usual Jacobian matrix of f . Using the bijections i we defined in (1.1), the diagram Ω1 × Rd i

f0

T Ω1

/ Ω2 × Rn

Tf

i

/ T Ω2

commutes, too. Indeed, we have (T f ◦ i)(x, e) = T f ([cx,e ]x ) = [f ◦ cx,e ]f (x) , (i ◦ f 0 )(x, e) = i(f (x), ∇f (x)e) = [cf (x),∇f (x)e ]f (x) and the curves t 7→ f (x+te) and t 7→ f (x)+t∇f (x)e are tangent at f (x) by definition of the derivative ∇f . In this situation, we identify f 0 and T f without mentioning the bijections i. Let f : M → N and g : N → K be C 1 -mappings between manifolds. Using local representatives, we can check immediately that g◦f : M → K is a C 1 -mapping with T (g◦f ) = T g◦T f , since T (g ◦ f )([c]m ) = [g ◦ f ◦ c]g◦f (m) = T g([f ◦ c]f (m) ) = (T g ◦ T f )([c]m ). The latter property implies that for a diffeomorphism f : M → N , the tangent T f : T M → T N is a bijection with inverse (T f )−1 = T (f −1 ).

1.2. Vector Bundles Suppose E and F are (finite dimensional) vector spaces and U ⊂ E is an open subset. Then the product U × F is called a local vector bundle. U is called the base space and can be identified with the zero section U × {0}. The fiber over u ∈ U is the set {u} × F , endowed with the vector space structure of F . The map π : U × F → U , π(u, f ) := u is called the projection of U × F . Local vector bundle maps are maps between local vector bundles that are compatible with the vector space structure of the respective fibers. That is, a map ϕ : U × F → U 0 × F 0 between local vector bundles is said to be a C r local vector bundle map if it has the form ϕ(u, f ) := (ϕ1 (u), ϕ2 (u)f ),

(1.4)

where ϕ1 : U → U 0 and ϕ2 : U → L(F, F 0 ) are C r . A local vector bundle isomorphism is a local vector bundle map whose inverse is a local vector bundle map, too. As we already mentioned, if f : Ω1 → Ω2 is a C r+1 -mapping between open subsets of finite dimensional Euclidean spaces, then the tangent map T f : T Ω1 → T Ω2 is an example of a local vector

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bundle map, see (1.3). Following the patterns of the definition of a manifold we can define a vector bundle by first defining charts. Let therefore S be a set and let U × F be a local vector bundle. A local bundle chart is a subset W ⊂ S together with a map ϕ : W ⊂ S → U × F onto the local vector bundle. A vector bundle atlas on S is a family B = (Wi , ϕi ) | i ∈ I of local bundle charts such that S is covered by the union of the sets Wi and such that the overlap maps ϕji := ϕj ◦ ϕ−1 i restricted to ϕ(Wi ∩ Wj ) are C ∞ local vector bundle isomorphisms, provided that Wi ∩ Wj 6= ∅. Here, we also require the sets ϕ(Wi ∩ Wj ) to be local vector bundles. As in the case of manifolds we call two vector bundle atlases B1 and B2 vector bundle equivalent if their union is a vector bundle atlas. A vector bundle E is a pair (S, V) where S is a set and V is a vector bundle structure, which is defined as an equivalence class of vector bundle atlases. In this sense, any vector bundle E is a manifold with differentiable structure induced by V. Especially, the tangent bundle T M of a manifold M is itself a manifold. In fact, if A = { (Ui , ϕi ) | i ∈ I } is an atlas on M , then T ϕi : T Ui → T (ϕi (Ui )), T ϕi ([c]u ) := (ϕi (u), (ϕi ◦ c)0 (0)) defines a bijection for any i ∈ I since any ϕi is a diffeomorphism. Hence, it makes sense to call T A := { (T Ui , T ϕi ) | (Ui , ϕi ) ∈ A } the natural vector bundle atlas on T M . The fact that all overlap maps are smooth follows from the identity T ϕj ◦ T ϕ−1 = i T (ϕj ◦ ϕ−1 ). i The tangents of charts on a manifold M can further be used to construct a basis on each fiber Tm M . Since this construction is local, we first choose a chart (U, ϕ) ∈ A, m ∈ U . The tangent T ϕ : T U → T (ϕ(U )) is a smooth bijection. Since ϕ(U ) ⊂ Rd is open, we may identify T (ϕ(U )) and ϕ(U ) × Rd via the bijection i. As { (ϕ(m), ej ) | j = 1, .., d } is a basis at any fiber of the form {ϕ(m)} × Rd ∈ ϕ(U ) × Rd , the set n o i((ϕ(m), ej )) = ([cϕ(m),ej ]ϕ(m) ), j = 1, . . . , d is a basis of the fiber Tϕ(m) (ϕ(U )). As T ϕ yields isomorphisms on fibers, we get a basis of Tm U by setting ∂ (m) := Tϕ(m) ϕ−1 ([cϕ(m),ej ]ϕ(m) ). ∂xj Here, Tϕ(m) ϕ−1 denotes the restriction of T ϕ−1 to the fiber Tϕ(m) (ϕ(U )). Using the fact that T A is an atlas for T M , it follows that the local construction of the basis is actually global. Since all the mappings are smooth, the basis smoothly depends on the base points. For a d-dimensional manifold M there is another naturally associated bundle, namely the cotangent bundle T ∗ M . A local chart ϕ on M induces local basis fields ∂x∂ j on the tangent bundle T M . It can be shown that the definition ∗ dxi (m) ∈ Tm M := (Tm M )∗ , ∂ dxi (m) (m) := δji j ∂x

yields smooth coordinate covector fields dxi , i = 1, . . . , d. Here, (Tm M )∗ denotes the dual space of the fiber Tm M and δji is the usual Kronecker delta.

11

1.3 Flows

In this sense, an atlas on M not only induces a natural atlas on T M but also on the cotangent bundle [ ∗ T ∗ M := Tm M. m∈M

If E = (S, V) is a vector bundle, then the base B is defined by n o B := e ∈ E ∃(W, ϕ) ∈ V, ∃u ∈ U : e = ϕ−1 (u, 0) As any such W can be identified with a local vector bundle by means of ϕ : W → U × F , B is the union of all zero sections of the local vector bundles. It can be shown that B is a submanifold in E and that there is a surjective C ∞ -map π : E → B called projection. Therefore, a vector bundle E is sometimes just denoted by ”π : E → B”. In case of the tangent bundle T M , the base can be identified with M and the projection is given by τM . Now, a vector bundle can be thought of as a manifold with a vector space attached to each point. With this picture in mind, we can define mappings from the manifold into these vector spaces. Such mappings can be regarded as generalizations of the graph of a function. Let π : E → B be a vector bundle. A section of π is a map η : U → E, U ⊂ B an open subset, such that π(η(b)) = b, that is, η takes its values in the fiber over b. Two examples which make this definition clearer are vector fields and covector fields, which are defined as sections of the tangent bundle and of the cotangent bundle, respectively. Using the basis fields we already constructed, any (smooth) vector field X : M → T M may be written as X=

d X i=1

Xi

∂ ∂xi

for (smooth) functions X i : M → R. Similarly, any (smooth) covector field ω : M → T ∗ M is given by ω=

d X

ωi dxi

i=1

for (smooth) functions ωi : M → R. In the sequel, we will use X r (M ) to denote the set of C r vector fields on a manifold M and X (M ) := X ∞ (M ). Also, X ∗r (M ) denotes the set of C r covector fields and we set X ∗ (M ) := X ∗∞ (M ).

1.3. Flows Any vector field f : Ω ⊂ Rd → Rd gives rise to a dynamical system, for it can serve as the right-hand side of a system of ordinary differential equations. Using local charts, we can transfer this construction to manifolds. If M is a d-dimensional manifold and X ∈ X (M ), a C 1 -curve c : I → M , 0 ∈ I, is called integral curve of X at m ∈ M if c(0) = m and if for any t ∈ I there holds c0 (t) = X(c(t)).

(1.5)

12


For a C 1 -curve c : I → M , the mapping c0 : I → T M is defined by c0 (t) := T c(t, 1). Here, we implicitly use the identification of L(R, F ) with F , given by ϕ 7→ ϕ(1). In case of a curve c, this shows c0 (t) ∈ Tc(t) M . Using local charts, we can express (1.5) in terms of the local representatives. Suppose (U, ϕ) is a chart such that c(I) is contained in U . The local representative of c is then given by cϕ = ϕ ◦ c while the local representative of c0 with respect to the natural atlas on T M is given by (c0 )ϕ (t) = T ϕ ◦ c0 (t) = T ϕ ◦ T c(t, 1) = T (ϕ ◦ c)(t, 1) = (cϕ )0 (t). For the local representative of X ◦ c we have (X ◦ c)ϕ = T ϕ ◦ X ◦ c = T ϕ ◦ X ◦ ϕ−1 ◦ ϕ ◦ c = Xϕ ◦ cϕ . It follows that c is an integral curve of X if and only if cϕ is an integral curve of Xϕ . The last statement takes the familiar form c0i (t) = Xi (c1 (t), . . . , cd (t)), i = 1, . . . , d, where ci and Xi denote the component functions of the local representatives. It is proved in calculus, that smoothness of the local representative of X implies the existence, uniqueness and smoothness of integral curves at any m ∈ M . This result is formalized in the concept of a local flow box of X at m. A local flow box of X (at m) is a triple (U0 , τ, F X ) such that U0 ⊂ M is open, m ∈ U0 , and τ is a real number or τ = ∞ such that the following assertions are valid: • F X : Iτ × U0 → M is smooth, where Iτ denotes the interval (−τ, τ ). • The restriction of F X to Iτ × {u} is an integral curve of X at u for any u ∈ U0 . • The restriction of F X to {t} × U0 , t ∈ Iτ , is such that FtX (U0 ) ⊂ M is open and FtX is a diffeomorphism. Existence, uniqueness and smoothness of integral curves are in fact equivalent to the existence of unique flow boxes. The domain of definition of the integral curves t 7→ FtX (u) := F X (t, u), u ∈ U0 , is called the lifetime of the integral curves and a vector field X is called complete if each of its integral curves is defined on (−∞, ∞). In this case, the mapping F X is called the flow of X. There are two useful criteria that ensure completeness of a vector field: If either M is compact or X has compact support, then X is complete. Another consequence of the existence and uniqueness of flow boxes is that the induced diffeomorphisms have the semi-group property X , FtX ◦ FsX = Fs+t

as long as either of the two sides is well-defined. All these results can be generalized to time-dependent vector fields on a manifold M . Here, a smooth time-dependent vector field is a smooth map X : R × M → T M such that X(t, m) ∈ Tm M holds for any (t, m) ∈ R × M . We also use the notation Xt (·) to denote

13

1.4 The Lie Derivative

X and is defined by the requirement X(t, ·). The time-dependent flow of X is denoted by Ft,s that

d X X F (m) = X(t, Ft,s (m)), dt t,s X Fs,s (m) = m. That is, the flow is again defined to be an integral curve of X at a point m ∈ M . In case of time-dependent vector fields, the integral curves are still unique on their domain of definition but satisfy the generalized semi-group property X X X Ft,r ◦ Fr,s = Ft,s .

1.4. The Lie Derivative Vector fields on manifolds not only generate dynamical systems, they also induce differential operators on the set of smooth functions and smooth vector fields. This construction will lead to the definition of the Lie derivative of a function or a field, which is a basic tool in continuum mechanics. Before we give the definition of the Lie derivative we introduce push-forwards and pull-backs of functions and vector fields. Let F r (M ) denote the set of C r -functions and set F(M ) := F ∞ (M ). Furthermore, suppose that ϕ : M → N is a C r -diffeomorphism of manifolds. The pull-back of f ∈ F r (N ) by ϕ is given by ϕ∗ f = f ◦ ϕ ∈ F r (M ). For X ∈ X (M ) the push-forward of X by ϕ is the smooth vector field on N defined by ϕ∗ X := T ϕ ◦ X ◦ ϕ−1 . This vector field can alternatively be defined by requiring the diagram ϕ

M X

/N

TM

Tϕ

ϕ∗ X

/ TN

to commute. Using local coordinates on the manifolds, one can compute the local representative of ϕ∗ X, by means of the diagram Rn ⊃ η(U ) o Xη

η

M X

η(U ) × Rn o

Tη

ϕ

TM

/N

Tϕ

ψ

/ ψ(V ) ⊂ Rn

ϕ∗ X

/ TN

(ϕ∗ X)ψ

/ ψ(V ) × Rn .

Tψ

In fact, it shows that ((ϕ∗ X)ψ )(v) = (T ψ ◦ (ϕ∗ X) ◦ ψ −1 )(v) = (v, ∇(ψ ◦ ϕ ◦ η −1 )(u)Xη (u)), where u = (η◦ϕ−1 ◦ψ −1 )(v). If xi are the local coordinates on M and y j are local coordinates on N the preceding formula can be expressed as (ϕ∗ X)jψ (y) =

X ∂(ϕηψ )j i

∂xi

(x)Xiη (x)

14


with y = ϕηψ (x). If there is no danger of confusion we will refrain from mentioning the chart explicitly, but instead identify it with the local coordinates it induces. Since ϕ : M → N was supposed to be a diffeomorphism, we can also push and pull along the inverse ϕ−1 . Thus, if Y is a smooth vector field on N , the pull-back of Y is given by ϕ∗ Y := (ϕ−1 )∗ Y = T (ϕ−1 ) ◦ Y ◦ ϕ. The push-forward of a function is defined analogously. The flows of vector fields provide a special class of diffeomorphisms on a manifold which are used to define the Lie derivative. We start with the case of time-independent vector fields. In order to measure how a vector field Y ∈ X (M ) (or even a general tensor field) changes along a curve c : I → M with c(0) = m and c0 (0) = Z ∈ Tc(0) M , it does not make sense to consider the limit for t → 0 of Y(c(t)) − Y(c(0)) t since Y(c(t)) ∈ Tc(t) M and Y(c(0)) ∈ Tc(0) M lie in different fibers. Now suppose X ∈ X (M ) is such that X(c(0)) = Z. Then we can alternatively use the flow generated by X, a diffeomorphism at least for small t, to pull back Y(c(t)) to the fiber Tc(0) M and then differentiate. That is, if F X denotes the flow of X we consider (LX Y)(m) : =

d X ∗ (F ) Y (m) t dt t=0 ((T FtX )−1 ◦ Y ◦ FtX )(m) − Y(m) . t→0 t

= lim

LX Y is called the Lie derivative of Y with respect to X. Note that (LX Y)(m) is welldefined since (FtX )∗ Y (m) and Y(m) both lie in Tm M by definition of the pull-back. For f ∈ F(M ) we set (LX f )(m) :=

d X ∗ (F ) f (m). t dt t=0

(1.6)

X denotes its flow we set If Xt ∈ X (M ), t ∈ R, is a time-dependent vector field and if Ft,s

(LXs Y)(m) :=

d X ∗ ) Y (F (m). t,s dt t=s

The Lie derivative of a function with respect to a time-dependent vector field is defined analogously. If the functions or vector fields under consideration smoothly depend on time, the dynamic or time-dependent Lie derivative LX is given by LX = ∂t + LX .

(1.7)

with the obvious modifications for time-dependent vector fields Xt . Furthermore, it follows by a simple calculation that the mappings (X, f ) → LX f and (X, Y) → LX Y are linear in

15

1.4 The Lie Derivative

both arguments. To compute the Lie derivative of a function in coordinates, we first of all introduce the differential of a smooth function f . For f ∈ F(M ) we have Tf : TM → TR = R × R Tm f := T f : Tm M → R × R. Tm M If we let P2 : R × R → R denote the projection onto the second component, then the "vector part" df (m) := P2 ◦ Tm f is called the differential of f . Let η : U → η(U ) be a chart on M and let x = η(m) denote local coordinates. Unwinding the definitions we see that ∂(fη ) (x) ∂xi and that the linear mapping df (m) : Tm M → R is given by d(fη )i (x) =

df (m)(X(m)) =

d X ∂(fη ) i=1

Thus, for f ∈

F r (M )

∂xi

(x)Xiη (x).

the differential df : M → T ∗ M is a C r−1 -covector field given by df =

d X ∂(fη ) i=1

∂xi

dxi .

Using local coordinates and the linearity of the Lie derivative in the vector field a simple computation shows (LX f )(m) = df (m)(X(m)),

(1.8)

which, in the Euclidean case, is just the derivative of f in the direction X. To compute the Lie derivative of a vector field one may again write down the local representatives of the vector fields to get d X ∂(Yηj ) ∂(Xjη ) i LX Y η (x) := X, Y η (x) := Xiη (x) (x) − Y (x) ej η ∂xi ∂xi

(1.9)

i,j=1

which is equivalent to

LX Y (m) := X, Y (m) =

d X

Xi (m)

i,j=1

∂ ∂(Yj ) ∂(Xj ) i (m) − Y (m) (m) (m) ∂xi ∂xi ∂xj

and additionally shows that L : X (M ) × X r (M ) → X r−1 (M ). Moreover, the bilinear mapping [·, ·] : X (M ) × X (M ) → X (M ) is called the Lie-bracket. Another useful property of the Lie derivative is that LX is natural with respect to diffeomorphisms. That is, if Z(M ) denotes either of the spaces X (M ) or F(M ) on a manifold M and if ϕ : M → N is a diffeomorphism between manifolds, then the diagram Z(M ) LX

ϕ∗

Z(M )

/ Z(N )

ϕ∗

Lϕ∗ X

/ Z(N )

16


commutes, i.e. Lϕ∗ X ϕ∗ T = ϕ∗ LX T for T ∈ Z(M ). Also, the Lie derivative is natural with respect to restrictions, which means that the following diagram commutes as well Z(M ) LX

|U

Z(M )

/ Z(U )

|U

(1.10)

LX

/ Z(U ).

We cite yet another property of the Lie derivative which turns out to be useful for extending it to k-forms or even arbitrary tensor fields. To this end, we define the wedge product between functions and vector fields by ∧ : F(M ) × F(M ) → F(M ), (f ∧ g)(m) := f (m)g(m), and extend it to ∧ : F(M ) × X (M ) → X (M ) by (f ∧ X)(m) := f (m)Xm . Then, it can be easily verified by direct computation that for any X ∈ X (M ) the mappings LX : F(M ) → F(M ) and LX : X (M ) → X (M ) are R-linear and satisfy LX (f ∧ g) = LX f ∧ g + f ∧ LX g, LX (f ∧ Y) = LX f ∧ Y + f ∧ LX Y.

(1.11)

One says that LX is a derivation on {F(M ), X (M )}. We will come back to these properties in the next section. The time derivative that was used to define the Lie derivative can also be evaluated at t 6= 0, or t 6= s in case of time-dependent flows. Let T ∈ Z(M ) denote either a smooth vector field or a smooth function and suppose that Xt ∈ X (M ), t ∈ R, is a smooth timeX . Using the generalized semi-group property of the flow dependent vector field with flow Ft,s we have ∗ d d d X ∗ X X ∗ (Ft,s ) T = Fr,t (Fr,t ◦ FtX T = (FtX )∗ ) T = (FtX )∗ LXt0 T . 0 0 0 ,s 0 ,s 0 ,s dt t=t0 dr r=t0 dr r=t0 Of course, if FtX is the flow of a time-independent vector field X then the semi-group property of the flow implies in this case d d )∗ T = (FtX )∗ LX T . (FtX )∗ T = (FtX 0 +r 0 dt t=t0 dr r=0 These two formulas can again be generalized to time-dependent functions or vector fields. Once more, this is a straightforward application of the semi-group properties of the respective flows. If Tt , t ∈ R, denotes either a smooth time-dependent function or a smooth timeX denotes a time-dependent flow we have dependent vector field and if Ft,s d X ∗ (Ft,s ) Tt = (FtX )∗ LXt0 Tt0 + (FtX )∗ (∂t T )t0 . 0 ,s 0 ,s dt t=t0

(1.12)

If FtX denotes the flow of a time-independent vector field X the last formula can be compactly written as d X ∗ X ∗ (F ) T = (F ) L T . (1.13) t X t t 0 dt t=t0 These formulas will play a significant role for transport theorems.

17

1.5 k-Forms and the Bundle Λk (M )

1.5. k-Forms and the Bundle Λk (M ) The exterior derivative generalizes the differential of functions to (smooth) multilinear mappings. Also, the construction of the Piola transformation which will be used extensively in this work, is a lot more lucid, if we formulate it with the help of the exterior derivative and the Lie derivative. However, we have to work our way through some definitions first. Let V be a finite dimensional real vector space. By Lk (V ) we denote the space of all k-linear mappings ω : |V × ·{z · · × V} → R. k copies

The subspace Λk (V ) ⊂ Lk (V ) consists of all k-linear mappings ω that are skew symmetric in the sense that ω(vσ(1) , . . . , vσ(k) ) = sgn(σ) ω(v1 , . . . , vk ) holds for any permutation σ in the symmetric group Sk . Any element ω ∈ Λk (V ) is called k-form on V . There is a bilinear mapping ⊗ : Lk (V ) × Ll (V ) → Lk+l (V ) called the tensor product which is defined by (η ⊗ µ)(v1 , . . . , vk , vk+1 , . . . , vk+l ) = η(v1 , . . . , vk )µ(vk+1 , . . . , vk+l ) on arbitrary vectors vi ∈ V , i = 1, . . . , k + l. Any η ∈ Lk (V ) can be skew-symmetrized by applying Alt : Lk (V ) → Λk (V ) which is defined by (Alt η)(v1 , . . . , vk ) :=

1 X sgn(σ) η(vσ(1) , . . . , vσ(k) ). k! σ∈Sk

The mapping Alt is called alternating mapping, with the property that it is the identity mapping when restricted to Λk (V ). Combining the tensor product and the alternating mapping, we get another bilinear map ∧ : Lk (V ) × Ll (V ) → Λk+l (V ) called the wedge product ω ∧ η :=

(k + l)! Alt (ω ⊗ η). k! l!

1 n Now suppose that dim V = ni with basis {e1 , . . . , en } and let {e , . . . , e } denote the dual i basis defined by e , ej = δj . Also, if we let I(k, n) denote the set of all multiindices j = (j1 , . . . , jk ) with the property that 1 ≤ j1 < j2 < · · · < jk ≤ n, then the set of k-forms { ej := ej1 ∧ · · · ∧ ejk | j ∈ I(k, n) } is a basis of Λk (V ). Therefore any ω ∈ Λk (V ) can be written as X ω= ωj e j . j∈I(k,n)

If j = (j1 , . . . , jk) ∈ I(k, n) is arbitrary, the factor ωj is given by wj = ω(ej1 , . . . , ejk ). Thus, dim Λk (V ) = nk , Λ1 (V ) = V ∗ , the algebraic dual space of V , and Λn (V ) = span(e1 ∧ · · · ∧

18


en ). Finally, for any ω ∈ Λk+1 V and a fixed vector v ∈ V , the interior product iv ω ∈ Λk V is given by (iv ω)(v1 , . . . , vk ) := ω(v, v1 , . . . , vk ). Before we come to define k-forms on manifolds, we need to take a look at the effect of isomorphisms on spaces of k-forms. Suppose that V and V 0 are finite dimensional, real vector spaces and that ϕ ∈ L(V, V 0 ) is an isomorphism. If α ∈ Λk (V 0 ), then the pull-back ϕ∗ α ∈ Λk (V ) is given by ϕ∗ α(v1 , . . . , vk ) := α(ϕ(v1 ), . . . , ϕ(vk )), v1 , . . . , vk ∈ V. For ω ∈ Λk (V ) the push-forward ϕ∗ ω is defined by ϕ∗ ω := (ϕ−1 )∗ ω. Furthermore, one can show that an isomorphism ϕ ∈ L(V, V 0 ) induces an isomorphism ϕ∗ : Λk (V ) → Λk (V 0 ) and (ϕ∗ )−1 = (ϕ−1 )∗ . The tensor algebra of k-forms can be extended to the tangent bundle of a manifold by using local charts. To begin with, let U × F be a local vector bundle, F being a d-dimensional vector space. Now U × Λk (F ) is also a local vector bundle by virtue of dim Λk (F ) = kd . Any local vector bundle isomorphism ϕ : U × F → U 0 × F 0 induces a local vector bundle isomorphism ϕ∗ : U × Λk (F ) → U 0 × Λk (F 0 ), ϕ∗ (u, ω) := (ϕ(u), (ϕu )∗ ω). Here, ϕu means the restriction of the second component of the local vector bundle isomorphism ϕ to the fiber {u} × Λk (F ). It follows that (ϕu )∗ is an isomorphism. Now let τM : T M → M be the tangent bundle of a d-dimensional manifold M and U ⊂ M a chart domain. Define [ Λk (M ) = Λk (Tu M ), Λk (T M ) := Λk (M ) := Λk (M ) U M u∈U

and define π k : Λk (M ) → M by π k (m, ωm ) := m. If T A denotes the bundle atlas on T M , then for any (T ϕ, T U ) ∈ T A, the mapping (T ϕ∗ ) : Λk (M ) → (U, Λk (Rd )) U

defines a local vector bundle isomorphism and therefore yields a local bundle chart for the bundle π k : Λk (M ) → M . Any such bundle chart is called a natural chart and the set of all natural charts constitutes a vector bundle atlas on Λk (M ). The bundle Λk (M ) is called the bundle of exterior k-forms on the tangent spaces of M . Also, let Ωk (M ), k ∈ N, denote the set of all smooth sections of the bundle Λk (M ) with the convention Ω0 (M ) = F(M ). Any ω ∈ Ωk (M ) is called k-form, especially, any α ∈ Ω1 (M ) = X ∗ (M ) is called one-form. Using the bundle atlas, we can show that locally, any ω ∈ Ωk (M ) can be written as X ω= ωj dxj . j∈I(k,d)

19

1.6 Extension of the Lie Derivative

Here we used the notation dxj := dxj1 ∧ · · · ∧ dxjk for j = (j1 , . . . , jk ) ∈ I(k, d) and basis vectors of the cotangent bundle dxji , ji ∈ {1, . . . , d}. The factors ωj are smooth functions on M since ωj (m) = ω( ∂x∂j1 (m), . . . , ∂x∂jk (m)). Operations on k-forms on a fixed vector space have a natural extension to spaces of forms on a fixed manifold. In case of the wedge product, we set for ω ∈ Ωk (M ) and η ∈ Ωl (M ) (ω ∧ η)(m) := ω(m) ∧ η(m), to obtain a bilinear mapping ∧ : Ωk (M ) × Ωl (M ) → Ωk+l (M ). If X ∈ X (M ) is a smooth vector field and ω ∈ Ωk+1 (M ) is a smooth k + 1-form, a smooth k-form iX ω ∈ Ωk (M ) is given by (iX ω)m (X1 , . . . , Xk ) := ωm (X(m), X1 , . . . , Xk ). For functions f ∈ F(M ) = Ω0 (M ) we set iX f := 0. It can be shown that the mapping iX : Ωk+1 (M ) → Ωk (M ) is a R-linear antiderivation in the sense that for any ω ∈ Ωk (M ) and any η ∈ Ωl (M ) there holds iX (ω ∧ η) = (iX ω) ∧ η + (−1)k ω ∧ (iX η).

(1.14)

If F : M → N is a smooth diffeomorphism of manifolds, its tangent T F : T M → T N induces a smooth vector bundle isomorphism, which means that the local representative is a local bundle isomorphism on chart domains. For this reason, it is natural to use F to pull back k-forms on N . That is, for ω ∈ Ωk (N ) and a diffeomorphism between manifolds F : M → N , we get a new k-form on M by setting (F ∗ ω)m (v1 , . . . , vk ) := ωF (m) (Tm F v1 , . . . , Tm F vk ). F ∗ ω ∈ Ωk (M ) is called the pull-back of ω ∈ Ωk (N ). The push-forward is defined by setting F∗ = (F −1 )∗ . It easily follows from direct computation that the interior product behaves natural with respect to diffeomorphisms which means that the following diagram commutes Ωk (N ) iY

F∗

Ωk−1 (N )

/ Ωk (M )

F∗

iF ∗ Y

/ Ωk−1 (N ).

In fact, using the tensor characterization lemma, see [Lee97], it is even clear that if F : M → N is a smooth diffeomorphism of manifolds, ω ∈ Ωk (N ) and X1 , .., Xk ∈ X (M ) are smooth vector fields, then m 7→ ωF (m) (Tm F X1 (m), . . . , Tm F Xk (m)) ∈ F(M ).

1.6. Extension of the Lie Derivative Up to now, we have only defined and computed the Lie derivative of functions and vector fields. In the next section we shall also apply the Lie derivative to k-forms. As we do not

20


need the description in coordinates of these objects but merely use their algebraic properties, we cite a general theorem that enables us to extend the Lie derivative to k-forms on a manifold or even to the full tensor algebra. If M is a d-dimensional manifold, the direct sum of the spaces Ωk (M ), k = 1, . . . , d, together with the wedge product is denoted by Ω(M ) and is called the Grassmann algebra. A differential operator on Ω(M ) is a map D : Ω(U ) → Ω(U ) for any open subset U ⊂ M such that • D is R-linear and for any ω, η ∈ Ω(U ) there holds D(ω ∧ η) = Dω ∧ η + ω ∧ Dη. • D is natural with respect to restrictions in the sense that (Dη)|U = D(η|U ) for any η ∈ Ω(M ) and any open subset U ⊂ M . Now suppose that for any open subset U ⊂ M there are derivations EU : F(U ) → F(U ) and FU : X (U ) → X (U ). By Willmore’s theorem, see [AMKK67], there is a unique differential operator D on Ω(M ) such that D = EU on F(U ) and D = FU on X (U ). Appealing to (1.10) and (1.11) we may apply this theorem with EU = LX|U and FU = LX|U . The unique extension of LX to Ω(M ) is again called the Lie derivative and we therefore use the same notation. Remark 1.1. We have not used Willmore’s theorem in its full generality. In fact, it can be used to extend the Lie derivative to the full tensor algebra on a manifold. For our purpose though, the extension to Ω(M ) is sufficient.

1.7. The Exterior Derivative and Cartan’s Magic Formula k-forms have a rich structure which we will now try to elaborate by introducing the exterior derivative d : Ωk (M ) → Ωk+1 (M ). The exterior derivative extends the differential of a function to k-forms and is intimately connected to the Lie derivative by virtue of "Cartan’s magic formula". Theorem 1.2 (Exterior Derivative) For any smooth d-dimensional manifold M there is a unique family of R-linear mappings d : Ωk (M ) → Ωk+1 (M ), k = 0, . . . , d, called the exterior derivative on M with the following properties: • For f ∈ Ω0 (M ) = F(M ), df coincides with the differential of f . • d2 := d ◦ d = 0. • d is a ∧-antiderivation in the sense that for any ω ∈ Ωk (M ) and any η ∈ Ωl (M ) there holds d(ω ∧ η) = dω ∧ η + (−1)k ω ∧ dη.

(1.15)

21

1.7 The Exterior Derivative and Cartan’s Magic Formula

• d is natural with respect to restrictions, that is, if U ⊂ M is open and ω ∈ Ωk (M ) then (dω)|U = d(ω|U ). We skip the proof which can be found for example in [AMKK67], since we are merely interested in the properties of d. As we have already seen, most of the operations we have introduced so far behave natural with respect to diffeomorphisms between manifolds. Also, having extended the Lie derivative to k-forms, one could ask for the interplay between LX and the exterior derivative. The following two commuting diagrams show that d behaves natural with respect to both operations: Ωk (N ) d

F∗

Ωk+1 (N )

/ Ωk (M )

F∗

Ωk (M )

d

/ Ωk+1 (M )

d

LX

Ωk+1 (M )

/ Ωk (M )

LX

d

/ Ωk+1 (M )

Especially, if f ∈ F(M ) is a smooth function then (1.16)

LX f = df (X) = iX df,

by (1.8) and the definition of the interior product. The latter formula will be generalized below. We give two more examples that highlight the relevance of the exterior derivative. If U ⊂ Rd is a local manifold, X : U → Rd is a vector field and ω = dx1 ∧ · · · ∧ dxd , then iX ω =

d X

(1.17)

dj ∧ · · · ∧ dxd , (−1)j−1 Xj dx1 ∧ · · · ∧ dx

j=1

dj means that this factor is left out. The latter identity in turn implies that where dx d(iX ω) = (div X) ω. If Y : U → R3 is vector field and the one-form α is defined by α = one can show that dα = irot Y dx1 ∧ dx2 ∧ dx3 . Since d2 = 0 we get

P3

i=1 αi dx

i,

αi = Y i ,

0 = d2 α = d(irot Y dx1 ∧ dx2 ∧ dx3 ) = (div rot Y)(dx1 ∧ dx2 ∧ dx3 ), thereby showing that div rot Y vanishes identically. By the same method, one can show ∂f that curl gradf = 0, where grad f denotes the vector with components ∂x i . Thus, the basic operations and relations of vector calculus in Euclidean spaces can be formulated entirely in terms of the exterior derivative. We now come to the proof of "Cartan’s magic formula". What is magic about this rule is that it elegantly links the exterior derivative and the interior product to the Lie derivative. This may help avoiding messy calculations in coordinates. Theorem 1.3 (Cartan’s Magic Formula) Let M be a smooth manifold and let X ∈ X (M ) be a smooth vector field on M . For any smooth k-form α ∈ Ωk (M ) there holds LX α = iX dα + diX α.

(1.18)

22


Proof. The proof proceeds by induction on k. In fact, for k = 0 and f ∈ Ω0 (M ) the formula reduces to df (X) = LX f = iX df + diX f = iX df , which is true due to (1.16). Remember also that iX f = 0. Now suppose that theP assertion holds true for ω ∈ Ωk (M ). Any (k + 1)nα 0 k form α may locally be written as α = i=1 dfi ∧ ωi for fi ∈ Ω (M ) and ωi ∈ Ω (M ), i = 1, ..., nα . By linearity it is therefore sufficient to verify the formula for α = df ∧ ω. Since LX is a derivation we have LX (df ∧ ω) = (LX df ) ∧ ω + df ∧ (LX ω). On the other hand, due to the fact that iX and d are ∧-antiderivations, see (1.14) and (1.15), and since d2 = 0, we have iX d(df ∧ ω) + diX (df ∧ ω) = iX (ddf ∧ ω − df ∧ dω) + d(iX df ∧ ω − df ∧ iX ω) = −iX (df ∧ dω) + d(iX df ∧ ω − df ∧ iX ω) Using the same properties of iX and d once more we get iX d(df ∧ ω) + diX (df ∧ ω) = − iX df ∧ ω + df ∧ iX dω + diX df ∧ ω + iX df ∧ ω + df ∧ diX ω = df ∧ iX dω + df ∧ diX ω + diX df ∧ ω = df ∧ (iX dω + diX ω) + dLX f ∧ ω = df ∧ (iX dω + diX ω) + LX df ∧ ω, where, for the last equality, we also used the fact that the Lie derivative and the exterior derivative commute. By induction hypothesis there holds df ∧ (iX dω + diX ω) = df ∧ LX ω. Hence, the assertion now follows since LX is a derivation.

1.8. The Piola Transform Vector fields describing continuum mechanical processes usually have an additional constraint on their divergence. Here, we first think of the divergence as the sum of certain partial derivatives of the vector field under consideration and clarify its geometric meaning later. For example, if Ω ⊂ Rd is an open subset and u : Ω → Rd is the velocity field that describes the flow of an incompressible fluid in Ω, it has to satisfy the equation div u =

d X ∂ui i=1

∂xi

= 0.

If div u = 0, the vector field u is called divergence-free, solenoidal or incompressible. Now suppose ϕ : Ω → Ω0 is a smooth diffeomorphism. We may think of ϕ as a change of coordinates, for example. Then a natural question is, how to transform a solenoidal vector field on Ω to a vector field on Ω0 in such way that the resulting field is divergence-free with respect to the coordinates on Ω0 induced by ϕ. The Piola transform answers this question and yields a concrete transformation rule to map solenoidal vector fields to solenoidal vector

23

1.8 The Piola Transform

fields. Before we come to its definition we first have to develop some more theory. As we have already seen in the foregoing sections, for any linear isomorphism ϕ : E → F between finite dimensional vector spaces its pull-back ϕ∗ : Λk (F ) → Λk (E) induces a linear isomorphism between the corresponding spaces of k-forms. We also saw that if E is ddimensional then dim Λd (E) = 1. Consequently, if µ0 is a basis for Λd (E) and ϕ ∈ L(E, E) is an isomorphism, then for any µ = c µ0 there holds ϕ∗ µ = cϕ∗ µ0 = bµ0

(1.19)

with a unique constant b ∈ R. The unique constant det ϕ such that (1.19) holds for any µ ∈ Λd (E) is called the determinant of ϕ. Evidently, this definition does not depend on the choice of the basis in E and it may be checked that it is consistent with the usual definition of the determinant. The non zero elements in Λd (E) are called volume elements and have natural global counterparts on manifolds. In fact, the non zero elements in Ωd (M ) are called volume forms on a manifold M . A manifold is called orientable if there exists a volume form on M . Any local manifold U ⊂ Rd is orientable since any basis {vi | i = 1, . . . , d } of Rd yields a volume form ω = v 1 ∧ · · · ∧ v d . If M is a connected d-dimensional manifold it can be shown that M is orientable if and only if there exists a fixed volume form µ ∈ Ωd (M ) such that any η ∈ Ωd (M ) may be written as η = f µ for a fixed function f ∈ F(M ). Now suppose that ϕ : U ⊂ Rd → V ⊂ Rd is a smooth diffeomorphism between local ddimensional manifolds. Then, its tangent T ϕ : T U → T V induces a bundle isomorphism between the respective tangent bundles and Tu ϕ : {u}×Rd → {ϕ(u)}×Rd is an isomorphism on fibers for any u ∈ U . Consequently, ϕ induces a bundle isomorphism ϕ∗ : Λd (V ) → Λd (U ) by pulling back smooth volume forms on V to smooth volume forms on U . The global version of the determinant is now defined as the unique smooth function Jϕ (µU , µV ) ∈ F(U ) such that ϕ∗ µV = Jϕ (µU , µV ) µU

(1.20)

holds for any µU ∈ Ωd (U ) and µV ∈ Ωd (V ). The function Jϕ (µU , µV ) is called the Jacobian determinant of ϕ with respect to the volume forms µU and µV . The diffeomorphism is called volume preserving if Jϕ (µU , µV ) = 1. If µU = µV = e1 ∧ · · · ∧ ed is the volume form induced by the standard basis { ei | i = 1, . . . , d } of Rd , the Jacobian determinant of a diffeomorphism ϕ will be denoted by Jϕ . Also, the standard volume form e1 ∧ · · · ∧ ed on a d-dimensional Euclidean space will be denoted by dx := dx := e1 ∧ · · · ∧ ed . This is the conventional notation used in integration theory. If we write out (1.20) for this case we see that Jϕ (x) = det (Tx ϕ). It follows that the Jacobian determinant of a diffeomorphism ϕ : U → V cannot vanish. We now take a look at the identity d(iX dx) = (div X) dx once more. Using Cartan’s formula (1.18), this implies LX dx = iX ddx + diX dx = diX dx = (div X) dx

24


Thus, the divergence of a vector field X may be seen a measure for the rate of change of the volume form along the flow of X. To make things clearer, assume that X : U ⊂ Rd → Rd is a smooth complete vector field with flow ϕ. Then, the flow of X is volume preserving (with respect to dx) if and only if div X = 0. In fact, Jϕt = 1 means that ϕ∗t dx = dx for any t. But then the mapping t 7→ ϕ∗t dx is constant and therefore div X dx = LX dx = 0. The other direction essentially follows along the lines by noting that ϕ∗0 = id∗ = id. Now we finally come to the definition of the Piola transformation. Definition 1.4 (Piola Transform) Let ϕ : U → V be a smooth diffeomorphism between local manifolds with positive Jacobian determinant. Furthermore, let Y : V → Rd be a vector field on V . The vector field X : U → Rd , X := Pϕ Y := Jϕ ϕ∗ Y is called the Piola transform of Y with respect to ϕ. The Piola transform is the usual pull-back of vector fields, scaled with the Jacobian determinant in order to take into account the change of volume forms caused by the diffeomorphism. The effect of this scaling is the content of the next proposition. Proposition 1.5 (A Characterization Using Forms) Let Y : V → Rd be a vector field and let ϕ : U → V be an orientation preserving diffeomorphism, i.e. Jϕ > 0. A vector field X : U → Rd is the Piola transform of Y if and only if ϕ∗ (iY dx) = iX dx. Proof. First of all, we notice that the mapping ∗ : X (V ) → Λd−1 (V ) ∗Y := iY dx is one-to-one. This follows from (1.17) since (iY dx)(e1 , . . . , ebj , .., ed ) = (−1)j−1 Y j . As the interior product is natural with respect to diffeomorphisms there holds ϕ∗ (iY dx) = iϕ∗ Y ϕ∗ dx. Hence, X = Pϕ Y is satisfied if and only if ϕ∗ (iY dx) = iϕ∗ Y ϕ∗ dx = iϕ∗ Y Jϕ dx = i(Jϕ ϕ∗ Y) dx = i(Pϕ Y) dx = iX dx.

The most important property of the Piola transform for our purposes is the fact that is preserves the divergence of vector fields. Proposition 1.6 (Piola Transform and Divergence) If X : U → Rd is the Piola transform of Y : V → Rd with respect to ϕ : U → V then div X = Jϕ (div Y) ◦ ϕ.

(1.21)

25

1.9 The Reynolds Transport Theorem

Proof. Using Cartan’s formula (1.18), we have div X dx = LX dx = d(iX dx). As the exterior derivative is natural with respect to diffeomorphisms it follows that div X dx = d(iX dx) = d(ϕ∗ (iY dx)) = ϕ∗ (d(iY dx)) = ϕ∗ (div Y dx) = Jϕ ((div Y) ◦ ϕ)dx.

Remark 1.7. The Piola transform maps solenoidal vector fields to solenoidal vector fields. Furthermore, all of the above mentioned properties of the Piola transform hold true for diffeomorphisms between general d-dimensional Riemannian manifolds (M, G) and (N, g). Here, G = (Gij ) and g = (gij ) denote the respective Riemannian metrics and the Riemannian volume form µG ∈ Ωd (M ) is given by p µG = det(G) dX 1 ∧ · · · ∧ dX d , where X i , i = 1, . . . , d, are coordinate functions on M . The volume form µg is defined similarly. Then of course, the divergence has to be taken with respect to the corresponding Riemannian metrics, for example d X ∂ p j divG Y = p det(G)Y µG . ∂X j det(G)

1

j=1

The divergence theorem shows another property of vector fields that are related by the Piola transform. Let again ϕ : U → V denote a diffeomorphism between local manifolds and Ω0 = ϕ(Ω) for some bounded subset Ω ⊂ U with sufficiently smooth boundary. Let Y : V → Rd be a vector field and let X := Pϕ Y denote its Piola transform. Also, let n, n0 denote the unit outward normal fields on ∂Ω and ∂Ω0 , respectively, and let da denote the surface measure. The divergence theorem and the change of variables formula, see Proposition A.4, then imply Z Z Z X · n da = div X dx = Jϕ (div Y) ◦ ϕ dx ∂Ω

Ω

Ω

Z =

Z div Y dx =

Ω0

Y · n0 da.

(1.22)

∂Ω0

Since the last equality is true for any sufficiently smooth subset Ω ⊂ U we may conclude that the flux of X across ∂Ω is equal to the flux of Y across ∂Ω0 . Consequently, the Piola transform preserves the respective fluxes.

1.9. The Reynolds Transport Theorem In this section we want to give a proof of the Reynolds transport theorem. Its relevance stems from the fact that it yields a formula for the rate of change of time-dependent quantities that are integrated over domains which may explicitely depend on time, too. In this sense, it may be understood as a rule for interchanging the time derivative and the operation

26


of integration over time-dependent domains. In some situations though, it can even serve as a generalized version of the fundamental theorem of calculus as we will see. A typical application of the transport theorem are energy balance laws that lead to the classical equations in continuum mechanics. It will therefore come as no surprise that Reynolds’ theorem will play a crucial role for deriving energy estimates for the systems of equations we are interested in. As it turns out, the transport theorem is also a nice application of the Lie derivative formalism we have already introduced. We will formulate the theorem in terms of smooth functions but not without mentioning that this formulation is not optimal with regard to (spatial) regularity. However, by appropriate density arguments, more general situations may be reduced to this smooth setting. Theorem 1.8 (Transport Theorem I) Suppose that X : Rd → Rd is a complete vector field and let ϕ := ϕX : R × Rd → Rd denote the flow of X. If Ω0 ⊂ Rd is an open, bounded subset, let Ω(t) := ϕt (Ω0 ) denote the diffeomorphic image of Ω0 under ϕt . Also, assume that f : R × Rd → R is C 1 with respect to both variables. Then there holds Z Z d f (t, x) dx = ∂t f (s, x) + div (f X)(s, x) dx dt t=s Ω(t)

Ω(s)

Z

(1.23)

Z

=

f (s, x)X(x) · n(s, x) da,

∂t f (s, x) dx + Ω(s)

∂Ω(s)

where n(s, ·) denotes the unit outward normal field on ∂Ω(s). Proof. For the sake of readability we will suppress the variables in the following calculations. Using the change of variables formula we first remove the time dependence of the domain of integration to get Z Z Z ∗ f dx = ϕt (f dx) = ϕ∗t f Jϕt dx. Ω0

Ω(t)

Ω0

Since the integrand is supposed to be smooth we can pull the time derivative under the integral which yields Z Z d d f dx = ϕ∗t f Jϕt dx dt t=s dt t=s Ω0

Ω(t)

Z = Ω0

d d ϕ∗t f Jϕs + ϕ∗s f Jϕt dx. dt t=s dt t=s

Using property (1.13) of the Lie derivative we see that d ∗ ∗ ∗ ϕ f = ϕ L f = ϕ ∂ f + df · X , t X t s s dt t=s

27

1.9 The Reynolds Transport Theorem

while the rate of change of the Jacobian determinant can similarly be computed as d ∗ ∗ Jϕt dx = ϕ dx = ϕ L dx X t s dt t=s dt t=s

d

= ϕ∗s div X dx = Jϕs div X ◦ ϕs dx. Combining these two identities yields d ϕ∗t f Jϕt =ϕ∗s ∂t f + df · X Jϕs + ϕ∗s f div X ◦ ϕs Jϕs dt t=s =ϕ∗s ∂t f + df · X Jϕs + ϕ∗s f div X) Jϕs . Integrating both sides and using the change of variables formula once again we get Z Z d d f dx = ϕ∗t f Jϕt dx dt t=s dt t=s Ω0

Ω(t)

Z ∂t f + df · X + f div X dx

= Ω(s)

Z =

∂t f + div f X dx.

Ω(s)

Now, if ∂Ω(s) is sufficiently smooth we can apply the divergence theorem to the last integral to prove the remaining assertion. For time-dependent vector fields X : R×Rd → Rd , we can formulate an analogue transport theorem. The proof essentially follows along the lines of the proof of Theorem 1.8 if one uses formula (1.12) instead of formula (1.13). Corollary 1.9 (Transport Theorem II) Let X : R × Rd → Rd be a smooth time-dependent vector field and let ϕ : R × Rd → Rd denote its flow. If Ω(t) again denotes the diffeomorphic image under ϕt of some open, bounded subset Ω0 ⊂ Rd , then Z Z d f (t, x) dx = ∂t f (s, x) + div (f X)(s, x) dx dt t=s Ω(t)

Ω(s)

Z =

∂t f (s, x) dx + Ω(s)

(1.24)

Z f (s, x)X(s, x) · n(s, x) da

∂Ω(s)

holds for any sufficiently smooth function f : R × Rd → R. R The transport theorem states that the rate of change of t 7→ Ω(t) f (t, x) dx can be quantified as the sum of two effects. The first term on the right-hand side of (1.23) takes into account the effect caused by the time dependence of the integrated quantity f . The second

28


one describes the effect of the evolution of the domain of integration. However, the second identity in (1.23) shows that this effect is completely encoded in terms of the restriction of f X to the boundary. Since X is the vector field that generates the flow, the restriction of f X to the boundary is just the velocity of the boundary weighted with f . Thus, if either f or X vanishes on ∂Ω(s) the time dependence of the domain has no effect at time s and the transport theorem reduces to a rule for differentiation under the integral sign. Also, if the vector field X is tangential at ∂Ω(s), i.e. X · n(s) = 0, then the boundary integral does not contribute to the rate of change either. If the function f does not depend on time the situation is reversed: The rate of change then solely depends on the weighted flux f X in the direction of the outward unit normal n(s). R The fact that the rate of change of t 7→ Ω(t) f (t, x) dx splits into a derivative with respect to time and a boundary integral has another obvious yet important implication. To make things clearer, let I = (s, s0 ) ⊂ R denote a bounded time interval and suppose that the smooth function f : R × Rd → Rd vanishes on the lateral boundary of [ Q = {t} × Ω(t) ⊂ Rd+1 , t∈I

that is, we suppose that for any t ∈ I there holds (1.25)

f (t, ·)|∂Ω(t) = 0.

R Setting Ef (t) := Ω(t) f (t) dx, the transport theorem yields a generalization of the fundamental theorem of calculus in the following form: Ef (s0 ) − Ef (s) =

Zs0 s

d Ef (t) dt = dt

Zs0 E∂t f (t) dt. s

In applications, this identity will be interpreted as an energy equality. Such an energy equality not only serves as a starting point towards a priori estimates for the nonlinear unsteady problems we are going to consider but it will also be of major importance for the identification of weak limits in nonlinear terms. Of course, these identities heavily depend on the smoothness of the functions involved and have to be transferred to functions that only have derivatives in a generalized or weak sense. We also notice that it is not enough to require a certain smoothness in the time variable of f as we used the spatial information (1.25) to make sure that we do not have to deal with surface integrals.

2. Function Spaces on Non-cylindrical Domains 2.1. Non-cylindrical Domains Evolution equations in fluid dynamics are usually formulated on a cylindrical domain I ×Ω0 , where I ⊂ R is a time interval and Ω0 ⊂ Rd is a possibly unbounded but fixed subset in Euclidean space. In recent years though, mostly stimulated by fluid-structure-interaction (FSI) problems, there has been a growing interest in evolution equations that do not fit in the geometric framework of cylindrical domains. One of the most challenging problems that comes to mind is the mathematical modelling of blood flow in the human cardiovascular system. The physical domain under consideration in this case would be some portion of an artery. Viewing an artery as a rigid subset of R3 does not make sense. Indeed, the diameter of an artery can strongly vary and its profile may deform (owing on one hand to the pressure and motion of the blood, and to elastic strains on the vessel walls on the other hand). Further difficulties arise from the permeability of the vessel walls. For clarity, we will however neglect these effects from our motivation. This example also illustrates a mathematical specificity of FSI problems. The solution of the actual fluid equation enters the structure dynamics through coupling the respective equations by appropriate boundary conditions on the vessel wall. Conversely, the solution of the structure equations determines the solution of the fluid equations through these boundary conditions as well. Therefore, the physical domain itself depends on the solutions of these equations. As a consequence, one typically deals with a highly nonlinear, strongly coupled system of partial differential equations. One possible way of mathematically handling such a system is to decouple the equations for the fluid and the structure by trying to apply a fixed-point-type argument. As a first simplification, one may assume that the deformation of the physical domain is known in advance. To further motivate this procedure we might consider an artery that is located in the vicinity of the heart, where the vessel motion is assumed to be mainly determined by the (hopefully predictable!) action of the heart. Nevertheless, this approach requires solving an evolution equation on a non-cylindrical domain, thus leading to new mathematical problems. If the deformation is assumed to be smooth enough (healthy arteries usually do not crack!) one could transform the fluid equation to a fixed cylindrical domain, thereby altering characteristic properties of the equation at hand. To retain the latter, we shall refrain from transforming the problem onto a fixed domain. This will require that we develop a suitable functional framework in which to seek a potential solution. As the spatial domain is timedependent, the classical Bochner spaces of functions with values in a fixed, time-independent Banach space do not provide the appropriate functional setting. We will therefore introduce suitable substitute spaces and establish their relevant properties, once the aforementioned non-cylindrical domains have been defined.

29

30

Chapter 2. Function Spaces on Non-cylindrical Domains

To begin with, we assume that v is a (possibly time-dependent) smooth vector field on with compact support contained in some open and bounded subset U ⊂ Rd . If v is time-dependent we assume that supp(v(t, ·)) ⊂⊂ U for all t ∈ R. As we saw in the previous chapter, under these assumptions, ODE theory ensures the existence of the induced flow ϕ := ϕv : R×Rd → Rd . Moreover, any ϕt is a diffeomorphism. The regularity of the flow with respect to time and space variables depends on the smoothness of v. However, if v is assumed to be smooth, the diffeomorphisms ϕt are smooth as well. One particular important quantity connected with the flow is the Jacobian determinant of ϕt . As it smoothly depends on the partial derivatives of the flow, which in turn are controlled in terms of derivatives of the vector field, for any compact time interval I¯ ⊂ R there exist constants c, C ∈ (0, ∞) such that

Rd

00 ⊂ C0∞ (I × Rd )d in H 1 (I, L2 (Ω(t))d ). Such a sequence can be constructed by convolution with a mollifying kernel. Let then ϕ ∈ C0∞ (I × Rd )d be arbitrary. Since η · ϕ has compact support we see Z Z d η · ϕ dxdt 0= dt I Rd Z Z Z Z d d = η (t) · ϕ(t) dxdt + η (t) · ϕ(t) dxdt. dt dt I

I

Ω(t)

Rd \Ω(t)

Since the lateral boundary of Q moves with velocity v, the transport theorem yields Z Z Z Z d ∂t η (t) · ϕ(t) + η (t) · ∂t ϕ(t) dxdt η (t) · ϕ(t) dxdt = dt I

I Ω(t)

Ω(t)

Z

Z

η (t) · ϕ(t) v(t) · n(t) dadt.

+ I ∂Ω(t)

Passing to the limit → 0, the boundary integral vanishes as η ∈ Lp (I, W01,1 (Ω(t))d ) and we infer Z Z Z Z d η(t) · ϕ(t) dxdt = ∂t η(t) · ϕ(t) + η(t) · ∂t ϕ(t) dxdt. dt I

I Ω(t)

Ω(t)

We can similarly show Z Z Z d η(t) · ϕ(t) dxdt = dt I

Z ∂t η(t) · ϕ(t) + η(t) · ∂t ϕ(t) dxdt,

I Rd \Ω(t)

Rd \Ω(t)

so that adding these identities shows Z Z Z Z η · ∂t ϕ dxdt = − ∂t η · ϕ dxdt. I Rd

I Rd

Here, ∂t η is identified with the zero extension of ∂t η ∈ L2 (I, L2 (Ω(t))d ). We now come to the non-cylindrical analogue of the time derivative in the sense of vectorvalued distributions. We give the definition for the case of vector fields, the modifications for scalar functions being obvious.

43

2.3 Generalized Bochner Spaces

Definition 2.12 (Generalized Time Derivative) ¯ let Vq (t) denote either W 1,q (Ω(t))d or W 1,q (Ω(t))d , 1 < q < ∞, and suppose For any t ∈ I, 0 0,σ d for every t ∈ I. ¯ Then, u ∈ Lr (I, Xp (t)) has a generalized time derivative Xp (t) ,→ L2 (Ω(t)) ∗ ∗ in Ls (I, Vq (t)) if and only if there exists Lf ,F ∈ Ls (I, Vq (t)) such that Z Z

u(t), ∂t η(t) L2 (Ω(t)) dt Lf (t),F(t) , η(t) Vq (t) dt = − I

I

holds for any η ∈ H 1 (I, L2 (Ω(t))d ) ∩ Ls (I, Vq (t)) with η(T, ·) = η(0, ·) = 0.3 We then denote Lf ,F by dt u. Appealing to (2.17), we see that Definition 2.12 indeed reflects the sense of a time derivative. Moreover, it contains the Gelfand triple setting as a special case. In fact, using the 0 notation we introduced above, let Vq (t), H(t), Vq (t) t∈I¯ be a family of Gelfand triples where ¯ Then, we H(t) ⊆ L2 (Ω(t))d is equipped with the L2 (Ω(t))d inner product for every t ∈ I. ∗ q q say that u ∈ L (I, Vq (t)) has a generalized time derivative in L (I, Vq (t)) if there exists ∗ dt u ∈ Lq (I, Vq (t)) such that Z Z

dt u(t), η(t) Vq (t) dt = − u(t), ∂t η(t) L2 (Ω(t)) dt (2.18) I

I

holds for any η ∈ H 1 (I, L2 (Ω(t))d ) ∩ Lq (I, Vq (t)) with η(T, ·) = η(0, ·) = 0. Remark 2.13. A few more words about the spaces of test functions in the above definitions are in order: By η ∈ H 1 (I, L2 (Ω(t))d ) ∩ Ls (I, Vq (t)) we mean η ∈ L2 (I, L2 (Ω(t))d ) ∩ Ls (I, Vq (t)) such that ∂t η ∈ L2 (I, L2 (Ω(t))d ). Applying the Piola transform it is easily verified that H 1 (I, L2 (Ω(t))d ) ∩ Lq (I, Vq (t)) is isomorphic to H 1 (I, L2 (Ω0 )d ) ∩ Lq (I, Vq (Ω0 )). Even though it would be desirable, it does not make sense to impose any constraint such as div ∂t η = 0 on the time derivative of the respective test functions. For example, div η = 0 does not imply div (∂t (Pϕ η)) = 0 as may be seen by direct calculation. That is to say, the spatial and the temporal properties of the test functions cannot be decoupled entirely as in the case of Bochner spaces. This is the main justification for the hybrid character of the generalized time derivative which is further reflected in the choice of the test functions. However, our definition of the generalized time derivative coincides with the time derivative in the sense of vector valued distributions if the generalized Bochner spaces coincide with their respective classical analogues. This is the case if, for example, the Piola transform ¯ This fact may be obtained by reduces to the identity mapping, i.e. ϕt = idRd for any t ∈ I. unwinding the definition of the generalized time derivative and together with the fact that temporal and spatial properties of test functions can be decoupled from each other if the spatial domain stays fixed in time. Similar reasoning shows that (2.18) then reduces to the convenient characterization of the time derivative as stated in (2.15). 3

Of course we assume that the integrals exist. This can be guaranteed through restrictions on the exponents of integrability. However, we suppress these restrictions in the respective definitions for the sake of readability.

44

Chapter 2. Function Spaces on Non-cylindrical Domains

3. Unsteady Monotone Problems on Non-cylindrical Domains 3.1. Informal Discussion of Relevant Examples and Results We turn our attention to a class of initial boundary value problems of the form ∂t u + Au = f

in Q,

u=g

on Γ, in Ω0 .

u(0) = u0

The set Q is supposed to be a non-cylindrical subset of Rd+1 , d ≥ 1, generated by the flow a smooth, (possibly) time-dependent vector field v : R × Rd → Rd . That is, we assume that Q=

[

[

{t} × Ω(t) :=

t∈I

{t} × ϕt (Ω0 ),

t∈I

where I = (0, T ) denotes a bounded time interval, Ω0 ⊂ Rd is a bounded domain with Lipschitz boundary and (ϕt )t∈I denotes the flow (with inverse (ψt )t∈I ) of v. By Γ we will denote the lateral boundary of Q which is given by [ [ Γ := {t} × ∂Ω(t) := {t} × ϕt (∂Ω0 ) t∈I

t∈I

and thus moves with velocity v. The operator A is assumed to be a (possibly nonlinear) elliptic second order differential operator in divergence form. Before we specify the precise properties of A we want to give some examples for such problems on non-cylindrical domains. An obvious example that belongs to this category is the heat equation with homogeneous Dirichlet boundary conditions ∂t u − ∆ u = f

in Q,

u=0

on Γ,

u(0) = u0

(3.1)

in Ω0 .

Replacing the Laplacian by the p-Laplacian operator leads to an equation that describes more general diffusion processes in domains that vary in time: ∂t u − div |∇u|p−2 ∇u = f in Q, u=0 u(0) = u0

on Γ, in Ω0 .

45

46

Chapter 3. Unsteady Monotone Problems on Non-cylindrical Domains

As our main motivation for equations on non-cylindrical domains are FSI problems we give two more examples from fluid dynamics. The unsteady, non-cylindrical p-Stokes problem with homogeneous boundary conditions asks for a velocity vector field u : Q → Rd and a pressure function π : Q → R such that ∂t u − div |∇u|p−2 ∇u + ∇π = f

in Q,

div u = 0

in Q,

u=0

on Γ, in Ω0 .

u(0) = u0

Although this system is physically meaningful only to a limited extent it contains the typical incompressibility constraint div u = 0. Moreover, this form of the Stokes problem can be regarded as a simplification of the full noncylindrical p-Navier–Stokes system. The latter describes the motion of an incompressible non-Newtonian fluid of constant density which is contained in a time-dependent domain. Again, the problem is to find a velocity vector field u and a pressure π that solve ∂t u + (u · ∇)u − div |Du|p−2 Du + ∇π = f

in Q,

div u = 0

in Q,

u=v

on Γ,

u(0) = u0

(3.2)

in Ω0 .

Notice that the non-homogeneous boundary condition u = v on Γ reflects the fact that the fluid particles stick to the moving boundary. Also, Du denotes the symmetric part of the gradient ∇u. The usual weak formulation of such problems is guided by the idea of interpreting the (weak form of) differential operators as mappings between suitable function spaces. The characteristic common feature of the above examples is that the elliptic parts of the respective equations induce a family of operators on function spaces that vary in time. This circumstance is in sharp contrast to problems on cylindrical domains, where one mapping on a fixed function space usually meets the requirements. To clarify this point and some further aspects of the time-dependency of the spatial domain, let us first consider a smooth solution u of the non-cylindrical heat equation (3.1). Multiplying (3.1)1 with a smooth function η ∈ C ∞ (Q) and integrating over Ω(t) yields 4 Z

Z ∂t u(t)η(t) dx +

Ω(t)

Ω(t)

Z ∇u(t) · ∇η(t) dx =

f (t)η(t) dx. Ω(t)

Notice that we integrated by parts in the elliptic term using also the homogeneous Dirichlet boundary condition u = 0 on Γ. To calculate the integral containing the time derivative, we 4

We suppress the x-dependency of the functions for the sake of readability

47

3.1 Informal Discussion of Relevant Examples and Results

apply the transport theorem to the function uη to obtain Z

Z

Z ∂t (u(t)η(t)) dx −

∂t u(t)η(t) dx = Ω(t)

Ω(t)

=

Ω(t)

Z

Z

d dt

u(t)∂t η(t) dx

u(t)η(t) v(t) · n(t) da

u(t)η(t) dx − ∂Ω(t)

Ω(t)

Z −

u(t)∂t η(t) dx

Ω(t)

d = dt

Z

Z u(t)η(t) dx −

u(t)∂t η(t) dx.

Ω(t)

Ω(t)

Here, the boundary integral vanishes due to the homogeneous boundary conditions. Integrating the resulting identity from 0 to T with respect to t we infer that a weak solution of the non-cylindrical heat equation should satisfy ZT Z −

ZT Z ∇u(t) · ∇η(t) dxdt

u(t)∂t η(t) dxdt + 0 Ω(t)

0 Ω(t)

ZT Z =

(3.3) f (t)η(t) dxdt + u(0), η(0) L2 (Ω0 )

0 Ω(t)

for any sufficiently smooth function η vanishing at t = T . At next, we rewrite the last equality: Setting H(t) := L2 (Ω(t)), H = L2 (Ω0 ) and V (t) = W01,2 (Ω(t)) we obtain Z − 0

T

u(t), ∂t η(t) H(t) dt +

ZT

A(t, u(t)), η(t) V (t) dt

0

(3.4)

ZT =

f (t), η(t)

H(t)

dt + u(0), η(0)

H

.

0

Based on the last identity, let us work out some more distinctive peculiarities. First of all, we recognize that in general, we will have to deal with a time-dependent family of operators A(t, ·) t∈I in order to derive a familiar formulation of the problem. Then, if we use identity (3.4) to define weak solutions, we will have to clarify the sense of the initial condition. The point is that a continuity condition like lim u(t) = u0 in L2 (Ω0 )

t→0

48


does not make sense a priori as u(t) will typically live in L2 (Ω(t)) whereas u0 is defined on Ω0 . This problem is connected to the question whether the mapping [ u:I→ L2 (Ω(t)), t∈I¯

t 7→ u(t) ∈ L2 (Ω(t)), is well-defined. As a first step towards answering this question, we formally multiply equation (3.1) with the solution u itself. Under reasonable assumptions on f and u0 we may deduce the familiar energy estimate

u 2

u ∞ ≤ C. (3.5) + L (I,V (t)) L (I,H(t)) However, notice that the usual Bochner spaces have to be replaced by generalized Bochner spaces. Equation (3.3) serves as an equation for the time derivative of u. In fact, for a solution u in the natural energy space L∞ (I, H(t)) ∩ L2 (I, V (t)) we formally obtain ∗ dt u ∈ L2 (I, W01,2 (Ω(t))) (3.6) for the generalized time derivative dt u, see Definition 2.12. Furthermore, since for any t ∈ I¯ there holds dense

V (t) ,→ H(t)

injective

,→

V (t)0 ,

(3.7)

the spaces V (t), H(t), V (t)0 t∈I¯ form a family of Gelfand triples. In the cylindrical setting where Ω(t) ≡ Ω0 , we know that the energy estimate (3.5), the information about the time derivative in (3.6) and the Gelfand triple structure of the target spaces (3.7) yield L2 continuity of u by an abstract embedding result for function spaces. In order to justify initial values we need a non-cylindrical analogue of this result. This technical tool is particularly relevant when it comes to nonlinear differential operators such as the p-Laplacian operator on Vp (t) := W01,p (Ω(t)). This is due to the fact that the continuity result we just mentioned is intimately connected with an abstract integration by parts formula. In terms of function spaces this point can be formulated as follows: If u ∈ Lp (I, Vp (t)), ∗ has a generalized time derivative dt u ∈ Lp (I, Vp (t)) in the sense of Definition 2.12, is the S evaluation u : I¯ → s∈I¯ H(s), s 7→ u(s), then well-defined and does the identity Zs

2 1 dt u(t), u(t) Vp (t) dt = u(s) H(s) − 2

1

u(0) 2 H(0) 2

(3.8)

0

hold for any s ∈ I¯ ? The familiar cylindrical version of this formula is essential for proving existence results for parabolic problems with pseudomonotone elliptic parts. At this point we would like to refer to [Lio69], [GGZ74], [Zei90], [Růž04] and [Rou13] where this technique is presented in great detail. The justification of (3.8) will be the main result of the current chapter. The presence of lower order perturbations such as the convective term in the p-Navier–Stokes system (3.2) requires compactness methods. As a first step in this context we are going to generalize a result of Landes and Mustonen in Chapter 4. Despite its easy applicability, this

49

3.2 The p-Stokes System as a Model Problem

result seems to be rather unknown even in the cylindrical framework. To our best knowledge, its generalization to non-cylindrical problems is a new result. By contrast, a fairly well-known tool is the Aubin–Lions lemma. In Chapter 5 we will formulate a version of this lemma that is adapted to the functional framework of generalized Bochner spaces. A similar approach has been succesfully applied in [LR14]. However, we think that our formulation is more convenient in the sense that its proof proceeds along the lines of the proof of the classical version of the Aubin–Lions lemma. In Chapter 6 and 7 this result is then combined with the so-called solenoidal Lipschitz truncation technique to obtain existence of weak solutions in the context of non-Newtonian fluids in moving domains.

3.2. The p-Stokes System as a Model Problem We now focus on the non-cylindrical p-Stokes system

∂t u − div |∇u|p−2 ∇u + ∇π = f

in Q,

div u = 0

in Q,

u=0

on Γ,

u(0) = u0

(3.9)

in Ω0 .

The physical meaning of this system will be postponed to a later chapter about nonNewtonian fluids. For the moment we confine ourselves with the fact that the p-Stokes system represents a model problem for the theory of monotone operators, since the p-Laplacian operator is one of the most common nonlinear pseudomonotone operators in applications. A second point we want to stress is the incompressibility constraint div u = 0. In the cylindrical case, this constraint can be captured in the functional setting without changing the relevant abstract properties of the spaces. In the non-cylindrical setting though, this constraint represents one of the major problems as we will see. Apart from that, we will treat the case of homogeneous Dirichlet boundary conditions first, even if they do not seem to be meaningful from the physical point of view. At this stage though, we are rather interested in a general theory for nonlinear monotone operators than in general boundary conditions. The methods we develop will easily carry over to scalar problems such as the non-cylindrical heat equation or the non-cylindrical unsteady p-Laplacian equation. To begin with, we show how the structural conditions imposed by the p-Laplacian determine the function spaces we will have to use. An important point in the theory of monotone operators is that the space where the solution is sought in and the space of test functions coincide. As a byproduct, we can eliminate the pressure in the weak formulation by choosing the following functional setting: We define for

50


t ∈ I¯ V (t) := W01,p (Ω(t))d = C0∞ (Ω(t))d

k·kV (t)

,

u(t) := ∇u(t) Lp (Ω(t)) , V (t)

V := V (0), H(t) := L2 (Ω(t))d ,

u(t) := u(t) L2 (Ω(t)) , H(t)

H := H(0), 1,p ∞ (Ω(t))d Vσ (t) := W0,σ (Ω(t))d = C0,σ ∞ (Ω(t))d Hσ (t) := C0,σ

Then we also define

k·kH(t)

k·kV (t)

,

.

5

A : Vσ (t) → Vσ (t)0 , Z

A(u(t)), η(t) Vσ (t) := |∇u(t)|p−2 ∇u(t) : ∇η(t) dxdt, Ω(t)

to recover the typical properties of the operator associated to the p-Laplacian: For any t ∈ I¯ and for any u(t), η(t) ∈ Vσ (t) there holds • Growth:

A(u(t))

Vσ

(t)0

p−1 ≤ u(t) Vσ (t) .

• Coercivity:

A(u(t)), u(t)

Vσ (t)

p = u(t) Vσ (t) .

• Monotonicity:

A(u(t)) − A(η(t)), u(t) − η(t) Vσ (t) ≥ 0. With a slight abuse of notation we set 0 A : Lp (I, Vσ (t)) → Lp (I, Vσ (t)) , Z

Au, η := A(u(t)), η(t) Vσ (t) dt, I

0 to obtain a monotone, hemicontinuous, coercive operator A : Lp (I, Vσ (t)) → Lp (I, Vσ (t)) , see also Definition A.24. From the same calculation as in case of the heat equation, we infer that any sufficiently smooth solution of (3.9) satisfies the energy equality

1

u(s) 2 + H(s) 2

Zs

1 2 A(u(t)), u(t) Vσ (t) dt = u0 H + 2

0 5

We suppress the time-dependency of the operator A.

Zs 0

f (t), u(t) Vσ (t) dt.

51

3.2 The p-Stokes System as a Model Problem

Under reasonable assumptions on f and u0 this energy equality and the coercivity of A immediately yield the formal energy estimate

u ∞

u p + ≤ C, L (I,H(t)) L (I,Vσ (t)) with a constant C only depending on u0 and f . The aim of this chapter is to construct weak solutions in this regularity class by using the theory of monotone operators directly on the non-cylindrical domain Q. Now that we have specified the natural functional setting of the problem we define in what sense a weak solution is to be understood. Definition 3.1 (Weak Solution) ∗ Suppose f ∈ Lp (I, V (t)) , p ≥ 2, and u0 ∈ Vσ (0) ,→ Hσ (0). A vector field u ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) is a weak solution of the non-cylindrical p-Stokes system (3.9) if ZT −

u(t), ∂t η(t)

ZT

H(t)

dt+

0

A(u(t)), η(t)

Vσ (t)

dt

0

ZT

f (t), η(t) Vσ (t) dt − u0 , η(0) H

= 0

holds for any η in the space of test functions n o X := ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) ϕ(T, ·) = 0 . Remark 3.2. The time traces in H(t) of these ∗ test functions are well-defined according to Lemma 2.11. Moreover, for f ∈ Lp (I, V (t)) and η ∈ Lp (I, Vσ (t)) the duality

f, η

Z

Lp (I,Vσ (t))

:= 0

T

f (t), η(t) Vσ (t) dt

is well-defined due to the embedding (2.12). Remark 3.3. Note that the initial value is required to be more regular than in the cylindrical case. There, u0 ∈ Hσ (0) would be the natural choice. Also, the necessity for the lower bound p ≥ 2 is not quite obvious as it is not required in the cylindrical case. However, for the approximation procedure we will apply below we will need both assumptions. The standard process to prove existence of weak solutions for (cylindrical) unsteady problems combines some finite dimensional approximation scheme with subsequent (weak) compactness methods, in order to pass to the limit in the approximation. If the equation contains nonlinear terms, the weak limit usually cannot be identified as a weak solution directly. Considering a problem on a prescribed non-cylindrical domain, it is convenient to transform it to an auxiliary problem on a fixed, cylindrical domain. One could then essentially try to apply the above mentioned techniques to solve the new system.

52


Although the transformed problem may have similar characteristics, it would certainly contain artificial terms that are solely affiliated to the transformation. Besides esthetical flaws, this method does neither take into account the covariant or coordinate invariant character that is shared by most equations in mathematical physics. Therefore, we are going to construct a Galerkin approximation scheme directly on the noncylindrical domain by using a special set of ansatz functions. To identify the weak limit in the nonlinear elliptic term, we will prove an energy equality that serves as a generalization of the common "integration by parts" formula to the non-cylindrical setting. The interesting feature of this approach is that diffeomorphisms are not used to transform the equation, but instead they are used to construct approximations.

3.3. Galerkin Approximation As Vσ (0) is separable it contains a countable and dense subset which we denote by n o Dσ (0) := Wk k∈N ⊂ Vσ (0) . Since ψt : Ω(t) → Ω0 is a diffeomorphism, the Piola transform Pψt : Vσ (0) → Vσ (t) yields an isomorphism as we have shown in Lemma 2.4. Therefore, the set o n Dσ (t) := wk (t) k∈N wk (t) := Pψt Wk , k ∈ N ⊂ Vσ (t) is countable and dense in Vσ (t). Lemma 3.4 The set X G :=

n nX

o αl wl (·) n ∈ N, αl ∈ C01 ([0, T )), wl (·) ∈ Dσ (·)

l=1

is dense in the set of test functions n o X = ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) ϕ(T, ·) = 0 . Proof. By virtue of the isomorphism Pϕ : H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) → H 1 (I, H) ∩ Lp (I, Vσ (0)) the assertion is equivalent to the density of the set n nX l=1

o αl Wl n ∈ N, αl ∈ C01 ([0, T )), Wl ∈ Dσ (0)

(3.10)

53

3.3 Galerkin Approximation

in the corresponding space over the cylindrical domain, namely in n o η ∈ H 1 (I, H) ∩ Lp (I, Vσ (0)) η(T, ·) = 0 . ˜ ∈ C0∞ ([0, T ), Vσ (0)) Any function η in the latter space can be approximated by some η ˜ by mollifying a suitable truncation of η in time. We may subsequently approximate ∂t η P in Lp (I, Vσ (0)) ,→ L2 (I, H) by some ϕn = nl=1 αl Wl as the following argument shows: Suppose there exists F ∈ (Lp (I, Vσ (0)))0 , F 6= 0, such that Z hF, αl Wl iLp (I,Vσ (0)) = αl (t)hF(t), Wl iVσ (0) dt = 0 I

for every αl ∈ C01 ([0, T )) and every Wl ∈ Dσ (0). It follows that hF(t), Wl iVσ (0) = 0 for almost every t ∈ I and every Wl ∈ Dσ (0). By density of Dσ (0) in Vσ (0) we get F(t) = 0 almostP everywhere in I, thereby contradicting our assumption. Thus, there exists a sequence ˜ . Setting ϕn = nl=1 αl Wl approximating ∂t η T

Φn (t) := −

n Z X

αl (s) ds Wl

l=1 t

Pn l ˜ as n → ∞. The fundamental theorem of calculus in yields ∂t Φn = l=1 α Wl → ∂t η Bochner spaces combined with the continuous embedding Vσ (0) ,→ H immediately shows ¯ H) and also in C(I, ¯ Vσ (0)). This ˜ in C(I, that, as n tends to infinity, we get Φn → η 2 p ˜ in L (I, H) ∩ L (I, Vσ (0)) for n → ∞. eventually implies Φn → η Applying similar arguments as in Lemma 3.4 we can show Corollary 3.5 The embedding H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) ,→ Lp (I, Vσ (t)), p ≥ 2, is dense. Having defined a suitable set of ansatz functions, we now construct the actual Galerkin approximation. We look for vector fields of the form un (t, x) =

n X

αnl (t)wl (t, x)

l=1

which solve for any 1 ≤ k ≤ n the ordinary differential equation

∂t un (t), wk (t) H(t) + A(un (t)), wk (t) Vσ (t) = f (t), wk (t) Vσ (t) ,

(3.11)

54


subject to the initial condition un (0) = u0n . (3.12) Here, u0n ∈ span Wk 1 ≤ k ≤ n is a sequence which strongly converges to u0 in Hσ (0). The latter is possible due to the density of the embedding Vσ (0) ,→ Hσ (0). Using the symbol αn for the solution vector, the system (3.11) can then be rewritten as

˙ n (t) = F(t) αn (t) + G(t) + H(t), E(t) α

(3.13)

with the definitions Z wk (t) · wl (t) dx,

Ekl (t) := Ω(t)

Z wk (t) · (∂t wl )(t) dx,

Fkl (t) := − Ω(t)

Z

|∇un (t)|p−2 ∇un (t) : ∇wk (t) dx,

Gk (t) := Ω(t)

Z f (t) · wk (t) dx.

Hk (t) := Ω(t)

Notice that the matrix P E(t) is invertible for any t as it is symmetric and positive. In fact, for λ ∈ Rn and v(t) = nl=1 λl wl (t) we get Z

2

T λ E(t)λ = |v(t)|2 dx = v(t) H(t) ≥ 0. Ω(t)

This implies the positivity of E(t) since { wl (t), 1 ≤ l ≤ n } is linearly independent in Hσ (t). Thus, (3.13) may further be transformed into ˙ n (t) = F (t, αn (t)), α where F : I × Rn → Rn is a vector field which is measurable in the time variable and continuous in the spatial variable. The measurability of F is a consequence of Fubini’s theorem since F is integrable. Thus, for every n ∈ N, Carathéodory’s theorem implies the short-time existence of a continuous solution αn which is almost everywhere differentiable and satisfies the initial condition (3.12). Global in time existence follows from the standard a priori estimates we will now derive. To this end, we multiply (3.11) with the coefficients of the solution, sum up and integrate in time. Taking also into account the coercivity of A we get

max un (s) Hσ (s) + un Lp (I,Vσ (t)) ≤ C, s∈I¯

where C > 0 only depends on the data but is independent of n. Thanks to the growth conditions of the p-Laplacian operator, we also see Z Z p0

0

p p−1

Aun p p0

un p = |∇u (t)| dxdt = , n L (I,Vσ (t)) L (Q) I Ω(t)

55


which yields

Aun p0 ≤ C. L (Q) These estimates and the weak compactness of bounded sets in the underlying function spaces, 0 see Corollary 2.9, imply the existence of u ∈ Lp (I, Vσ (t)) ∩ L∞ (I, Hσ (t)), χ ∈ Lp (Q)d×d and u∗ ∈ Hσ (T ) such that un * u in Lp (I, Vσ (t)), ∗

un * u in L∞ (I, Hσ (t)), un * u in L2 (I, Hσ (t)),

(3.14)

0

Aun * χ in Lp (Q)d×d , un (T ) * u∗ in Hσ (T ), un (0) → u0 in Hσ (0)

holds for a not relabeled subsequence n → ∞. Consequently, we can pass to the limit in the Galerkin system using also the density of the ansatz functions in the space of test functions, see Lemma 3.4. We then obtain that u ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) satisfies ZT −

u(t), ∂t η(t)

ZT

H(t)

dt+

0

χ(t), η(t)

Vσ (t)

dt

0

ZT =

(3.15)

f (t), η(t) Vσ (t) dt + u0 , η(0) H

0

for every η in the space of test functions. We can further localize the latter identity for almost every s ∈ I, thereby abandoning the restriction η(T, ·) = 0. Lemma 3.6 For almost every s ∈ I and for every η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) there holds

u(s), η(s) H(s) − u0 , η(0) H =

Zs

f (t), η(t)

0

Vσ (t)

Zs dt − 0

Zs +

χ(t), η(t)

Vσ (t)

dt (3.16)

u(t), ∂t η(t) H(t) dt.

0

Proof. First fix an arbitrary s ∈ I and k ∈ C ∞ (R) such that k(t) ≡ 1 if t ≤ 0 and k(t) ≡ 0 if t ≥ 1. For δ > 0 we define the smooth cut-off function kδs (·) := k( ·−s δ ). It follows that s s s kδ (t) ≡ 1 if s ≤ t, kδ (t) ≡ 0 if s ≥ t + δ and kδ (·) → χ[0,s] as δ tends to zero. Moreover, as R s ∂t kδs (t) = 1δ k 0 ( t−s δ ) a simple change of variables implies − R ∂t kδ (t) = 1 for any s ∈ I.

56


If η is a test function then kδs η still is an admissible test function if δ is sufficiently small (depending on s). Plugging kδs η into (3.15) yields ZT −

u(t), ∂t (kδs (t)η(t)) H(t) dt − u0 , η(0) H

0

ZT =

f (t), kδs (t) η(t) Vσ (t) dt

0

ZT −

χ(t), kδs (t) η(t) Vσ (t) dt

0

and especially ZT −

u(t), ∂t (kδs (t)η(t)) H(t) dt = −

0

ZT

u(t), kδs (t) ∂t η(t) H(t) dt

0

ZT −

u(t), ∂t kδs (t) η(t) H(t) dt.

0

It follows from dominated convergence that those integrals that do not contain the term ∂t kδs converge as δ tends to zero to their respective restrictions to the interval (0, s). If u was continuous we had ZT −

u(t), ∂t kδs (t) η(t) H(t) dt → u(s), η(s) H(s) as δ → 0

0

for any s ∈ I by Lebesgue’s differentiation theorem. However, this property is also valid for every Lebesgue point s ∈ I of u. Since u ∈ L2 (Q)d the assertion holds for almost every s ∈ I. By Lemma 2.11, the evaluation of η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) at s ∈ I¯ only lies in = H(s), although η(s) ∈ Vσ (s) holds for almost every s ∈ I. Moreover, it is ¯ can be realized as the trace at time s unclear whether any arbitrary field η s ∈ H(s), s ∈ I, 1 p ¯ any of some η ∈ H (I, H(t)) ∩ L (I, Vσ (t)). Nevertheless, we can show that for every s ∈ I, 1 p ξ s ∈ Vσ (s) is the trace at time s of some ξ ∈ H (I, H(t)) ∩ L (I, Vσ (t)). L2 (Ω(s))3

Lemma 3.7 For any s ∈ I¯ and every ξ s ∈ Vσ (s), there exists ξ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) such that ξ(s) = ξ s in Vσ (s). Proof. Let s ∈ I¯ and ξ s ∈ Vσ (s) be arbitrary. We obtain ξ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) with ξ(s) = ξ s by setting ξ := Pψ Pϕs ξ s , (3.17) ¯ ξ(t) := Pψ Pϕs ξ s , t ∈ I. t

57


The fact that ξ thus constructed lies in Lp (I, Vσ (t)) follows from the transformation formula for integrals an the fact that the Piola transform induces isomorphisms between the respective Sobolev spaces of incompressible vector fields, see also Lemma 2.4. The differentiability of ξ in time is a consequence of the chain rule, as the Piola transform acts on Pϕs ξ s by concatenation. Using these special test function, we get the following corollary which is a direct consequence of Lemma 3.6. Corollary 3.8 For almost every s ∈ I and for any ξ s ∈ Vσ (s) there holds u(s), ξ s

H(s)

− u0 , ξ(0)

Zs

H

u(t), ∂t ξ(t) H(t) dt

− 0

Zs =

f (t), ξ(t)

Zs

Vσ (t)

0

dt −

(3.18)

χ(t), ξ(t) Vσ (t) dt.

0

Here, the test function ξ is defined by (3.17). We now come to the remaining identification 0

χ = Au in Lp (Q)d×d . The same identification has to be accomplished in the cylindrical setting. However, we first show up the similarities and differences between the cylindrical and the non-cylindrical case. The most important similarity of the limit equations in both cases is that they characterize the structure and the regularity of a certain time derivative of the weak limit u. Note that we cannot characterize the weak time derivative since D(Q)d is not contained in the set of test functions by virtue of the incompressibility constraint. However, we may conclude the following: For any test function η ∈ X which additionally satisfies η(0) = 0, we have the identity Z T Z T Z T

− u(t), ∂t η(t) H(t) dt + χ(t), η(t) Vσ (t) dt = f (t), η(t) Vσ (t) dt , 0

0

0

which by Definition 2.12 implies ∗ dt u ∈ Lp (I, Vσ (t)) . Since the embedding n o η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) η(0, ·) = η(T, ·) = 0 ,→ Lp (I, Vσ (t)) 0 is dense,6 we also get a bound on dt u ∈ Lp (I, Vσ (t)) in terms of f and χ. Altogether, the limit function u ∈ Lp (I, Vσ (t)) has a generalized time derivative in the dual of its 6

This in turn can be seen by applying a transformation to the cylindrical setting. There, the assertion follows by mollification. See also Lemma 3.4 and Corollary 3.5

58


own regularity class. This is the same information as the one obtained in the standard (cylindrical) setting. At this point, there is a useful result we cite for the sake of completeness. Lemma 3.9 (Integration by Parts in Bochner Spaces) Suppose (V, H, V 0 ) is a Gelfand triple and 1 < p < ∞. Then, the embedding ¯ V ) ,→ W 1,p,p0 (V, V 0 ) = C (I, ∞

n

dv o 0 0 v ∈ L (I, V ) ∈ Lp (I, V ) ∼ = Lp (I, V 0 ) dt p

is dense. Moreover, there is a continuous embedding 0

¯ H). W 1,p,p (V, V 0 ) ,→ C(I, 0

That is, after modifying u ∈ W 1,p,p (V, V 0 ) on a subset of I¯ of measure zero it can be 0 identified with a continuous function u : I¯ → H. Also, any u ∈ W 1,p,p (V, V 0 ) satisfies for ¯ s0 < s, the integration by parts formula any s0 , s ∈ I,

1

u(s) 2 − 1 u(s0 ) 2 = H H 2 2

Zs

du (t), u(t) V dt. dt

s0

The proof of the last lemma can be found in [GGZ74] and in [Rou13]. With the noncylindrical analogue of Lemma 3.9 at hand, we will be able to identify χ by applying Minty’s trick in Proposition 3.16 below. Let us make some comments on Lemma 3.9 to motivate our next steps. An inspection of the proof of Lemma 3.9 shows that the result boils down to the construction of a smooth sequence ¯ V ) approximating u in W 1,p,p0 (V, V 0 ). The latter construction is straightforward in C ∞ (I, in the setting of Bochner spaces: After extending v ∈ Lp (I, V ) to v ∈ Lp (R, V ) by reflection ¯ V ). The and truncation in time, mollification in time yields a sequence (v )>0 ⊂ C ∞ (I, p convergence lim→0 v = v in L (I, V ) is standard. Since convolution is a self-adjoint operation (at least for even mollifiers) and since it commutes with (time) derivatives, the fact that 0 p0 0 ∼ p the respective classical time derivatives strongly converge to dv dt ∈ L (I, V ) = L (I, V ) follows without essential difficulties. Notice also that the property of v and dv dt of being V -valued is always fulfilled if the smoothing operation solely applies to the time variable, i.e. mollification in time does not manipulate "spatial" properties. This point is of particular importance if constraints such as incompressibility are captured in the target space V . In the non-cylindrical setting however, we cannot just mollify in time since this operation is not well-defined in general, the reason being that the smoothing parameter cannot be chosen uniformly. Due to the embedding Lp (I, Vσ (t)) ,→ Lp (Q)d one could try to apply mollification in both variables. In order to preserve zero boundary values one is led to additionally use (spatial) cut-off functions or partitions of unity, thereby altering the divergence. We will avoid these technical difficulties by transforming u to a fixed cylindrical domain by means of the Piola transform. The latter preserves the incompressibility constraint as well as vanishing traces on the spatial boundary. With the help of a suitable mollification we obtain an approximate sequence that can be transformed back to the non-cylindrical

59

3.4 Density Result

domain. There, it is readily seen to converge at least in the natural energy norm. However, the Piola transform does not commute with time derivatives so that we cannot expect strong convergence of the sequence of approximate time derivatives. Moreover, the lack of commutativity keeps us from transforming the problem (3.9) to the fixed reference cylinder I × Ω0 by means of the Piola transform. On a final note, the Piola transform does not induce a self-adjoint operation in general. Nevertheless, a thorough inspection of the proof of Lemma 3.9 reveals the fact that strong convergence in the energy norm together with boundedness of the approximate time derivatives in the respective dual space is sufficient to deduce a result similar to that of Lemma 3.9.

3.4. Density Result Applying the Piola transform to the weak limit u ∈ Lp (I, Vσ (t)) we obtain a vector field U := Pϕ u ∈ Lp (I, Vσ (0)), where U (t) = Pϕt u(t) = Jϕt ϕ∗t u(t). ¯ Vσ (0)), is a family of continuous smoothing Now, if (F )>0 , F : Lp (I, Vσ (0)) → C ∞ (I, operators such that for any V ∈ Lp (I, Vσ (0)) there holds F V → V in Lp (I, Vσ (0)) as → 0, we deduce that R u := Pψ ◦ F ◦ Pϕ u → u in Lp (I, Vσ (t)) as → 0, since the respective Piola transforms are isomorphisms. We have a certain freedom of choice concerning F . One usually takes F as the convolution in time with an even mollifier. Then, F is an L2 self-adjoint operator. However, it does not preserve time traces at t = 0 or t = T in general. This problem can be partly circumvented by using a mollifier with a shifted axis Rof symmetry. We fix a function ρ ∈ C0∞ ((0, 1)) with R ρ ds = 1. Setting ρ (s) := −1 ρ(s/) we define for V ∈ Lp (I, Vσ (0)) and t ∈ I Zt

F V(t) := ρ ∗ V (t) =

ρ (t − s)V(s) ds t−

(3.19)

Z0 =

ρ (−s)V(s + t) ds, −

where the last equality follows from a change of variables. Suppose that the evaluation of V at 0 is well-defined. Then we may extend V to (−T, 0) by V(0). In this case, (3.19) shows that F V(0) = V(0). The L2 adjoint operator F0 has similar properties. In fact, by

60


definition of the adjoint there holds V, F0 W L2 (I)

= F V, W

Z Zt

L2 (I)

ρ (t − s)V(s) ds W(t) dt

= t−

ZI Z

χ(t−,t) (s)ρ (t − s)V(s) ds W(t) dt

= ZI ZI

χ(s,s+) (t)ρ (t − s)V(s) ds W(t) dt

= I

I

Z Zs+ = ρ (t − s)W(t) dt V(s) ds I

s

Z Zs+ = ρˇ (s − t)W(t) dt V(s) ds, I

s

where ρˇ (·) := ρ (−·). Therefore the adjoint operator F0 is given by mollification with ρˇ : F0 W(t)

Zt+ = ρˇ ∗ W (t) = ρˇ (t − s)W(s) ds

t t+ Z

ρ (s − t)W(s) ds.

= t

Especially, F0 preserves the values at t = T if the field W is extended by W(T ) to, say, (T, 2 T ) whenever its evaluation at T is well-defined. Extending U = Pϕ u by zero7 outside of I we obtain R u(0) = 0. Thus, (R u(0))>0 cannot converge to u0 6= 0 and we will have toR take into account the initial value separately. t For ρ defined above we set λ (t) := 0 ρ (s) ds. Also, let χ[0,) denote the characteristic function of [0, ). Then we define a continuous cut-off function by κ = 1 − λ χ[0,) . The initial value u0 ∈ Vσ (0) is extended in time by setting ˆ 0 (t) := Pψt u0 ∈ Vσ (t). u From the smoothness of the Piola transform and using also the transformation formula for ˆ 0 ∈ Lp (I, Vσ (t)). Eventually we define integrals we infer u ˆ 0 (t) := κ (t)ˆ S u u0 (t). ˆ 0 (0) = u ˆ 0 (0) = u0 , S u ˆ 0 ∈ Lp (I, Vσ (t)) and S u ˆ 0 → 0 in Lp (I, Vσ (t)) It follows that S u as → 0. The latter is a consequence of Lebesgue’s theorem since κ is bounded on I and 7

We will always assume that Pϕ u is extended by zero without mentioning it explicitly in the sequel

61

3.4 Density Result

vanishes almost everywhere on I as tends to zero. We now define a smooth approximation in time of u and summarize its properties in the next proposition. Proposition 3.10 Let u ∈ Lp (I, Vσ (t)) satisfy (3.15) and extend u by u := 0 outside of I. For (3.20)

ˆ 0, u := R u + S u there holds • u ∈ Lp (I, Vσ (t)) ,→ L2 (I, H(t)). • u (0) = u0 in Vσ (0) ,→ Hσ (0). • u → u in Lp (I, Vσ (t)) as → 0.

• u is differentiable with respect to the time variable and ∂t u ∈ L2 (I, H(t)) for any fixed > 0. Proof. Thanks to the properties of R and S we infer u ∈ Lp (I, Vσ (t)). The embedding Lp (I, Vσ (t)) ,→ L2 (I, H(t)) clearly holds true for p ≥ 2. Since u is extended by zero we ˆ 0 (0) = u0 the second point follows. As we already know know that R u(0) = 0 and as S u ˆ 0 → 0 in Lp (I, Vσ (t)) as → 0, the third assertion is obtained from the fact that that S u R u converges to u in Lp (I, Vσ (t)) as tends to zero. To prove the remaining claim, we give a more detailed expression for R u, that is R u(t, x) = Pψt F (Pϕ u) (t) (x) Zt ρ (t − s)Jϕs Tψs (x)u(s, x) ds.

= Jψt Tϕt (ψt (x)) t−

It follows that Zt

∂t R u(t, x) = ∂t Jψt Tϕt (ψt (x))

ρ (t − s)Jϕs Tψs (x)u(s, x) ds

t−

= ∂t (Jϕt Tϕt (ψt (x)))

Zt

ρ (t − s)Jϕs Tψs (x)u(s, x) ds

t−

Zt + Jϕt Tϕt (ψt (x)

∂t ρ (t − s)Jϕs Tψs (x)u(s, x) ds.

t−

Keeping in mind that the diffeomorphisms and the kernel ρ are smooth, we infer ∂t R u ∈ ˆ 0 since L2 (I, H(t)), > 0, from the last identity. A similar argument applies to ∂t S u ˆ 0 (t) = ∂t κ Pψ u0 (t) = ∂t κ Jψt (Tϕt u0 ) ◦ ψt ∂t S u = −ρ χ[0,) Jψt (Tϕt u0 ) ◦ ψt + κ (t)∂t Jψt (Tϕt u0 ) ◦ ψt

62


Remark 3.11. Although the smoothing operator R is intuitively accessible, the convergence (in a suitable sense) ∂t R u → dt u for → 0 is not obvious, even if dt u was more regular. To clarify this point, suppose that u ∈ Lp (I, Vσ (t)) has a weak time derivative ∂t u ∈ Lr (Q)d , 1 < r < ∞. For the sake of simplicity, we assume that Q is generated by the flow of a smooth, compactly supported and incompressible vector field w. Denoting the respective flow again by (ϕt )t∈I and the inverse flow by (ψt )t∈I , the Piola transform reduces to the pull-back of vector fields and we obtain u := R u = ψ ∗ (ρ ∗ ϕ∗ u). We neglect the initial value of u in what follows. Recall that the flow property of the Lie derivative, see (1.13), yields8 ∂t (ϕ∗ u) = ϕ∗ (Lw u), where Lw := ∂t + Lw denotes the dynamic Lie derivative and Lw u denotes the Lie bracket [w, u], see (1.9). With this notation we get ∂t u = ψ ∗ L−w (ρ ∗ ϕ∗ u) = ψ ∗ ∂t (ρ ∗ ϕ∗ u) − Lw (ρ ∗ ϕ∗ u) . Notice that spatial regularity of u is necessary to justify this calculation because the Lie bracket contains gradients. Since ϕ∗ u ∈ Lp (I, Vσ (0)), the time derivative of ∂t (ρ ∗ ϕ∗ u) commutes with mollification and we therefore obtain ∂t u = ψ ∗ ρ ∗ ∂t (ϕ∗ u) − Lw (ρ ∗ ϕ∗ u) = ψ ∗ ρ ∗ ϕ∗ (∂t u) + ρ ∗ ϕ∗ (Lw u) − Lw (ρ ∗ ϕ∗ u) = ψ ∗ ρ ∗ (ϕ∗ ∂t u) + ψ ∗ ρ ∗ Lϕ∗ w ϕ∗ u − Lψ∗ w ψ ∗ (ρ ∗ ϕ∗ u) . Here, we also used the fact that the Lie bracket behaves natural with respect to diffeomor ∗ ∗ phisms. While the convergence for → 0 of ψ ρ ∗ (ϕ ∂t u) follows from the properties of the pull-backs and the regularity of ∂t u, the question whether the remaining terms cancel is not trivial. This cancellation however is suggested by the fact that ∂t u = ∂t (ψ ∗ ϕ∗ u) = ∂t u + Lw u − Lw u = ∂t u. With regard to our problem, the situation is even more subtle as the generalized time derivative dt u is only characterized as a functional on a dense subset of the energy space Lp (I, Vσ (t)). Keeping in mind the latter remark, our next goal is to prove boundedness of the sequence (∂t u )>0 in the sense

sup ∂t u (Lp (I,Vσ (t)))0 ≤ C. >0

8

We suppress the concrete evaluation in time for the sake of readability.

63

3.4 Density Result

The idea is to work directly with the limit equation of the Galerkin approximation. However, due to the structure of the smoothing operator R , we will have to deal with artificial terms related to the Piola transform and its adjoint. We therefore treat these additional terms related to the adjoint R0 first. An easy computation shows that the adjoints of the Piola transforms at time t are given by Pψ0 t v = Jψt (TϕTt v) ◦ ϕt and Pϕ0 t V = Jϕt (TψtT V) ◦ ψt , respectively. Here TϕTt and TψtT denote the transposed matrices of Tϕt and Tψt . Since we already know that the adjoint of F is given by convolution with ρˇ we can give the following expression for R0 0 R0 η(t, x) = Pψ ◦ F ◦ Pϕ (η)(t, x) = Pϕ0 ◦ F0 ◦ Pψ0 (η)(t, x) =

Jϕt TψtT (x)

Zt+ ρ (s − t) Jψs TϕTs (x) η(s, x) ds. t

This representation shows that R is not self-adjoint, even if we had used an even mollifying kernel. For R to be self-adjoint we would require Tϕt (ψt (x)) = TψtT (x) and Jϕt = Jψt ≡ 1. But since Tψt = (Tϕt )−1 this would be the case if and only if TϕTt = (Tϕt )−1 . That means, R is self-adjoint provided that ρ is symmetric and that Tϕt is an orthogonal movement, i.e. a rotation or a true displacement. However, this need not be the case for the general situation we consider here. Moreover, the expression for R0 shows that the adjoint operator does not preserve the divergence constraint in general. This fact will complicate the computation since the generalized time derivative we are interested in is only defined on solenoidal test functions. As a consequence we cannot just shift the smoothing operator to the test function by simple transposition. This issue does not occur in the framework of classical Bochner spaces. We also notice that neither R nor R0 commutes with the time derivative because the Piola transform itself depends on time, see also Remark 3.11. As it acts on space variables by concatenation, taking the time derivative results in spatial derivatives due to the chain rule. In order to control these terms properly, we have to estimate the commutator [∂t , R0 ] which is defined by h

i ∂t , R0 η := ∂t (R0 η) − R0 (∂t η)

and measures to which ∂t and R0 fail to commute. the extent For any η ∈ ϕ ∈ H 1 (I, H(t))∩Lp (I, Vσ (t)) ϕ(T, ·) = 0 9 we get, using also the expression 9

Test functions are always extended constantly in time

64


for R0 , h

i ∂t , R0 (η)(t, x) :=∂t (R0 η)(t, x) − R0 (∂t η)(t, x) =(∂t (Jϕt TψtT (x)))

Zt+ ρ (s − t) Jψs TϕTs (x) η(s, x) ds t

Zt+ T + Jϕt Tψt (x) ρ (s − t) ∂s Jψs TϕTs (x) η(s, x) ds t

Zt+ − Jϕt TψtT (x) ρ (s − t) Jψs TϕTs (x)∂s η(s, x) ds. t

Using the product rule, the last identity implies Zt+ h i T 0 ∂t , R (η)(t, x) =(∂t (Jϕt Tψt (x))) ρ (s − t) Jψs TϕTs (x) η(s, x) ds t

Zt+

ρ (s − t) ∂s (Jψs TϕTs (x)) η(s, x) ds

+ Jϕt TψtT (x)

t

= ∂t , Jϕt TψtT (x)

Zt+ ρ (s − t) TϕTs (x) ∂s η(s, x) ds t

+ TψtT (x)

Zt+

ρ (t − s) ∂s , Jψs TϕTs η(s, x) ds.

t

The first identity shows that the time derivative only applies to terms related to the adjoint of the Piola transform. Since these terms are smooth we infer

∂t , R0 (η) p ≤ c η Lp (I,Vσ (t)) L (I,Vσ (t)) for any test function η. The constant c depends on ρ and the flows ϕ and ψ but is independent of . We introduce yet another smoothing operator corresponding to mollification "forward in time". For η ∈ X we set 0 R+ η(t, x) : = (Pψ ◦ F ◦ Pϕ )(η)(t, x) Zt+ = Jψt Tϕt (ψt (x)) ρ (s − t) Jϕs Tψs (x) η(s, x) ds. t

Since F0 leaves T -traces invariant and only acts in time we see that R+ maps the set of test functions into itself. We now have all the necessary tools, to prove

65

3.4 Density Result

Proposition 3.12 (Boundedness of (∂t u )>0 ) 0 The sequence of time derivatives (∂t u )>0 is uniformly bounded in Lp (I, Vσ (t)) , with a bound only depending on the data. Proof. As X = ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) | ϕ(T, ·) = 0 is dense in Lp (I, Vσ (t)), it is sufficient to estimate the norm of ∂t u on this set of test functions. Furthermore, this allows to invoke the limit equation of the Galerkin approximation which in turn characterizes the generalized time derivative dt u.

For ease of notation we use the bracket ·, · to refer to the duality in Lp (I, Vσ (t)) if it can be expressed by integrals over the non-cylindrical domain Q. Also, we use ·, · to denote the L2 -inner product on Rd . For any test function η ∈ X the transport theorem yields

ˆ 0, η ∂t u , η = ∂t R u, η + ∂t S u

ˆ 0 , η + R u(T ), η(T ) − R u(0), η(0) . = − R u, ∂t η + ∂t S u The temporal boundary terms vanish since R u(0, ·) = η(T, ·) = 0. Thus, by definition of the adjoint operator R0 we see

ˆ 0, η . ∂t u , η = − u, R0 (∂t η) + ∂t S u Using the commutator ∂t , R0 the first term on the right may be rewritten as

− u, R0 (∂t η) = − u, ∂t (R0 η) + u, [∂t , R0 ]η

+ 0 = − u, ∂t (R0 η − R+ η) − u, ∂t (R η) + u, [∂t , R ]η . + Since R+ η is an admissible test function we can substitute hu, ∂t (R η)i by using (3.15), i.e. the limit equation of the Galerkin approximation, which yields

+ 0 ∂t u , η = f , R+ η − χ, R η + u, [∂t , R ]η + κ ∂t (Pψ u0 ), η

(3.21) + − u, ∂t (R0 η − R+ η) + u0 , R η(0) − ρ χ[0,) Pψ u0 , η .

Using the integral representation for the brackets, the first three terms on the right hand side of the last equation can be estimated by a constant multiple of kηkLp (I,Vσ (t)) . The fourth term can similarly be estimated by

κ ∂t (Pψ u0 ), η ≤ c u0

η p . Vσ (0) L (I,Vσ (t)) To see this we essentially have to use the higher regularity for the initial value, i.e. u0 ∈ Vσ (0), as the time derivative induces spatial derivatives due to the chain rule. Thanks to the assumption 2 ≤ p < ∞ we may then apply Hölder’s inequality together with the fact that κ is bounded to get the assertion. In order to treat the remaining terms we recall the representations for R0 and R+ : R0 η(t, x)

=

Jϕt TψtT (x)

Zt+ ρ (s − t) Jψs TϕTs (x) η(s, x) ds. t

R+ η(t, x)

Zt+ = Jψt Tϕt (ψt (x)) ρ (s − t) Jϕs Tψs (x) η(s, x) ds. t

66


Thus, their difference is given by ∂t R0 η

−

∂t R + η

(t, x)

=∂t Jϕt TψtT (x)

Zt+ ρ (s − t) Jψs TϕTs (x) η(s, x) ds t

Zt+ − Jϕt TψtT (x) (∂t ρ (s − t)) Jψs TϕTs (x) η(s, x) ds t

− ∂t Jψt Tϕt (ψt (x))

Zt+ ρ (s − t) Jϕs Tψs (x) η(s, x) ds t

Zt+ − Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) Jϕs Tψs (x) η(s, x) ds. t

Exploiting the smoothness of the transformations, the first and the third term on the righthand side of the last identity can be estimated by a constant multiple of kηkLp (I,Vσ (t)) when tested against u. Notice that ∂t ρ behaves as −2 , so that we need to bound the two remaining terms. Since Jψt Tϕt (ψt (x))Jϕt Tψt (x) = J(ψt ◦ϕt ) T(ϕt ◦ ψt )(x) = id(x) and Jϕt TψtT (x)Jψt TϕTt (x) = J(ψt ◦ϕt ) T(ϕt ◦ ψt )T (x) = id(x), the remaining difference can be rewritten as Zt+ − Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) Jϕs Tψs (x) η(s, x) ds t

Zt+ T − Jϕt Tψt (x) (∂t ρ (s − t)) Jψs TϕTs (x) η(s, x) ds t

Zt+ = − Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) Jϕs Tψs (x) η(s, x) ds t

Zt+ + Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) Jϕt Tψt (x) η(s, x) ds t

Zt+ T − Jϕt Tψt (x) (∂t ρ (s − t)) Jψt TϕTt (x) η(s, x) ds t

Zt+ T − Jϕt Tψt (x) (∂t ρ (s − t)) Jψs TϕTs (x) η(s, x) ds. t

We confine ourselves with estimating the first difference on the right-hand side because the

67

3.4 Density Result

second one may be treated analogously. Thanks to the smoothness of Jϕ Tψ we see that Zt+ Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) Jϕt Tψt (x) η(s, x) ds t

Zt+ − Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) Jϕs Tψs (x) η(s, x) ds t

Zt+ =Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) η(s, x) Jϕt Tψt (x) − Jϕs Tψs (x) ds t

Zt+ Zt d =Jψt Tϕt (ψt (x)) (∂t ρ (s − t)) η(s, x) Jϕτ Tψτ (x) dτ ds. dτ s

t

Since |t − s| ≤ the latter identity yields the estimate Zt+ Zs d ∂t ρ (s − t) η(s, x) Jψτ Tψτ (x) dτ ds Jψt Tϕt (ψt (x)) dτ t

t

Zt+ ≤ c(ψ, ϕ) −1 |η(s, x)| ds ≤ c M τ η (t, x), t

where M τ denotes the temporal maximal function. The continuity of the maximal function, see also Theorem A.6 and Theorem A.7 in the Appendix, and the embedding Lp (I, Vσ (t)) ,→ Lp (Q)d then provide the estimate

η) u, ∂t (R0 η − R+ ≤ c u Lp (I,Vσ (t)) η Lp (I,Vσ (t)) . For the two remaining terms in (3.21) we use the transformation formula to compute

u0 , R+ η(0) − ρ χ[0,) Pψ u0 , η Z Z = u0 (x) ρ (s) Jϕs Tψs (x) η(s, x) dx ds 0 Ω0 Z

Z

−

ρ (s) Jψs Tϕs (ψs (x)) u0 (ψs (x)) η(s, x) dx ds 0 Ω(s)

Z =

Z u0 (x) ρ (s) Jϕs Tψs (x) η(s, x) dx ds

0 Ω0 Z Z

−

ρ (s) Tϕs (x) u0 (x) η(s, ϕs (x)) dx ds 0 Ω0

68


Again, these two integrals can be estimated independently of by applying the same trick we already used above. Plugging all our estimates together we eventually infer

∂t u , η ≤ C. sup sup ∂t u (Lp (I,Vσ (t)))0 = sup >0

>0

η∈X

kηkLp (I,V (t)) ≤1 σ

Here, the constant C only depends on the data.

3.5. Integration by Parts Formula We now have all the estimates to prove an integration by parts formula for the weak limit of the Galerkin approximation. This formula corresponds to an energy identity and is one of the key tools for the identification of χ. Proposition 3.13 (Integration by Parts for Weak Limits) ∗ 0 Suppose f ∈ Lp (I, V (t)) , 2 ≤ p < ∞, χ ∈ Lp (Q)d×d and u0 ∈ Vσ (0). Let u ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) satisfy ZT −

u(t), ∂t η(t) H(t) dt+

0

ZT

χ(t), η(t)

Vσ (t)

dt

0

ZT =

(3.22)

f (t), η(t) Vσ (t) dt + u0 , η(0) H

0

for any test function η ∈ X. Then, after redefining the zero extension in space of u on a ¯ L2 (Rd )d ) and subset of I¯ of measure zero, there holds u ∈ C(I,

(3.23) max u(s) H(s) ≤ C. s∈I¯

¯ s0 < s, u satisfies The constant C only depends on the data. Furthermore, for any s0 , s ∈ I, the energy equality

1

u(s) 2 − H(s) 2

1

u(s0 ) 2 0 = H(s ) 2

Zs

f (t), u(t) Vσ (t) dt −

s0

Zs

χ(t), u(t)

Vσ (t)

dt

(3.24)

s0

¯ s0 < s, the identity (3.24) and the mapping s 7→ ku(s)kH(s) is continuous. For any s0 , s ∈ I, is equivalent to

1

u(s) 2 − H(s) 2

1

u(s0 ) 2 0 = H(s ) 2

Zs

dt u(t), u(t)

Vσ (t)

dt.

(3.25)

s0

Proof. In Proposition 3.10 and Proposition 3.12 we showed that the approximate sequence (u )>0 ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) defined in (3.20) is such that u → u in Lp (I, Vσ (t))

69

3.5 Integration by Parts Formula

0 for → 0 and that the sequence of time derivatives is uniformly bounded in Lp (I, Vσ (t)) , that is,

sup ∂t u (Lp (I,Vσ (t)))0 ≤ C. >0

Proceeding as in Lemma 2.11, we see that any u is continuous as a mapping with values in L2 (Rd )d . We also mentioned in that lemma that any u ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) can be approximated in H 1 (I, H(t)) by smooth fields. As a consequence, we can apply the transport theorem to u − uδ to obtain

Zs Z

∂t (u − uδ ), χ(s0 ,s) (u − uδ ) =

∂t (u − uδ ) · (u − uδ ) dxdt s0

=

Ω(t)

Zs

∂t u (t) − uδ (t) , u (t) − uδ (t)

H(t)

dt

s0

1 = 2

Zs

(3.26)

d

u (t) − uδ (t) 2 dt H(t) dt

s0

2

2 1 1 = u (s) − uδ (s) H(s) − u (s0 ) − uδ (s0 ) H(s0 ) . 2 2 The assumption 2 ≤ p < ∞ trivially implies ¯ Vσ (s) ⊂ V (s) ,→ H(s) for every s ∈ I. The constants of embedding can be shown to be independent of s. This is due to the fact that they depend on diam Ω(s) by virtue of Poincaré’s inequality. However, this quantity is globally bounded due to (2.2). Thus, we can rearrange the terms in (3.26) and integrate with respect to s0 to infer

u (s)−uδ (s) 2 d = u (s) − uδ (s) L (R ) H(s)

≤ c u − uδ Lp (I,Vσ (t)) + ∂t (u − uδ ) (Lp (I,Vσ (t)))0 u − uδ Lp (I,Vσ (t)) . ¯ L2 (Rd )d ) and therefore The latter estimate shows that (u )>0 is a Cauchy sequence in C(I, ¯ L2 (Rd )d ). However, the strong convergence lim→0 u = u in ˜ ∈ C(I, converges to some u p 2 ˜ coincides with the zero extension of u. Thus, we L (I, Vσ (t)) ,→ L (I, H(t)) implies that u 2 d d ¯ have shown that u ∈ C(I, L (R ) ). Passing to the limit in the estimate

u (s) ≤ c u Lp (I,Vσ (t)) H(s) we deduce (3.23). 0 Since Lp (I, Vσ (t)) is a reflexive Banach space, see Corollary 2.9, its dual space Lp (I, Vσ (t)) is reflexive, too. By weak compactness of bounded sets in reflexive Banach spaces, the bound 0 on the sequence of time derivatives yields the existence of some functional L ∈ Lp (I, Vσ (t)) such that 0 ∂t u * L in Lp (I, Vσ (t)) ,

70


at least for a not relabeled subsequence → 0. As tends to zero we obtain

L, η ←− ∂t u , η = −

Z u (t), ∂t η(t)

Z

H(t)

dt −→ −

I

u(t), ∂t η(t) H(t) dt

I

for any η in the dense subspace ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) | ϕ(0, ·) = ϕ(T, ·) = 0 of Lp (I, Vσ (t)). On the other hand, u satisfies the identity (3.22) which in turn implies Z −

u(t), ∂t η(t) H(t) dt = Lf ,−χ , η

I

for any η ∈ deduce

ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) | ϕ(0, ·) = ϕ(T, ·) = 0 ∗ dt u = Lf ,−χ in Lp (I, Vσ (t)) .

and especially we (3.27)

Upon noticing that

lim ∂t u , χ(s0 ,s) u = Lf ,−χ , χ(s0 ,s) u

→0

we can let tend to zero in the identity

2

1 1

u (s) 2 − u (s0 ) H(s0 ) = ∂t u , χ(s0 ,s) u H(s) 2 2 to deduce the energy equality (3.24). As the integrands on the right-hand side of (3.24) are in L1 (I) we know that ku(s)kH(s) tends to ku(s0 )kH(s0 ) for s → s0 . Thus, the mapping s 7→ ku(s)kH(s) is continuous. By (3.27), the identities (3.24) and (3.25) are equivalent. Remark 3.14. Note that the results of Proposition 3.13 are weaker than the results of Lemma 3.9 where analogue assertions are summarized for the cylindrical case. In fact, the 0 ¯ H), where (V, H, V 0 ) is an arbitrary Gelfand continuous embedding W 1,p,p (V, V 0 ) ,→ C(I, triple, cannot be recovered directly in Proposition 3.13 but is substituted by an embedding ¯ L2 (Rd )d ) together with the well-defined evaluation t 7→ u(t) which is to be interinto C(I, S preted as a mapping from I¯ to t∈I¯ H(t). At first sight one would expect an embedding ¯ Hσ (Rd )) together with a well-defined evaluation I¯ → S ¯ Hσ (t), t 7→ u(t). This into C(I, t∈I apparent discrepancy can be perceived as follows: While Lemma 3.9 is essentially based ¯ V ) ,→ W 1,p,p0 (V, V 0 ), the sequence (u )>0 we had on the density of the embedding C ∞ (I, constructed only lives in H 1 (I, H(t)) ∩ Lp (I, Vσ (t)). That is, the evaluation of the time derivative ∂t u does not lie in Vσ (t) but merely in H(t) = L2 (Ω(t))d . The very root for this circumstance is the (time-dependent) Piola transform that is involved in the approximation process. Therefore, we must implicitly use the Gelfand triples V (t), H(t), V (t)0 t∈I¯ instead of the natural underlying Gelfand triples Vσ (t), Hσ (t), Vσ (t)0 t∈I¯. This distinction can be abandoned for problems without incompressibility constraint.

71

3.5 Integration by Parts Formula

Corollary 3.15 The weak limit u ∈ Lp (I, Vσ (t)) satisfies u(s), η(s)

− u(0), η(0) H(s)

Zs

H

=

f (t), η(t) Vσ (t) dt −

0

Zs

χ(t), η(t) Vσ (t) dt

0

Zs +

u(t), ∂t η(t)

H(t)

(3.28)

dt

0

for every s ∈ I¯ and every η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)). Proof. Let (u )>0 again denote the approximate sequence we used in Proposition 3.13. Then we can pass to the limit in the identity10

∂t u , χ[0,s] η

Lp (I,Vσ (t))

+ u , χ[0,s] ∂t η

L2 (Q)

= u (s), η(s)

H(s)

− u (0), η(0)

H

due to Proposition 3.13 to get the result. Notice that the identity (3.28) contains the evaluation of u at 0. By the above arguments we know that this quantity is well-defined in H(0) = H but we now prove that u(0) = u0 ¯ we use the test actually holds in Hσ (0). Choosing an arbitrary ξ T ∈ Vσ (T ) and κ ∈ C 1 (I) function κ ξ := κ Pψ PϕT ξ T

to infer u(T ), κ(T ) ξ T

H(T )

− u(0), κ(0) ξ(0)

H

= u∗ , κ(T ) ξ T

H(T )

− u0 , κ(0) ξ(0) H

from the identities (3.28) and (3.18). For κ with κ(0) = 0 and κ(T ) = 1 we deduce u(T ) − u∗ , ξ T

H(T )

=0

for any ξ T in the dense subspace Vσ (T ) ,→ Hσ (T ). Thus, u(T ) = u∗ in Hσ (T ). Similar reasoning then shows the identity u(0) = u0 in Hσ (0). If we furthermore compare (3.28) and the localized identity (3.16) from Lemma 3.6 we get u(0) − u0 , η(0) for any η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)). 10

This identity follows from the transport theorem.

H

=0

72


3.6. Minty’s Trick 0

Now we can finally identify the weak limit χ ∈ Lp (Q)d×d by using a variant of Minty’s trick. For convenience we cited a basic version of Minty’s trick in Lemma A.25. Proposition 3.16 (Identification of χ) Let u ∈ Lp (I, Vσ (t) denote the weak limit of the sequence of Galerkin solutions (un )n∈N . Then there holds 0

χ = Au = |∇u|p−2 ∇u in Lp (Q)d×d Proof. Recall that the sequence of Galerkin solutions was characterized by solving

∂t un (t), wk (t) H(t) + A(un (t)), wk (t) Vσ (t) = f (t), wk (t) Vσ (t) un (0) = u0n . We test the system with a Galerkin solution un itself and integrate from 0 to T with respect to t to obtain Z Z

2

2

1 1 A(un (t)), un (t) Vσ (t) dt = f (t), un (t) Vσ (t) dt − un (T ) H(T ) + u0n H . 2 2 I

I

By the weak convergence properties of the sequence (un )n∈N , see (3.14), and weak lower semicontinuity of the norm we get Z Z

2

1 2 1 A(un (t)), un (t) Vσ (t) dt ≤ f (t), u(t) Vσ (t) dt − u(T ) H(T ) + u0 H . lim sup 2 2 n→∞ I

I

Owing to the energy equality (3.24), we can substitute the right-hand side of the latter inequality to obtain Z Z

lim sup A(un (t)), un (t) Vσ (t) dt ≤ χ(t), u(t) Vσ (t) dt. (3.29) n→∞

I

I

This inequality enables us to exploit the monotonicity of A by means of Minty’s trick. In fact, for any η ∈ Lp (I, Vσ (t)) we have Z

0≤ A(un (t)) − A(η(t)), un (t) − η(t) Vσ (t) dt = Aun − Aη, un − η Lp (I,Vσ (t)) I

from which we deduce

0 ≤ χ − Aη, u − η Lp (I,Vσ (t)) for any η ∈ Lp (I, Vσ (t)) by (3.29). Choosing η = u ± ξ, ξ ∈ Lp (I, Vσ (t)), > 0, and taking into account the hemicontinuity of A as → 0 we infer

0 ≤ χ − Au, ξ Lp (I,Vσ (t))

73

3.6 Minty’s Trick

From the last estimate we finally obtain p−2 0 χ = Au = ∇u ∇u in Lp (Q)d×d .

Corollary 3.17 As 2 ≤ p < ∞, the p-Laplacian operator A is not only monotone but even strongly monotone in the sense that

p

A(u(t)) − A(η(t)), u(t) − η(t) Vσ (t) ≥ c u(t) − η(t) Vσ (t) . (3.30) This property yields an alternative possibility of identifying χ. In fact, using (3.29) and the strong monotonicity we infer Z

p

A(un (t)) − A(u(t)), un (t) − u(t) Vσ (t) dt = 0. lim un − u Lp (I,Vσ (t)) ≤ c lim sup n→∞

n→∞

I

Thus, the sequence of Galerkin solutions (un )n∈N converges to u strongly in Lp (I, Vσ (t)). Passing to a subsequence if necessary, we obtain that almost everywhere in Q there holds lim ∇un = ∇u.

n→∞

This in turn implies p−2 p−2 lim ∇un ∇un = ∇u ∇u

n→∞

almost everywhere in Q. Together with the fact that (Aun )n∈N weakly converges to χ in 0 Lp (Q)d×d we may conclude p−2 χ = Au = ∇u ∇u because weak and pointwise limits coincide by Lemma A.1. For later purposes, we want to mention a third possibility to establish the identification of the weak limit χ under weaker assumptions on A. This result is due to Dal Maso and Murat and can be found in [DMM98]. Lemma 3.18 Suppose that A : Rd×d → Rd×d is continuous and strictly monotone in the sense that A(M) − A(N) : M − N > 0 for M, N ∈ Rd×d , M 6= N. If (Mn )n∈N ⊂ Rd×d is a sequence such that lim A(Mn ) − A(M) : Mn − M = 0 n→∞

holds for some M ∈ Rd×d , then it follows that lim Mn = M and lim A(Mn ) = A(M).

n→∞

n→∞

74


We summarize the result of this chapter in Theorem 3.19 (Existence and Uniqueness) ∗ For any f ∈ Lp (I, Vσ (t)) , 2 ≤ p < ∞, and u0 ∈ Vσ (0) there exists a unique weak solution u ∈ Lp (I, Vσ (t)) ∩ L∞ (I, Hσ (t)) of the non-cylindrical p-Stokes problem (3.9) in the sense that ZT −

u(t), ∂t η(t) H(t) dt+

0

ZT

A(u(t)), η(t)

Vσ (t)

dt

0

ZT =

f (t), η(t) Vσ (t) dt − u0 , η(0) H

0

holds for any test function η ∈ X = ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) ϕ(T, ·) = 0 . ¯ L2 (Rd )d ) and the evaluation Moreover, the zero extension (in space) of u lies in C(I, u : I¯ →

[

H(t), t 7→ u(t),

t∈I¯

is well-defined. For every s ∈ I¯ the field u satisfies the energy equality

1 1 2

u(s) 2 − u0 H = H(s) 2 2

Zs

f (t), u(t) Vσ (t) dt −

0

Zs

A(u(t)), u(t) Vσ (t) dt.

0

Proof. The existence of at least one weak solution with the above properties is already proved. In order to prove uniqueness of the weak solution we show that it continuously depends on the data. To this end, suppose that u1 , u2 ∈ Lp (I, Vσ (t)) ∩ L∞ (I, Hσ (t)) are two ∗ weak solutions of 2 p 1 the problems corresponding to the data (f1 , u0 ), (f1 , u0 ) ∈ L (I, V (t)) × Vσ (0). It follows that u1 − u2 satisfies u1 (s) − u2 (s), η(s) H(s) − u10 − u20 , η(0) H +

Zs

u1 (t) − u2 (t), ∂t η(t) H(t) dt

0

Zs = 0

f1 (t) − f2 (t), η(t) Vσ (t) dt −

Zs

A(u1 (t)) − A(u2 (t)), η(t) Vσ (t) dt

0

∗ for any s ∈ I¯ and any η ∈ X. However, the latter identity yields dt (u1 −u2 ) ∈ Lp (I, Vσ (t)) so that there is a sequence (u1 − u2 ) >0 ⊂ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) such that for → 0 (u1 − u2 ) → u1 − u2 in Lp (I, Vσ (t)), 0 ∂t (u1 − u2 ) * dt (u1 − u2 ) in Lp (I, Vσ (t))

75

3.6 Minty’s Trick

Hence, passing to the limit → 0 in the duality ∂t (u1 − u2 ) , (u1 − u2 ) χ[0,s] Lp (I,Vσ (t)) we infer Zs

2

1

u1 (s) − u2 (s) + A(u1 (t)) − A(u2 (t)), u1 (t) − u2 (t) Vσ (t) dt H(s) 2 0

2 1 = u10 − u20 H + 2

Zs

f1 (t) − f2 (t), u1 (t) − u2 (t)

Vσ (t)

dt.

0

By Hölder’s inequality and the strong monotonicity of A, see (3.30), we obtain the estimate

p

u1 (s) − u2 (s) 2 + χ(0,s) (u1 − u2 ) Lp (I,Vσ (t)) H(s)

2

p0 ≤ c u10 − u20 Hσ (0) + f1 − f2 (Lp (I,V (t)))∗ . Since s ∈ I¯ was arbitrary it follows that

p

2

max u1 (s) − u2 (s) H(s) + u1 − u2 Lp (I,Vσ (t)) s∈I¯

2

p0 ≤ c u10 − u20 H + f1 − f2 (Lp (I,V (t)))∗ . The results from this chapter may be generalized to problems that fall into a similar functional framework as the one determined by the non-cylindrical homogeneous p-Stokes problem we had considered. The key results of this chapter were the integration by parts formula and the necessary density result for incompressible vector fields having a generalized time derivative in the dual of the natural energy space. The following synoptic result serves as a non-cylindrical analogue of Lemma 3.9. Theorem 3.20 (Integration by Parts in Generalized Bochner Spaces) 1,p (Ω(t))d . Set V (t) := W01,p (Ω(t))d , t ∈ I¯ and 2 ≤ p < ∞, and denote by Vσ (t) the spaces W0,σ ¯ stand for L2 (Ω(t))d and Hσ (t) for the closure of C ∞ (Ω(t))d Similarly, let H(t), t ∈ I, 0,σ in H(t), respectively. It follows that V (t), H(t), V (t)0 t∈I¯ and Vσ (t), Hσ (t), Vσ (t)0 t∈I¯ are 0 families of Gelfand triples. Let furthermore W 1,p,p (Vσ (t), Vσ (t)0 ) be defined by n ∗ o 0 W 1,p,p (Vσ (t), Vσ (t)0 ) := v ∈ Lp (I, Vσ (t)) dt v ∈ Lp (I, Vσ (t)) . 0

Suppose u ∈ W 1,p,p (Vσ (t), Vσ (t)0 ) ∩ L∞ (I, Hσ (t)) satisfies Z Z

− u(t), ∂t η(t) H(t) dt = dt u(t), η(t) Vσ (t) dt + u0 , η(0) H I

I

for some u0 ∈ Vσ (0) and for every η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) with η(T ) = 0. Then, there exists a sequence (u )>0 ⊂ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) such that, as tends to zero, u → u in Lp (I, Vσ (t)), 0 ∂t u * dt u in Lp (I, Vσ (t)) .

76


Moreover, any such u can be identified with a continuous function with values in L2 (Rd )d , S ¯ the pointwise evaluation t 7→ u(t) is well-defined as a mapping from S I to t∈I¯ H(t) and u(0) = u0 holds in Hσ (0). Moreover, the evaluation u : I → t∈I¯ Hσ (t), t 7→ u(t) is well-defined for almost every t ∈ I. Eventually, u satisfies the integration by parts formula

2 1 1

u(s) 2 − u(s0 ) H(s0 ) = H(s) 2 2

Zs

dt u(t), u(t) Vσ (t) dt

s0

¯ s0 < s, thereby showing that the function t 7→ ku(t)kH(t) is continuous. for any s0 , s ∈ I, 0

Proof. Assuming that u ∈ W 1,p,p (Vσ (t), Vσ (t)0 ) ∩ L∞ (I, Hσ (t)) satisfies Z Z

− u(t), ∂t η(t) H(t) dt = dt u(t), η(t) Vσ (t) dt + u0 , η(0) H I

I

for some u0 ∈ Vσ (0) any for every η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) with η(T ) = 0, we infer that u meets the requirements of Proposition 3.10. In fact, in Proposition 3.10 the field under consideration was supposed to satisfy the related equation (3.15), i.e. the limit equation of a Galerkin approximation scheme. Therefore, it is straightforward to successively adapt the arguments of Proposition 3.10, Proposition 3.12 and Proposition 3.13. Remark 3.21. If the functional framework for the problem at hand is determined by the single family of Gelfand triples V (t), H(t), V (t)0 t∈I¯, the subscript σ referring to incompressibility can be dropped in Theorem 3.20 but the results still remain valid with obvious modifications. Of course, the assertions of Theorem 3.20 are not restricted to vector fields. The scalar case follows along the lines if the Piola transform is substituted for the pull-back of functions induced by the flow (ϕt )t∈I¯.

4. Unsteady Problems with Compact Perturbations In the previous chapter we established an abstract integration by parts formula for functions in generalized Bochner spaces. This result allows to treat monotone, nonlinear unsteady problems on non-cylindrical domains that do not contain nonlinear lower order terms. However, to successfully apply this abstract method to relevant physical examples we have to supplement it by compactness methods that are particularly necessary with regard to lower order nonlinear perturbations. A classical example illustrating the necessity of compactness results is the Navier–Stokes equation with its nonlinear convective term. The compactness method we develop in this chapter will allow us to prove existence of weak solutions for the non-cylindrical Navier– Stokes system ∂t u + (u · ∇)u − ∆u + ∇π = f

in Q,

div u = 0

in Q,

u=v

on Γ,

u(0) = u0

in Ω0 .

But before we come to this application we will present in detail a seemingly little-known compactness result of Landes and Mustonen. We extend their result and apply it to noncylindrical nonlinear problems of the form ∂t u + Au + G(u) = f

on Q

u=0

on Γ,

u(0) = u0

(4.1)

in Ω0 .

Here, G(u) denotes a nonlinear function of u which will be treated as a compact perturbation of the monotone operator A. As to the functional setting, we impose the following set of hypotheses: • The non-cylindrical domain Q ⊂ Rd+1 corresponds to the tube generated by the flow (ϕt )t∈I (with inverse (ψt )t∈I ) of a smooth, time-dependent, complete vector field v : R × Rd → Rd , i.e. [ [ Q = {t} × Ω(t) := {t} × ϕt (Ω0 ). t∈I

t∈I

Here I = (0, T ) denotes a time interval and Ω0 ⊂ Rd is a bounded domain with Lipschitz boundary. The lateral boundary is similarly given by [ [ Γ := {t} × Γ(t) := {t} × ϕt (∂Ω0 ). t∈I

t∈I

77

78

Chapter 4. Unsteady Problems with Compact Perturbations

¯ which is • We suppose that A is induced by an operator A : V (t) → V (t)0 , t ∈ I, ¯ i.e. we suppose that there is an exponent uniformly, strongly monotone for any t ∈ I, ¯ 2 ≤ p < ∞ such that for any t ∈ I and for any u(t), v(t) ∈ V (t) there holds

p

A(u(t)) − A(v(t)), u(t) − v(t) V (t) ≥ c u(t) − v(t) V (t) (4.2) with a uniform constant c > 0 that does not depend on time. • We further assume that A has the following growth and coercivity properties: There 0 exists a function h1 ∈ Lp (I) and a constant c1 > 0 such that

A(u(t))

u(t) p−1 ≤ c h (t) + (4.3) 1 1 0 V (t) V (t) holds for almost every t ∈ I. Also, we assume existence of a function h2 ∈ L1 (I) and a constant c2 > 0 such that

p

A(u(t)), u(t) V (t) ≥ c2 u(t) V (t) − h2 (t) (4.4) holds for almost every t ∈ I. In the previous assumptions V (t) denotes for any t ∈ I the Sobolev space W01,p (Ω(t))d . The exponent 2 ≤ p < ∞ is the same as in the structural conditions (4.2), (4.3) and (4.4). Denoting by H(t) the space L2 (Ω(t))d we see that V (t), H(t), V (t)0 t∈I¯ is a family of Gelfand triples. Following the results of Lemma 2.4 in Chapter 2, the Piola transform with respect to ϕt induces isomorphisms between the spaces V (t) and V := W01,p (Ω0 )d and H(t) and H = L2 (Ω0 )d , respectively. The inverse isomorphisms are given by Pψt . Moreover, we also have isomorphisms between the generalized Bochner spaces and their classical counterparts: Pϕ : Lp (I, V (t)) → Lp (I, V ), Pϕ : L2 (I, H(t)) → L2 (I, H), Pϕ u (t) := Pϕt (u(t)). The inverse is given by Pϕ−1 = Pϕ−1 = Pψ and is defined analogously. • We furthermore assume that the induced operator 0 A : Lp (I, V (t)) → Lp (I, V (t)) Au)(t) := A(u(t)) is well-defined and hemicontinuous. Concerning the nonlinearity G we assume the following: • G is a Nemyckii operator induced by a vector field g : Q × Rd → Rd which satisfies: • Carathéodory condition: g(·, ·, ζ) : (t, x)

7→ g(t, x, ζ) is measurable on Q for every ζ ∈ Rd ,

g(t, x, ·) :

7→ g(t, x, ζ) is continuous on Rd for any (t, x) ∈ Q.

ζ

(4.5)

4.1 The Compactness Principle of Landes and Mustonen

79

• Growth condition: There exists a real number r > 1 and a positive constant c such that for any (t, x, ζ) ∈ Q × Rd there holds g(t, x, ζ) ≤ c 1 + ζ r−1 .

(4.6)

• Sign condition: For every (t, x) ∈ Q there holds inf g(t, x, ζ) · ζ ≥ 0.

ζ∈Rd

(4.7)

The admissible range of the growth parameter r typically depends on the functional setting which in turn is determined by A. We are going to specify the definite range of r below. The sign condition is a common assumption that assures the validity of standard a priori estimates. However, that sign condition may surely be replaced by different conditions as long as they guarantee a priori estimates.

4.1. The Compactness Principle of Landes and Mustonen In the cylindrical case, existence of weak solutions for problems of type (4.1) can be established by pursuing the following approach. In a first step, the problem is approximated by a Galerkin scheme. The boundedness of the sequence of Galerkin solutions in the energy norm associated to the equation can be combined with a bound on their respective time derivatives. Then one can pass to the limit in the nonlinear term induced by G by means of the Aubin–Lions lemma. In the non-cylindrical setting however, this procedure is impracticable for mainly two reasons we would like to work out first. Since the domain is time-dependent one has to use time-dependent ansatz functions in order to construct a suitable Galerkin basis, see also the respective section in Chapter 3. Now let Pn (t) denote the projection onto the n-dimensional subspace of V (t) ,→ H(t) spanned by the first n ansatz functions { wk (t) | 1 ≤ k ≤ n }. It is readily checked that the relation ∂t un , ϕ = ∂t (Pn un ), ϕ = Pn (∂t un ), ϕ = ∂t un , Pn ϕ

(4.8)

which usually establishes the link between the sequence of time derivatives and the Galerkin system cannot be transferred if the projections Pn are replaced by time-dependent projections Pn (t). Moreover, the third equality in (4.8) does not even hold in general. The second point concerns the orthogonality of the ansatz functions. Uniform boundedness of the projections is usually established by using an orthogonal set of ansatz functions. However, this orthogonality relation cannot be maintained: If W k , W l ∈ H are orthogonal vector fields then Pψt W k , Pψt W l ∈ H(t) are not orthogonal in general if the Piola transform Pψt is induced by an arbitrary diffeomorphism ψt . The consequence of these two points is that we cannot derive uniform estimates for the time derivatives of Galerkin solutions in general. This circumstance complicates the possibility of obtaining strong compactness of Galerkin solutions. Instead of using a Galerkin scheme, our approximation will therefore consist of truncating and additionally linearizing G. To be more specific, the approximate problem is to find a weak solution u (in a sense to be

80


specified below) of ∂t u + Au + Gk (ξ) = f

on Q

u=0

on Γ,

u(0) = u0

(4.9)

in Ω0 ,

where ξ denotes some suitable linearization and k ∈ N denotes the level of truncation. As it turns out, these approximate problems can be handled by using the techniques of the previous chapter. Hence, the solution operators Lk : ξ 7→ u are well-defined and we will show that they admit fixed points (uk )k∈N . The existence of fixed points will follow from Schauder’s fixed point theorem. The necessary compactness of Lk in turn will be a consequence of a generalization of the Landes–Mustonen compactness principle. This technique seems to be new in this context. Before we treat problem (4.9) we present the proof of Landes’ and Mustonen’s theorem for the sake of completeness. Theorem 4.1 (Landes–Mustonen Compactness Principle, [LM87]) Let U ⊂ Rd be a bounded domain with smooth boundary and let I ⊂ R be a bounded time interval. Suppose that (uk )k∈N is a bounded sequence in Lp (I, W0m,p (U )d ) ∩ L∞ (I, L1 (U )d ), 1 < p < ∞, m ∈ N and m ≥ 1. By reflexivity of Lp (I, W0m,p (U )d ), there exists u ∈ Lp (I, W0m,p (U )d ) such that uk * u in Lp (I, W0m,p (U )d ), at least for a not relabeled subsequence k → ∞. If (uk )k∈N additionally satisfies for almost every s ∈ I uk (s) * u(s) in L1 (U )d as k → ∞, then lim uk = u in Lp (I, W0m−1,p (U )d ).

k→∞

Remark 4.2. This theorem differs from the Aubin–Lions lemma in that the usual bounded1 d k ness of ( du dt )k∈N in some distributional space is replaced by the weak convergence in L (U ) for almost every s ∈ I. Since L1 is not reflexive, weak convergence in L1 does not follow from the boundedness of (uk )k∈N but has to be required explicitly. However, for "standard" parabolic problems, all the assumptions on (uk )k∈N usually follow from a priori estimates, even on the Galerkin level. Especially in the cylindrical setting, this principle is therefore quite handy and may often be used instead of the Aubin–Lions lemma. In the non-cylindrical setting, the verification of the above assumptions is intimately connected with the results of Proposition 3.13 as we will see below. Proof. Let B1 (0) denote the unit ball in Rd . For ω ∈ C0∞ (B1 (0)) we define a sequence of mollifiers by ω (x) := −d ω(x/). For any u ∈ L1 (I × U )d we set Z u (s, x) := (u(s) ∗ ω )(x) = ω (x − y) u(s, y) dy Rd

81

4.1 The Compactness Principle of Landes and Mustonen

with the convention that any such function u is extended by zero outside of U . Also, we set uk, := (uk ) , k ∈ N, > 0. As k tends to infinity, we get for almost every (s, x) ∈ I × U and fixed > 0 Z u (s, x) − uk, (s, x) = ω (x − y) u(s, y) − uk (s, y) dy −→ 0. | {z } | {z } Rd

∈L∞ (Rd )

*0 in L1 (R)d

This in turn implies for k → ∞ (4.10)

uk, (s) − u (s) 1 →0 L (U )

for almost s ∈ I by dominated convergence. Moreover, by [DHHR11, Lemma 8.4.1] for any fixed k, any > 0 and for almost every s ∈ I there holds

uk (s) m,p .

uk, (s) − uk (s) m−1,p (4.11) ≤ (U ) (U ) W W 0

0

Due to the embeddings |α|,p

W0m,p (U )d ,→,→ W0

i

(U )d ,→ L1 (U )d

which are valid for any multiindex 0 ≤ |α| ≤ m − 1, Ehrling’s lemma, see also Lemma 5.2 in the next chapter, implies that for any δ > 0 there exists a constant cδ such that

α

m

∇ uk (s) p

∇ uk (s) p

uk (s) 1 (4.12) ≤ δ + c δ L (U ) L (U ) L (U ) holds for any uk (s) ∈ W0m,p (U )d . We will now show that ∇α uk is a Cauchy sequence in Lp (I, Lp (U )d ) for any 0 ≤ |α| ≤ m−1 which yields the assertion. The interpolation inequality (4.12) and the estimate (4.11) show that

α

∇ uk (s) − ∇α ul (s) p L (U )

≤ ∇α uk (s) − ∇α uk, (s) Lp (U ) + ∇α uk, (s) − ∇α ul, (s) Lp (U )

+ ∇α ul, (s) − ∇α ul (s) Lp (U )

≤ ∇m uk (s) Lp (U ) + ∇m ul (s) Lp (U ) + δ ∇m uk, (s) − ∇m ul, (s) Lp (U )

+ cδ uk, (s) − ul, (s) L1 (U ) . Integrating the latter inequality with respect to s we obtain

α

∇ uk − ∇α ul p p p L (I,L (U )) Z

α

p = ∇ uk (s) − ∇α ul (s) Lp (U ) ds I p

Z

m

∇ uk (s) p p ds + c p L (U )

≤c

I

+ cδ

Z

m

∇ ul (s) p p ds L (U )

I p

Z I

m

∇ uk, (s) − ∇m ul, (s) p p ds + cp δ L (U )

Z I

uk, (s) − ul, (s) p 1 ds, L (U )

(4.13)

82


where the constant c depends on p. Now let η > 0 be arbitrary. We can first fix > 0 such that Z Z

m

m

p p p

∇ ul (s) p p ds ≤ η ,

∇ uk (s) Lp (U ) ds + c c L (U ) 3 I

I

since by assumption, (uk )k∈N is bounded in Lp (I, W0m,p (U )d ). In the next step we choose δ > 0 in such a way that δp

Z

m

∇ uk, (s) − ∇m ul, (s) p p ds ≤ η . L (U ) 3

I

Concerning the last term in (4.13), notice that for any > 0 and any η > 0 there exists N (, η) such that k, l ≥ N (, η) implies that

η

uk, − ul, p 1 ≤ . L (I,L (U )) 3 In fact, it follows from (4.10) that for almost every s ∈ I we have

uk, (s) − ul, (s) 1 ≤ uk, (s) − u (s) L1 (U ) + u (s) − ul, (s) L1 (U ) → 0 L (U ) as k, l tend to infinity. Thus we may again conclude with Lebesgue’s theorem since for almost every s ∈ I there holds

uk ∞ 1

uk, (s) − ul, (s) 1 ≤ 2 c() sup L (I,L (U )) L (U ) k∈N

and thus

I 3 s 7→ uk, (s) − ul, (s) L1 (U ) ∈ Lp (I).

At next, we give two interpolation results that will be used for fixing the range of the growth parameter of the nonlinearity. Lemma 4.3 (Interpolation inequality) Suppose 1 ≤ p < d and q > 2, then there holds Lp (I, Lq (Ω(t))d ) ∩ L∞ (I, L2 (Ω(t))d ) ,→ Lα (I, Lβ (Ω(t))d ) for any p < α < ∞ and 2 < β < q that fulfill α≤ p

βq − 2β . βq − 2q

4.2 Existence of Solutions for the Truncated Problems

83

Proof. For u ∈ Lp (I, Lq (Ω(t))d ) ∩ L∞ (I, L2 (Ω(t))d ) we can interpolate on almost every time slice to obtain

u(t)

Lβ (Ω(t))

λ

1−λ ≤ u(t) Lq (Ω(t)) u(t) L2 (Ω(t))

for β1 = λq + 1−λ 2 , 0 < λ < 1. Then we raise the latter inequality to the power α and integrate with respect to t to get Z I

α(1−λ)

u(t) αβ dt ≤ u L∞ (I,L2 (Ω(t))) L (Ω(t))

Z I

u(t) λαq dt. L (Ω(t))

As we must require αλ ≤ p the assertion follows. Corollary 4.4 (Optimal Embedding) For the spaces V (t) and H(t) and every 1 ≤ α ≤ p d+2 d there holds Lp (I, V (t)) ∩ L∞ (I, H(t)) ,→ Lα (Q)d . Proof. By Sobolev embedding we get Lp (I, V (t)) ∩ L∞ (I, H(t)) ,→ Lp (I, Lq (Ω(t))d ) ∩ L∞ (I, H(t)) for every 1 ≤ q ≤

dp d−p .

If we require α = β in Lemma (4.3) we obtain the result.

Let 2 ≤ p < ∞ denote the coercivity parameter of A. We now fix the growth parameter of g to r< p

d+2 . d

The hypotheses we imposed on the structural conditions of A formally suggest existence of weak solutions of problem (4.1) in L∞ (I, H(t)) ∩ Lp (I, V (t)) ,→ Lα (Q)d , α ≤ p d+2 d . However, the reason for slightly lowering the optimal bound on the growth parameter r will become evident in the compactness argument we are going to apply.

4.2. Existence of Solutions for the Truncated Problems The L∞ -approximation of g is defined as follows: For any k ∈ N, any t ∈ I¯ and any ξ(t) ∈ Lr (Ω(t))d we set  g(ξ(t)) gk (ξ(t)) := g(ξ(t)) k |g(ξ(t))| We summarize the properties of gk in

if |g(ξ(t))| < k, if |g(ξ(t))| ≥ k.

(4.14)

84


Lemma 4.5 d there holds g (ξ(t)) ∈ L∞ (Ω(t))d ,→ For any k ∈ N, any t ∈ I¯ and every ξ(t) ∈ Lr (Ω(t)) k 0 r−1 V (t) and |gk (ξ(t))| ≤ |g(ξ(t))| ≤ c 1 + |ξ(t)| . The mapping 0

gk : Lr (Ω(t))d → Lr (Ω(t))d , ξ(t) 7→ gk (ξ(t)), is bounded and continuous and extends to a continuous mapping 0

Gk : Lr (Q)d → Lr (Q)d ,

ZT

Gk (ξ), η =

gk (ξ(t)), η(t) Lr (Ω(t)) dt,

ξ, η ∈ Lr (Q)d .

0

For every k ∈ N and any ξ ∈ Lr (Q)d , the functionals G(ξ) and Gk (ξ) lie in the dual space of L∞ (I, H(t)) ∩ Lp (I, V (t)). Moreover, G induces a bounded mapping 0 G : L∞ (I, H(t)) ∩ Lp (I, V (t)) → Lp (I, V (t)) Z

G(u), η := g(u(t)), η(t) V (t) dt. I

Proof. The first two points are obvious consequences of the definition of gk and the growth 0 condition (4.6). The fact that g : Lr (Ω(t))d → Lr (Ω(t))d is a Nemyckii operator implies the 0 continuity of gk : Lr (Ω(t))d → Lr (Ω(t))d . The properties of Gk follow from these assertions. The next point is clear from the embedding L∞ (I, H(t)) ∩ Lp (I, V (t)) ,→ Lr (Q)d which holds by virtue of Corollary 4.4. The last assertion is proved as follows: By Sobolev dp

embedding we have Lp (I, V (t)) ,→ Lp (I, L d−p (Ω(t))d ). Using (4.6) together with Hölder’s inequality we have for u ∈ L∞ (I, H(t)) ∩ Lp (I, V (t)), η ∈ Lp (I, V (t)) and r0 := p d+2 d ZT

g(u(t)), η(t) V (t) dt G(u), η = 0

≤ c

ZT

r −1 1 + u(t) 0 (r0 −1)dp L d(p−1)+p (Ω(t))

0

r −1 ≤ c 1 + u 0

η(t) dt V (t)

(r0 −1)dp 0 L(r0 −1)p (I,L d(p−1)+p (Ω(t))

η p L (I,V (t))

0 Since r0 = p d+2 d , we can now use Lemma 4.3 with α = (r0 − 1)p , β =

q=

(r0 −1)dp d(p−1)+p

dp d−p .

We now formulate the existence result for the linearized, truncated problem.

and

85

4.2 Existence of Solutions for the Truncated Problems

Lemma 4.6 (Linearized, Truncated Problem: Existence and Uniqueness) ∗ For every fixed k ∈ N and for every u0 ∈ V , f ∈ Lp (I, V (t)) , 2 ≤ p < ∞, and ξ ∈ Lr (Q)d there exists a unique weak solution u ∈ L∞ (I, H(t)) ∩ Lp (I, V (t)) of problem (4.9) in the sense that ZT −

ZT

u(t), ∂t η(t) H(t) dt +

0

A(u(t)), η(t) V (t) dt

0

ZT =

f (t), η(t)

ZT

dt − V (t)

0

gk (ξ(t)), η(t) V (t) dt − u0 , η(0) H

0

holds for every η ∈ X = η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)) η(T, ·) = 0 , see also Lemma 2.11. The weak solution satisfies the energy estimate

max u(t) H(t) + u Lp (I,V (t)) ≤ c(k). (4.15)

t∈I¯

The constant c(k) depends on the data and the truncation parameter k. The zero extension (in space) of u is continuous as a mapping u : I¯ → L2 (Rd )d , its pointwise evaluation [ u : I¯ → H(s), s 7→ u(s), s∈I¯

¯ u satisfies the is well-defined, u(0) = u0 holds as an identity in H and for every s ∈ I, energy equality

1

u(s) 2 − 1 u0 2 = H(t) H 2 2

Zs

Zs

f (t), u(t) V (t) dt −

0

gk (ξ(t)), u(t) V (t) dt

0

Zs −

A(u(t)), u(t)

V (t)

(4.16) dt.

0

Moreover, for every s ∈ I¯ and for every η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)) we have the identity u(s), η(s) H(s) − u0 , η(0) H =

Zs

f (t), η(t) V (t) dt −

0

Zs

gk (ξ(t)), η(t) V (t) dt

0

Zs +

u(t), ∂t η(t) H(t) dt −

0

Zs

A(u(t)), η(t)

V (t)

dt.

0

Proof. Defining ∗ fk := f + gk (ξ) ∈ Lp (I, V (t)) the existence of at least one weak solution with the above properties follows from modifications of the techniques we applied in the previous chapter. To be more specific: One

86


can construct an explicit Galerkin approximation scheme adapted to the system (4.9) linearized at some ξ ∈ Lr (Q)d . The time-dependent ansatz fields can be constructed similarly as in Chapter 3 by pushing forward a countable and dense subset of the separable space V = W01,p (Ω0 )d by means of the Piola transform Pψ . Since the system does not contain any nonlinear lower order terms, no strong compactness arguments are needed for the passage to the limit in the Galerkin approximation. However, the weak limit of the nonlinear elliptic term induced by the monotone operator A still has to be identified. The identification can be established by using the integration by parts formula from Theorem 3.20 in order to exploit the monotonicity of A. Notice that since we can work with the family of Gelfand triples V (t), H(t), V (t)0 t∈I¯, all the results of Theorem 3.20 pertain. In order to prove uniqueness of weak solutions we show that the solution continuously depends on the data for any fixed ∗ k ∈ N. r i p Let (fi , u0 , ξ i ) ∈ L (I, V (t)) × V × L (Q)d , i = 1, 2, denote two sets of admissible data and denote by ui , i = 1, 2, two corresponding weak solutions in L∞ (I, H(t)) ∩ Lp (I, V (t)). The difference u1 − u2 then satisfies the identity u1 (s) − u2 (s), η(s) H(s) − u10 − u20 , η(0) H Zs =

f1 (t) − f2 (t), η(t) V (t) dt −

0

Zs

gk (ξ 1 (t)) − gk (ξ 2 (t)), η(τ )

V (t)

dt

0

Zs u1 (t) − u2 (t), ∂t η(t)

+

0

Zs dt − H(t)

A(u1 (t)) − A(u2 (t)), η(t) V (t) dt

0

for any s ∈ I¯ and for any test function η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)). However, the latter identity also shows ∗ dt (u1 − u2 ) ∈ Lp (I, V (t)) . As u1 − u2 ∈ Lp (I, V (t)), we can invoke Theorem 3.20 and pass to the limit → 0 in the duality ∂t (u1 − u2 ) , χ(0,s) (u1 − u2 ) Lp (I,V (t)) to infer

1

u1 (s) − u2 (s) 2 + H(s) 2

Zs

A(u1 (t)) − A(u2 (t)), u1 (t) − u2 (t)

V (t)

dt

0

2 1 = u10 − u20 H + 2

Zs

f1 (t) − f2 (t), u1 (t) − u2 (t) V (t) dt

0

Zs +

gk (ξ 1 (t)) − gk (ξ 2 (t)), u1 (t) − u2 (t) V (t) dt.

0

Using the strong monotonicity (4.2) together with Hölder’s and Young’s inequality we get

p

u1 (s) − u2 (s) 2 + χ(0,s) (u1 − u2 ) Lp (I,V (t)) H(s)

2

p0

≤ c u10 − u20 H + f1 − f2 (Lp (I,V (t)))∗ + Gk (ξ 1 ) − Gk (ξ 2 ) Lr0 (Q) u1 − u2 Lr (Q) .

87

4.3 Fixed Point Argument

Thanks to the energy estimate (4.15) and Corollary 4.4 the third term on the right-hand side is bounded by

2 c(k) Gk (ξ 1 ) − Gk (ξ 2 ) Lr0 (Q) . Since s ∈ I¯ was arbitrary, we get for any fixed k ∈ N

2

p max u1 (s) − u2 (s) H(s) + u1 − u2 Lp (I,V (t)) s∈I¯

2

p0

≤ c¯(k) u10 − u20 H + f1 − f2 (Lp (I,V (t)))∗ + Gk (ξ 1 ) − Gk (ξ 2 ) Lr0 (Q) 0

from which the the result follows since Gk : Lr (Q)d → Lr (Q)d is continuous.

4.3. Fixed Point Argument Thanks to the preceding lemma, the solution operator for the truncated problem Lk : Lr (Q)d → Lr (Q)d ξ 7→ Lk ξ =: u, where u ∈ L∞ (I, H(t)) ∩ Lp (I, V (t)) ,→ Lr (Q)d denotes the unique weak solution of the system (4.9) linearized at ξ, is well-defined and single-valued. In addition, the energy estimate (4.15) provides a constant c(k) > 0 such that Lk maps the set n o Bk := ξ ∈ Lr (Q)d ξ Lr (Q) ≤ c(k) into itself. Since the set Bk is a bounded, closed and convex subset of the Banach space Lr (Q)d , we can apply Schauder’s fixed point theorem, see Theorem A.28, after proving Lemma 4.7 (Compactness of Lk ) For any k ∈ N, the solution operator Lk : Bk → Bk is continuous and compact. Proof. We prove the compactness assertion first. To this end, consider a sequence (ξ n )n∈N ⊂ Bk and let un := Lk ξ n denote the unique weak solutions of ∂t un + Aun + Gk (ξ n ) = f

on Q

un = 0

on Γ, in Ω0 .

un (0) = u0

We have to show that a subsequence of (un )n∈N strongly converges in Lr (Q)d . We already know that any un satisfies un (s), η(s) H(s) − u0 , η(0) H Zs =

f (t), η(t)

Zs

dt − V (t)

0

gk (ξ n (t)), η(t) V (t) dt

0

Zs +

un (t), ∂t η(t) 0

Zs

H(t)

dt − 0

A(un (t)), η(t) V (t) dt

(4.17)

88


for any s ∈ I¯ and for any η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)). Moreover, the sequence (un )n∈N also satisfies the a priori estimate

(4.18) max un H(t) + un Lp (I,V (t)) ≤ c(k), t∈I¯

where the constant c(k) is independent of n. By weak compactness we infer un * u in Lp (I, V (t)), ∗

un * u in L∞ (I, H(t)), un * u in L2 (I, H(t)), 0

Aun * χ in Lp (Q)d×d for a not relabeled subsequence n → ∞. Due to the growth condition (4.6) and the uniform 0 boundedness of the sequence (ξ n )n∈N ⊂ Bk the sequence (Gk (ξ n ))n∈N is bounded in Lr (Q)d 0 for any fixed k ∈ N. Using weak compactness of bounded sets in Lr (Q)d , there exists 0 0 γ k ∈ Lr (Q)d and possibly another subsequence such that Gk (ξ n ) * γ k in Lr (Q)d as n tends to infinity. In any case, there exists a subsequence which we denote by (un )n∈M1 , such that un * u in Lp (I, V (t)), ∗


Aun * χ in Lp (Q)d×d , 0

Gk (ξ n ) * γ k in Lr (Q)d as M1 3 n → ∞. Now, subtracting the identities (4.17) with un and ul , n, l ∈ M1 , respectively, we get un (s) − ul (s), η(s)

Zs

H(s)

un (t) − ul (t), ∂t η(t) H(t) dt

= 0

Zs +

A(ul (t)) − A(un (t)), η(t) V (t) dt

gk (ξ l (t)) − gk (ξ n (t)), η(t) V (t) .

(4.19)

0

Zs + 0

Due to the weak convergence results for (un )n∈M1 , (Aun )n∈M1 and (Gk (ξ n ))n∈M1 , the right hand side of the last equation converges to zero as M1 3 n, l → ∞ for any s ∈ I¯ and for every fixed η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)). Especially, the latter convergence is independent ¯ Now fix any s ∈ I, ¯ ξ s ∈ V (s) and κ ∈ C ∞ (I) ¯ with κ ≡ 1 in a neighborhood of s. of s ∈ I. Then, as in Lemma 3.7, Pϕs ξ s ∈ V and ξˆ := κ Pψ Pϕs ξ s ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)).

89


Moreover, we see that ˆ ξ(s) = ξs . Plugging ξˆ into (4.19) we obtain lim

M1 3n,l→∞

un (s) − ul (s), ξ s

H(s)

=0

for any arbitrary ξ s in the dense subspace V (s) ,→ H(s), thereby showing that (un (s))n∈M1 is a Cauchy sequence with respect to the weak topology in L2 (Ω(s))d = H(s). By (weak) ˆ (s) ∈ H(s). On the other hand, completeness of the space H(s) there exists a weak limit u we can combine this result with a localization procedure similar to Lemma 3.6, Lemma 3.7 and Corollary 3.8 to show that the sequences (un (s))n∈M1 weakly converge to u(s) ∈ H(s), at least for almost every s ∈ I. By uniqueness of weak limits and the continuous embedding L2 (Ω(s))d ,→ L1 (Ω(s))d we eventually deduce un (s) * u(s) in L1 (Ω(s))d for almost every s ∈ I. Then, using also the embedding L∞ (I, H(t)) ∩ Lp (I, V (t)) ,→ L∞ (I, L1 (Ω(t))d ) ∩ Lp (I, V (t)), the sequence (un )n∈M1 has the following properties: (un )n∈M1 is bounded in Lp (I, V (t)) ∩ L∞ (I, L1 (Ω(t))d ), un * u in Lp (I, V (t)), un (s) * u(s) in L1 (Ω(s))d for almost every s ∈ I, as M1 3 n → ∞. If we apply the isomorphism Pϕ we get a sequence (Un )n∈M1 , Un := Pϕ un , having essentially the same properties in Bochner spaces: (Un )n∈M1 is bounded in Lp (I, V ) ∩ L∞ (I, L1 (Ω0 )d ) and Un * U = Pϕ u in Lp (I, V ), Un (s) * U(s) in L1 (Ω0 )d as M1 3 n → ∞. The compactness principle from Theorem 4.1 now asserts limM3n→∞ Un = U in Lp (I × Ω0 )d , at least for a further infinite subsequence M ⊂ M1 . Without loss of generality we may even assume that limM3n→∞ Un = U holds almost everywhere in I × Ω0 . Since Pϕ is a linear isomorphism with inverse Pψ , the strong convergence in Lp (I × Ω0 )d implies lim un = lim Pψ Pϕ un = Pψ Pϕ u = u in Lp (Q)d . M3n→∞

M3n→∞

Without loss of generality we may again assume that limM3n→∞ un = u holds almost everywhere in Q. By interpolation combined with the a priori estimates for (un )n∈N , we know that the sequence (un )n∈M is bounded in Lp(d+2)/d (Q)d , too, and as it converges to u almost everywhere, we can can finally apply Vitali’s theorem, see Theorem A.2, to deduce Lk ξ n = un → u ∈ Bk in Lr (Q)d ,

90


as M 3 n → ∞. This proves compactness of Lk . We will now prove continuity of the solution operator Lk . Suppose that the sequence (ξ n )n ⊂ Bk converges to ξ in Lr (Q)d . As the Carathéodory function gk induces a contin0 uous Nemyckii operator with values in Lr (Q)d , the strong convergence of (ξ n )n∈N implies 0 Gk (ξ n ) → Gk (ξ) in Lr (Q)d as n → ∞. For any n ∈ N, let again un = Lk ξ n denote the corresponding unique weak solution to the corresponding truncated, linearized problem. Exploiting the a priori estimates for the sequence (un )n∈N and using also the previous compactness argument we may infer that there is a subsequence, which we again denote by (un )n∈M , such that un * u in Lp (I, V (t)), ∗


Aun * χ in Lp (Q)d×d , un → u in Lr (Q)d holds for M 3 n → ∞. Hence, passing to the limit M 3 n → ∞ in the identity ZT −

un (t), ∂t η(t) H(t) dt +

ZT

0

A(un (t)), η(t)

V (t)

dt

0

ZT =

f (t), η(t)

ZT

V (t)

dt −

0

gk (ξ n (t)), η(t) V (t) dt − u0 , η(0) H

0

we eventually obtain that ZT −

ZT

u(t), ∂t η(t) H(t) dt +

0

χ(t), η(t) V (t) dt

0

ZT =

f (t), η(t)

ZT

dt − V (t)

0

gk (ξ(t)), η(t) V (t) dt − u0 , η(0) H

0

holds for any test function η ∈ X. But now we can use Theorem 3.20 in order to justify the energy equality

1

u(s) 2 − 1 u0 2 = H(t) H 2 2

Zs

f (t), u(t) V (t) dt −

0

Zs

gk (ξ(t)), u(t) V (t) dt

0

Zs −

χ(t), u(t)

V (t)

dt

0

¯ Then, we can again use Minty’s trick to establish the remaining identification for any s ∈ I. χ = Au. In the end, this shows that u is a weak solution to the truncated problem, linearized

91


with respect to ξ. For the subsequence (un )n∈M we therefore deduce lim

un =

M3n→∞

lim M3n→∞

Lk ξ n = u = Lk ξ in Lr (Q)d .

Since this argument is valid for any convergent subsequence, we infer that actually the initial sequence converges to the same limit, thereby eventually proving the continuity of Lk . Remark 4.8. A crucial step in the compactness part of Lemma 4.7 is the identification of weak limits on time slices, i.e. the fact that un (s) * u(s) in H(s) holds for almost every s ∈ I as M1 3 n → ∞. Notice that this assertion does not just follow from the fact that the sequence (un )n∈M1 is bounded in L∞ (I, H(t)) since the weakly-∗ converging subsequences assured by boundedness would a priori depend on the time slice under consideration. However, the necessary uniformity can be enforced by noticing that the regularity of the generalized time derivatives (dt un )n∈M1 in fact increases the available regularity in time of the fields (un )n∈M1 . This fact is exploited explicitly by using the weak formulation in (4.19). If one is to apply the compactness principle of Landes and Mustonen in a corresponding cylindrical problem, the necessary identification on time slices can be deduced by using either the implications of Lemma 3.9 or, more generally, abstract interpolation theory in the spirit of the embedding ¯ (Y, X)1−1/r,r ), 1 < r < ∞. W 1,r,r (I, Z, Y ) ∩ Lr (I, X) ,→ C(I, See [Lun09, Corollary 1.14] and the definitions therein for the real interpolation space (Y, X)1−1/r,r . Now, as Lk : Bk → Bk is continuous and compact, Schauder’s fixed point theorem yields the existence of uk ∈ Bk such that uk = Lk uk . By definition of Lk , uk is a weak solution. Therefore, the next theorem follows immediately. Theorem 4.9 (Truncated Problem: Existence) ∗ For every k ∈ N, every f ∈ Lp (I, V (t)) , 2 ≤ p < ∞ and any u0 ∈ V there exists a weak solution uk ∈ L∞ (I, H(t)) ∩ Lp (I, V (t)) of the truncated problem in the sense that ZT −

uk (t), ∂t η(t) H(t) dt +

0

ZT

A(uk (t)), η(t) V (t) dt

0

ZT = 0

f (t), η(t)

V (t)

ZT dt −

(4.20)

gk (uk (t)), η(t) V (t) dt − u0 , η(0) H

0

holds for every η ∈ X. The mapping uk : I¯ → L2 (Rd )d is continuous, the pointwise evaluation [ uk : I¯ → H(s), s 7→ uk (s), s∈I¯

92


is well-defined and uk (0) = u0 holds as an identity in H for any k ∈ N. Every uk satisfies for any s ∈ I¯ the energy equality

1 2 1

uk (s) 2 − u0 H = H(s) 2 2

Zs

f (t), uk (t)

Zs

V (t)

dt −

0

gk (uk (t)), uk (t) V (t) dt

0

Zs −

(4.21)

A(uk (t)), uk (t) V (t) dt.

0

Furthermore, for every η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)) and any s ∈ I¯ there holds uk (s), η(s) H(s) − u0 , η(0) H Zs =

f (t), η(t) V (t) dt −

0

Zs

gk (uk (t)), η(t)

V (t)

dt

(4.22)

0

Zs +

uk (t), ∂t η(t) H(t) dt −

0

Zs

A(uk (t)), η(t)

V (t)

dt.

0

4.4. Passage to the Limit in the Truncated Nonlinearity In order to pass to the limit k → ∞ in (4.20), we first have to derive uniform a priori estimates for the sequence of fixed points (uk )k∈N . To this end, notice that the truncated nonlinearities (4.7). In fact, if we let Qk stand for (gk )k∈N inherit g’s coercivity property the set Q ∩ |g(uk )| < k and set Qck = Q ∩ |g(uk )| ≥ k we see11 ZT

gk (uk (t)), uk (t) V (t) =

0

Z gk (uk ) · uk dxdt Q

Z

Z gk (uk ) · uk dxdt +

= Qk

Z

Z g(uk ) · uk +

=

gk (uk ) · uk dxdt

Qck

Qk

k

Qck

g(uk ) · uk dxdt ≥ 0 |g(uk )|

thanks to (4.7) and the definition of gk , see (4.14). Note that all the integrals in the last equation are finite since gk satisfies the growth condition (4.6) and uk ∈ L∞ (I, H(t)) ∩ Lp (I, V (t)) ,→ Lr (Q)d . As a consequence, we can use the energy equality (4.21) to deduce the a priori estimate

max uk (t) + uk p ≤ C, t∈I¯

H(t)

L (I,V (t))

with a positive constant C that only depends on the data. Interpolation shows that (uk )k∈N is additionally bounded in Lp(d+2)/d ,→ Lr (Q)d . 11

For convenience, here and in the sequel we suppress the independent variables.

93

4.4 Passage to the Limit in the Truncated Nonlinearity 0

Thus, there exists a subsequence (uk )k∈M1 , u ∈ L∞ (I, H(t))∩Lp (I, V (t)) and χ ∈ Lp (Q)d×d such that for M1 3 k → ∞ there holds uk * u in Lp (I, V (t)), ∗

uk * u in L∞ (I, H(t)), uk * u in L2 (I, H(t)), 0

Auk * χ in Lp (Q)d×d and uk (0) * u∗ in H. Since |gk (uk )| ≤ |g(uk )| ≤ c 1 + |uk |r−1 , 0

the subsequence can be chosen in such way that (γ k )k∈M1 ⊂ Lr (Q)d , γ k := Gk (uk ), weakly 0 converges to γ ∈ Lr (Q)d . Using the identity (4.22) and the above convergence results, we can again show that u∗ = u0 and that uk (s) * u(s) in L1 (Ω(s))d as M1 3 k → ∞ for almost every s ∈ I. Hence, we can adapt the compactness argument from Lemma 4.7 to obtain a further subsequence (uk )k∈M , M ⊂ M1 , such that uk → u in Lp (Q)d and uk → u almost everywhere on Q for M 3 k → ∞. As (uk )k∈M is bounded in Lp(d+2)/d (Q)d , the fact that (uk )k∈M converges to u almost everywhere implies lim M3k→∞

uk = u in Lr (Q)d

by Vitali’s theorem. This turns out to be sufficient to prove γ = G(u). 0

Since G : Lr (Q)d → Lr (Q)d is continuous, it follows that G(uk ) strongly converges to G(u) 0 in Lr (Q)d . Using the triangle inequality the assertion follows if we manage to show 0

Gk (uk ) − G(uk ) → 0 in Lr (Q)d

(4.23)

for M 3 k → ∞. To do so, we notice that Z

0

Gk (uk ) − G(uk ) r r0 d = |gk (uk ) − g(uk )|r0 dxdt L (Q) Q

Z

r0

Z

|gk (uk ) − g(uk )| dxdt +

=

Qck

Qk

Z = Qck

0

|gk (uk ) − g(uk )|r dxdt

0

|gk (uk ) − g(uk )|r dxdt

94


since gk (uk ) = g(uk ) on Qk . It follows that Z Z 0 r0 |gk (uk ) − g(uk )| dxdt ≤ c |g(uk )|r dxdt Qck

Qck

Z 1 + |uk |

≤ c

0 r−1 r

Z

Z dxdt ≤ c

Qck

1+c

Qck

|uk |r dxdt.

Qck

Chebychev’s inequality yields

g(uk ) r0 d L (Q)

c −→ 0, k r0 for k → ∞. For the remaining integral we therefore get Z Z Z |uk |r dxdt ≤ c |uk − u|r dxdt + c |u|r dxdt |Qck | ≤

k r0

≤

Qck

Qck

Z ≤

Bk

|uk − u|r dxdt + c

Z

|u|r dxdt −→ 0

Qck

Q

since uk → u ∈ Lr (Q)d for M 3 k → ∞ and due to the absolute continuity of the integral. Thus, we obtain (4.23). Passing to the limit along the subsequence indexed by M we infer ZT −

u(t), ∂t η(t) H(t) dt +

0

ZT

χ(t), η(t)

dt

V (t)

0

ZT =

f (t), η(t) V (t) dt −

0

ZT

g(u(t)), η(t) V (t) dt − u0 , η(0) H

0

for every test function η ∈ X. Again, the last step consists of the identification 0

χ = Au in Lp (Q)d×d . ∗ However, the last assertion in Lemma 4.5 guarantees dt u ∈ Lp (I, V (t)) so that we can invoke the energy equality from Theorem 3.20. This in turn allows us to exploit the pseudomonotonicity of A and Minty’s trick in the end. The weak limit thus constructed then fulfills all the assumptions from Theorem 3.20, so that we have proved the following Theorem 4.10 (Existence) ∗ For every f ∈ Lp (I, V (t)) , 2 ≤ p < ∞, and u0 ∈ V there exists a weak solution u ∈ L∞ (I, H(t)) ∩ Lp (I, V (t)) of (4.1) in the sense that ZT −

u(t), ∂t η(t) H(t) dt +

0

ZT

A(u(t)), η(t)

V (t)

dt

0

ZT = 0

f (t), η(t) V (t) dt −

ZT 0

g(u(t)), η(t) V (t) dt − u0 , η(0) H

95

4.5 The Navier–Stokes Equations on Time-Dependent Domains

¯ L2 (Rd )d ), the pointwise evaluation holds for every η ∈ X. We have u ∈ C(I, [ u : I¯ → H(s), s 7→ u(s), s∈I¯

is well-defined and u(0) = u0 holds in H. The solution u satisfies for any s ∈ I¯ the energy equality

1 1 2

u(s) 2 − u0 H = H(s) 2 2

Zs

f (t), u(t) V (t) dt −

0

Zs

g(u(t)), u(t)

V (t)

dt

0

Zs −

A(u(t)), u(t) V (t) dt.

0

Also, for any s ∈ I¯ and for any η ∈ H 1 (I, H(t)) ∩ Lp (I, V (t)) we have the identity u(s), η(s) H(s) − u0 , η(0) H =

Zs

f (t), η(t) V (t) dt −

0

Zs

g(u(t)), η(t)

V (t)

dt

0

Zs +

u(t), ∂t η(t) H(t) dt −

0

Zs

A(u(t)), η(t)

V (t)

dt.

0

4.5. The Navier–Stokes Equations on Time-Dependent Domains We now apply the compactness result of this chapter to the Navier–Stokes equations in a time-dependent domain [ [ Q := {t} × Ω(t) = {t} × ϕt (Ω0 ) ⊂ R4 . (4.24) t∈I

t∈I

Here we assume that (ϕt )t∈I is the flow of a smooth, time-dependent, incompressible and complete vector field v : R × R3 → R3 and that Ω0 ⊂ R3 is an open and bounded domain with Lipschitz boundary. The lateral boundary of Q is then given by [ [ Γ := {t} × ∂Ω(t) = {t} × ϕt (∂Ω0 ). (4.25) t∈I

t∈I

The motion of an incompressible, viscous and homogeneous fluid which is contained in the impermeable, deforming domain Ω0 is modelled by means of the Navier–Stokes equations for the velocity field u and the pressure π: ρF ∂t u + (u · ∇)u = f + div S(σ, u) − π idR3 in Q, div u = 0

in Q,

u=v

on Γ,

u(0) = u0

in Ω0 .

(4.26)

96


Here, ρF denotes the fluid’s constant density and σ is the viscosity coefficient which is constant, too. It is assumed that external forces acting on the fluid are given by a force density f : Q → R3 . Moreover, S(σ, u) refers to the Cauchy stress tensor to be specified below. The incompressibility of the fluid is captured in the constraint div u = 0 and the impermeability of the domain’s boundary yields the non-homogeneous Dirichlet boundary condition (4.26)3 . Eventually, u0 : Ω0 → R3 is the initial velocity of the fluid. The structure of the Cauchy stress tensor determines whether a fluid is Newtonian or nonNewtonian. In both cases, the principle of conservation of angular momentum, see [MH94, Chapter 2], implies that the Cauchy stress tensor depends on u through the symmetric part of the velocity gradient Du :=

1 ∇u + ∇uT . 2

Newtonian fluids are characterized by a linear dependence of S on Du, whereas nonNewtonian fluids exhibit a nonlinear dependence. In this chapter we will confine ourselves with the linear case. However, generalized Navier–Stokes models for non-Newtonian fluids in non-cylindrical domains will be the content of the Chapters 6 and 7. Assuming that S(σ, u) = 2 σ Du, the incompressibility constraint div u = 0 yields div S(σ, u) = σ∆u which turns (4.26) into the standard Navier–Stokes equations on a non-cylindrical domain with inhomogeneous boundary conditions: ρF ∂t u + (u · ∇)u − σ∆u + ∇π = f in Q, div u = 0

in Q,

u=v

on Γ,

u(0) = u0

(4.27)

in Ω0 .

Since the given velocity field v is assumed to be incompressible, it represents a natural extension of the boundary values of u. Thanks to the linearity of (4.27)1 in the highest order term we can transform (4.27) to an equivalent problem by introducing the new unknown w := u − v. This transformation leads to a problem with homogeneous Dirichlet boundary conditions, but a different force density ¯f that additionally depends on v. Moreover, the transformation of the convective term creates terms that are either linear in the unknown velocity field or behave essentially as the term (u · ∇)u. Setting ρf = σ = 1 we therefore confine ourselves with the case of homogeneous Dirichlet conditions, that is, we consider ∂t u + (u · ∇)u − ∆u + ∇π = f

in Q,

div u = 0

in Q,

u=0

on Γ,

u(0) = u0

(4.28)

in Ω0 .

The first existence results for the non-cylindrical Navier–Stokes system (4.28) go back to [FS70], [IW77] and [MT82]. In [Boc77], Bock extended some earlier results obtained by

97


Ladyzhenskaya in [Lad68]. Miyakawa, Teramoto and Bock used the Piola transform in order to construct an auxiliary problem on a cylindrical domain which can then be solved by using a Galerkin approximation. Furthermore, the first two authors not only discuss the extensibility of prescribed boundary values but also present an interesting compactness result for the convective term, see [MT82, Lemma 2.5]. Nevertheless, we shall choose a different approach: As the convective term does not induce a well-defined operator on the natural energy space, an L∞ -approximation of that operator is constructed first. This method is inspired by the work of Wolf who used it in the context of non-Newtonian fluids in [Wol07]. Furthermore, this approximation makes the system accessible for the theory we had developed in the previous chapters. The passage to the limit is established by means of Theorem 4.1 and its non-cylindrical generalization. At this point, the Piola transform is essential. We also would like to mention the fact that Landes used Theorem 4.1 in [Lan86] to partly simplify the classical existence result for the Navier–Stokes equations of Leray and Hopf which is usually based on the Aubin–Lions lemma. Our method follows this line of ideas to some extent. We refer to [Lan86] for the details related to the cylindrical setting.

4.5.1. Functional Setting and Weak Formulation The functional setting for the problem is again determined by formally multiplying (4.28)1 with the solution itself and integrating the result over Ω(t) first. We obtain Z Z ∂t u(t) · u(t) dx + u(t) · ∇ u(t) · u(t) dx Ω(t)

Ω(t)

Z f (t) · u(t) dx −

=

(4.29)

Z

Ω(t)

∇u(t) · ∇u(t) dx.

Ω(t)

Notice that we used integration by parts and the homogeneous boundary conditions in the R elliptic term. The integral Ω(t) π div u dx vanishes due to the divergence constraint (4.28)2 . The convective term does not contribute either since Z Z Z 1 1 2 u(t) · ∇ u(t) · u(t) dx = u(t) · ∇|u(t)| dx = − div u(t) |u(t)|2 dx = 0. 2 2 Ω(t)

Ω(t)

Ω(t)

Reynolds’ transport theorem in turn implies Z Z 1d ∂t u(t) · u(t) dx = |u(t)|2 dx 2 dt Ω(t)

Ω(t)

since u = 0 on Γ. Integrating (4.29) from 0 to s with respect to t we therefore get the formal energy equality

1

u(s) 2 2 + L (Ω(s)) 2

Zs 0

1

∇u(t) 2 2

u0 2 2 dt = + L (Ω(t)) L (Ω0 ) 2

Zs Z f · u dxdt. 0 Ω(t)

98


That is, the total energy of the system at time s, consisting of the potential energy and the dissipated energy, is the sum of the initial potential energy and the energy provided by external forces. Under reasonable assumptions on f and u0 we infer

u ∞ 2 (4.30) + u L2 (I,W 1,2 (Ω(t))3 ) ≤ C L (I,L (Ω(t))3 ) 0

with a constant depending only on the data. Remark 4.11. The above energy estimates can only be derived on a formal level, the reason being that for a solution u in the natural energy space L∞ (I, L2 (Ω(t))3 )∩L2 (I, W01,2 (Ω(t))3 ) the integral Z Z u · ∇ u · u dxdt (4.31) I Ω(t)

is not well-defined. This issue is intimately connected with the uniqueness of solutions in this regularity class. Even in the cylindrical setting this is still an open problem. However, the construction of such weak solutions is still possible by choosing a suitable weak formulation of the problem. We will come back to this point below. We now define a functional setting that makes the weak formulation of (4.28) fit into the framework of this chapter. We set for t ∈ I¯ ∞ (Ω(t))3 V (t) := W01,2 (Ω(t))3 = C0,σ

k·kV (t)

,

u(t)

V (t)

:= ∇u(t) L2 (Ω(t)) ,

V := V (0), H(t) := L2 (Ω(t))3

u(t) 2

u(t) , := L (Ω(t)) H(t)

H := H(0), 1,2 ∞ (Ω(t))3 Vσ (t) := W0,σ (Ω(t))3 = C0,σ ∞ (Ω(t))3 Hσ (t) := C0,σ

k·kH(t)

k·kV (t)

,

.

To circumvent integrability issues it is standard to choose a weak formulation defined on smooth test functions. The definition of an operator G corresponding to the convective ¯ u(t) ∈ Vσ (t) and η(t) ∈ C ∞ (Ω(t))3 term requires a reformulation of that term. For t ∈ I, 0,σ integration by parts shows Z Z Z u(t) · ∇ u(t) · η(t) dx = − u(t) · η(t) div u(t) dx − u(t) ⊗ u(t) : ∇η(t) dx Ω(t)

Ω(t)

Z =−

Ω(t)

u(t) ⊗ u(t) : ∇η(t) dx =: g(u(t)), η(t) Vσ (t) .

Ω(t)

For u ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) and n o ∞ η ∈ D := ϕ ∈ C ∞ (Q)3 ϕ(t) ∈ C0,σ (Ω(t))3 , t ∈ I¯


99

the functional g is extended to

Z

G(u), η =

g(u(t)), η(t)

Vσ (t)

dt.

I

Note that the space D can be obtained by applying the Piola transform to the space of smooth, solenoidal vector fields vanishing on the lateral boundary of the cylindrical reference domain I × Ω0 . The linear operator induced by the Laplacian is on any time slice given by A : Vσ (t) → Vσ (t)0 , Z

∇u(t) : ∇η(t) dx A(u(t)), η(t) Vσ (t) := Ω(t)

and subsequently extended to a mapping 0 A : L2 (I, Vσ (t)) → L2 (I, Vσ (t)) , Z

Au, η := A(u(t)), η(t) Vσ (t) dt. I

It is readily checked that A is continuous. Since A is linear it follows that the continuity is preserved if we switch to the respective weak topologies. The functional setting and the formal energy estimate (4.30) motivate the following definition of weak solutions of problem (4.28). Definition 4.12 (Weak Solution of the Non-cylindrical Navier–Stokes System) ∗ For f ∈ L2 (I, V (t)) and u0 ∈ Vσ (0), a vector field u ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) is called a weak solution of (4.28) if ZT −

u(t), ∂t η(t) 0

ZT dt+ H(t)

A(u(t)), η(t) Vσ (t) dt +

0

ZT

g(u(t)), η(t) Vσ (t) dt

0

ZT =

f (t), η(t) Vσ (t) dt + u0 , η(0) H

0

holds for any η ∈ D with η(T ) = 0. Remark 4.13. Notice that the generalized time derivative of potential weak solutions does not lie in the dual of the space L2 (I, Vσ (t)). The reason is that G(u) does not define an 0 element in L2 (I, Vσ (t)) for fixed u ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)).

4.5.2. Existence of Approximate Solutions and Fixed Points The existence of solutions in the sense of Definition 4.12 will be proved by passing to the limit in a sequence of truncated problems. As the system is linear in its highest order term,

100


the very problem lies in the presence of the convective term. By means of an L∞ -approximation of the convective term, the equation is first of all converted into a problem which fits into the framework of the theory of monotone operators. Following [Wol07], let Φ ∈ C ∞ ([0, ∞)) be a non-increasing function that satisfies 0 ≤ Φ ≤ 1, Φ ≡ 1 on [0, 1], Φ ≡ 0 on [2, ∞] and 0 ≤ −Φ0 ≤ 2. Define a sequence (Φk )k∈N ∈ C ∞ ([0, ∞)), supp Φk ⊂⊂ [0, 2k], by Φk (t) := Φ(t/k), k ∈ N. It follows that Φk (·) → χ[0,∞)

(4.32)

¯ as k tends to infinity. The convective term is then approximated as follows: For t ∈ I, 2 3 u(t) ∈ L (Ω(t)) and η(t) ∈ Vσ (t) we set Z

u(t) ⊗ u(t) Φk (|u(t)|) : ∇η(t) dx. gk (u(t)), η(t) Vσ (t) := − Ω(t)

Likewise, we define 0 Gk : L2 (Q)3 → L2 (I, Vσ (t)) Z

Gk (u), η := gk (u(t)), η(t) Vσ (t) dt. I

In the beginning of this section we saw that the convective term does not contribute to the energy estimate. The L∞ -approximation defined above preserves this property as the next lemma shows. Lemma 4.14 The mappings gk : H(t) → Vσ (t)0 and 0 Gk : L2 (Q)3 → L2 (I, Vσ (t)) ¯ u(t) ∈ Vσ (t) and u ∈ L2 (I, Vσ (t)) there holds are well-defined. Moreover, for any t ∈ I,

gk (u(t)), u(t) Vσ (t) = 0 and

Gk (u), u = 0, respectively. Proof. We only prove the assertions for gk as a similar reasoning applies to Gk . For any k ∈ N and any u(t) ∈ H(t) the properties of Φk imply the pointwise estimate u(t) ⊗ u(t) Φk (|u(t)|) ≤ 4k 2 . It follows that gk : H(t) → L∞ (Ω(t))3 ,→ Vσ (t)0 is well-defined. It is sufficient to prove the ∞ (Ω(t))3 . In fact, for any u(t) ∈ V (t) there exists a sequence second result for u(t) ∈ C0,σ σ ∞ (Ω(t))3 such that (un (t))n∈N ⊂ C0,σ lim un (t) = u(t) in H(t) and lim ∇un (t) = ∇u(t) in L2 (Ω(t))3×3 ,

n→∞

n→∞


101

by Lemma 2.3. Without loss of generality, we can additionally assume that (un (t))n∈N and (∇un (t))n∈N converge almost everywhere in Ω(t) to their respective limits. The continuity of Φk yields lim un (t) ⊗ un (t) Φk (|un (t)|) : ∇un (t) = u(t) ⊗ u(t) Φk (|u(t)|) : ∇u(t)

n→∞

almost everywhere in Ω(t). The pointwise estimate un (t) ⊗ un (t) Φk (|un (t)|) : ∇un (t) ≤ 4k 2 ∇un (t) then allows us to conclude

lim gk (un (t)), un (t) Vσ (t) = gk (u(t)), u(t) Vσ (t) n→∞

by means of Lebesgue’s theorem. Since hk (t) := Φk (t)t is continuous with hk (t) = 0 if t ≥ 2k the function Zs Hk (s) =

hk (t) dt 0

∞ (Ω(t))3 we then infer is well-defined and satisfies Hk0 = hk . For any u(t) ∈ C0,σ Z

− gk (u(t)), u(t) Vσ (t) = u(t) ⊗ u(t) Φk (|u(t)|) : ∇u(t) dx Ω(t)

Z

1 = 2

u(t) Φk (|u(t)|) · ∇ u2 (t) dx

Ω(t)

=

Z

1 2

u(t) Φk (|u(t)|) · ∇ |u(t)|2 dx

Ω(t)

Z u(t) Φk (|u(t)|) |u(t)| · ∇ |u(t)| dx

= Ω(t)

. Using the properties of the function Hk and integrating by parts we get Z Z u(t) Φk (|u(t)|) |u(t)| · ∇ |u(t)| dx = u(t) Hk0 (|u(t)|) · ∇ |u(t)| dx Ω(t)

Ω(t)

Z

u(t) · ∇ Hk (|u(t)|) dx

= Ω(t)

Z =− Ω(t) ∞ (Ω(t))3 . for any u(t) ∈ C0,σ

div u(t) Hk (|u(t)|) dx = 0

102


We have the following existence result for the linearized, truncated system. Proposition 4.15 (Linearized, Truncated Problem: Existence and Uniqueness) ∗ For every k ∈ N, any f ∈ L2 (I, V (t)) , any u0 ∈ Vσ (0) and any ξ ∈ L2 (Q)3 , there exists a unique u ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) such that ZT −

u(t), ∂t η(t) H(t) dt +

0

ZT

A(u(t)), η(t)

ZT

dt + Vσ (t)

0

gk (ξ(t)), η(t)

Vσ (t)

dt

0

ZT =

(4.33)

f (t), η(t) Vσ (t) dt + u0 , η(0) H

0

holds for any η ∈ X :=

n

o ϕ ∈ H (I, H(t)) ∩ L (I, Vσ (t)) ϕ(T ) = 0 . 1

2

The mapping u : I¯ → L2 (R3 )3 is continuous, the evaluation of u, [

u : I¯ →

H(s), s 7→ u(s),

s∈I¯

is well-defined and u(0) = u0 holds as an identity in Hσ (0). Moreover, u satisfies the energy equality

1 1 2

u(s) 2 − u0 H = H(s) 2 2

Zs

f (t), u(t) Vσ (t) dt −

0

Zs

gk (ξ(t)), u(t) Vσ (t) dt,

0

Zs

−

(4.34)

A(u(t)), u(t) Vσ (t) dt,

0

as well as the a priori estimate (4.35)

u ∞ + u L2 (I,Vσ (t)) ≤ c(k). L (I,Hσ (t))

The constant c(k) depends on the data and the truncation parameter k. For any s ∈ I¯ and any η ∈ X there holds u(s), η(s) H(s) − u0 , η(0) H =

Zs

f (t), η(t) Vσ (t) dt −

0

Zs

gk (ξ(t)), η(t) Vσ (t) dt

0

Zs + 0

u(t), ∂t η(t) H(t) dt −

Zs 0

A(u(t)), η(t)

Vσ (t)

dt.

103


Proof. The existence of at least one solution for any fixed k ∈ N can be proved by using a Galerkin approximation scheme as in the previous chapter. Notice that the standard estimate for Galerkin solutions (un )n∈N ,

max un (t) Hσ (t) + un L2 (I,Vσ (t)) ≤ c(k), t∈I¯

still holds and is sufficient for the limiting process n → ∞ in the Galerkin system: If (un )n∈N weakly converges to u ∈ L2 (I, Vσ (t)), then 0 Aun * Au in L2 (I, Vσ (t)) as n → ∞ since A is linear and hence weakly continuous. For this reason, we do not need to invoke Theorem 3.20 to identify the weak limit of the elliptic term using pseudomonotonicity. However, we use the latter theorem toSjustify the continuity of the solution u and the pointwise in time evaluation, i.e. u : I¯ → s∈I¯ H(s), s 7→ u(s). In fact, the identity (4.33) shows that the weak limit of the Galerkin solutions has a generalized time derivative in (L2 (I, Vσ (t)))∗ . Thus, Theorem 3.20 implies the well-definedness of the pointwise in time evaluation of u. We now come to the uniqueness of approximate solutions. Let u1 , u2 ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) denote two weak solutions corresponding to the same data, the same linearization ξ and the same level of truncation k. The difference u1 − u2 then satisfies the assumptions of Theorem 3.20. Therefore we can use u1 − u2 as a test function to infer that for any s ∈ I¯ there holds

1

u1 (s) − u2 (s) 2 ≤ H(s) 2

Zs

f (t) − gk (ξ(t)), u1 (t) − u2 (t)

Vσ (t)

dt

0

But now the claim follows by applying Gronwall’s lemma, see Lemma A.26. Proposition 4.15 implies that the solution operator Lk : L2 (Q)3 → L2 (I, Vσ (t)) ,→ L2 (Q)3 ξ 7→ Lk ξ =: u, is well-defined and single-valued. An inspection of the calculations leading to the a priori estimate (4.35), shows that the constant c(k) does not depend on the choice of ξ ∈ L2 (Q)3 . It follows that Lk maps the closed, bounded and convex set n o Bk := ξ ∈ L2 (Q)3 ξ L2 (Q) ≤ c(k) into itself. In order to prove continuity and compactness of the nonlinear solution operator Lk , we may proceed similarly as in Lemma 4.7: Denoting by (un )n∈N a sequence of solutions corresponding to linearizations (ξ n )n∈N ⊂ Bk at the truncation level k, we first of all find that (un )n∈N satisfies the uniform estimate

un ∞ + un L2 (I,Vσ (t)) ≤ c(k). L (I,Hσ (t))

104


From weak compactness, Lemma 3.6 and Corollary 3.8 we may then infer the existence of a subsequence (un )n∈M1 ⊂ (un )n∈N with the additional property that lim un (s) − u(s), ξ s H(s) = 0 M1 3n→∞

holds for almost every s ∈ I and any ξ s ∈ Vσ (s). Here, u of course denotes the weak limit point of (un )n∈M1 in L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)). By density of the embedding Vσ (s) ,→ Hσ (s) we deduce for almost every s ∈ I that un (s) * u(s) in Hσ (s) as M1 3 n → ∞. The weak convergence un (s) * u(s) in L1 (Ω(s))3 , for almost every s ∈ I, which is necessary with regard to the non-cylindrical version of Theorem 4.1, is an immediate consequence if we manage to prove that un (s) * u(s) in H(s) = L2 (Ω(s))3

(4.36)

is satisfied for almost every s ∈ I as M1 3 n → ∞. However, the existence result of Proposition 4.15 shows that the sequence (un )n∈M1 actually takes its values in Hσ (s), at least for every s ∈ I outside a set of measure zero. It follows that (un (s) − u(s))n∈M1 is in fact a weak null sequence in Hσ (s) for almost every s ∈ I. By virtue of the Leray–Helmholtz decomposition ⊥

H(s) = L2 (Ω(s))3 = Hσ (s) ⊕ Hσ (s)⊥ ,

(4.37)

see also Theorem A.13, we can therefore conclude (4.36). The compactness principle of Landes and Mustonen allows us to proceed as in the compactness part of Lemma 4.7 to prove that a further subsequence (un )n∈M ⊂ (un )n∈M1 strongly converges in L2 (Q)3 . The continuity of the solution operator can be proved essentially as in Lemma 4.7. Schauder’s fixed point theorem hence implies the existence of a fixed point uk = Lk uk ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) which shares all the properties of the weak solution that are presented in Proposition 4.15. In particular, any fixed point uk of Lk satisfies for any s ∈ I¯ and for any η ∈ H 1 (I, H(t)) ∩ L2 (I, Vσ (t)) the identity uk (s), η(s) H(s) − u0 , η(0) H Zs =

f (t), η(t) Vσ (t) dt −

0

Zs

gk (uk (t)), η(t) Vσ (t) dt

0

Zs + 0

uk (t), ∂t η(t) H(t) dt −

Zs

(4.38)

A(uk (t)), η(t) Vσ (t) dt.

0

However, the a priori estimate (4.35) can be sharpened for the sequence of fixed points. In fact, plugging uk into (4.34) after replacing ξ by uk we can derive the uniform estimate

uk ∞

uk 2 ≤ C, (4.39) + L (I,Vσ (t)) L (I,Hσ (t)) with a constant independent of k. This is a direct consequence of the fact that the truncated convective term vanishes identically if we test it with uk , see Lemma 4.14.


105

4.5.3. Passage to the Limit in the Truncated Convective Term The limiting process in the integral ZT

gk (uk (t)), η(t)

Vσ (t)

dt, η ∈ D,

0

represents the major obstacle since we again need to prove compactness of (uk )k∈N in L2 (Q)3 . The compactness argument in the previous section demonstrates that it suffices to prove uk (s) * u(s) in Hσ (s)

(4.40)

for almost every s ∈ I and a suitable subsequence k → ∞. Just as in Lemma 4.7, the weak convergence in Hσ (s) can be deduced from the convergence of the integrals in the weak formulation. Here, we essentially need to show weak convergence of the term gk (uk ) = uk ⊗ uk Φk (|uk |). By definition of Φk we have the trivial estimate 2 gk (uk ) = uk ⊗ uk Φk (|uk |) ≤ uk .

(4.41)

This estimate implies that we need to control the sequence (u2k )k∈N in order to control (gk (uk ))k∈N . Using Corollary 4.4 we find the embedding L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) ,→ Lα (Q)3 , 2 for any 1 ≤ α ≤ 10 in L5/3 (Q)3 and consequently 3 . Thus, the sequence (uk )k∈N is bounded 0 5/3 3×3 5/2 3×3 (gk (uk ))k∈N is bounded in L (Q) = L (Q) as well. Hence, there exists G ∈ 5/3 3×3 L (Q) such that for any η ∈ D there holds

Z

Z gk (uk ) : ∇η dxdt −→

Q

G : ∇η dxdt

(4.42)

Q

for some subsequence M1 3 k → ∞. For the proof of (4.40) we follow the ideas of Lemma 3.7. That is, we first fix s ∈ I¯ and an ∞ (Ω(s))3 . Then arbitrary η s ∈ C0,σ ∞ Pϕs η s ∈ C0,σ (Ω0 )3 ,

and defining ˆ := Pψ Pϕs η s , η ˆ satisfies η ˆ ∈ D and η ˆ (s) = η s . These facts again follow from the smoothness of the flows, η the chain rule and the properties of the Piola transform. ˆ as a test function in (4.38) and subtract the respective identities In the next step we use η

106


for two fixed points uk and ul to get uk (s) − ul (s), η s

Zs

H(s)

ˆ (t) H(t) dt uk (t) − ul (t), ∂t η

= 0

Zs +

ˆ (t) V (t) dt A(ul (t)) − A(uk (t)), η σ

ˆ (t) gl (ul ) − gk (uk ), η

0

Zs +

Vσ (t)

dt.

0

However, the right-hand side converges to 0 for M1 3 k, l → ∞ at any time s ∈ I¯ and for ˆ . This is a direct consequence of the a priori estimate (4.39) and the weak any test function η convergence of the convective terms in (4.42). Since η s was arbitrary in the dense subset ∞ (Ω(s))3 ,→ H (s) we conclude that (u (s)) C0,σ σ k k∈M1 is a weak Cauchy sequence in Hσ (s) at least for almost every s ∈ I. As in the fixed point argument of the previous section, it follows that (uk (s))k∈M1 weakly converges to u(s) in Hσ (s) for almost every s ∈ I. Again, u ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) denotes the weak limit of (uk )k∈M1 . But now we can use the Piola transform and the compactness principle of Landes and Mustonen to show that we eventually obtain a further subsequence (uk )k∈M of the sequence of fixed points such that, for M 3 k → ∞, uk * u in L2 (I, Vσ (t)), 0 Auk * Au in L2 (I, Vσ (t)) , uk (0) * u∗ in Hσ (0), uk → u in L2 (Q)3 and uk → u almost everywhere in Q. With these results at hand we can now prove convergence of the approximated convective term. In fact, the pointwise convergence of (uk )k∈M implies the pointwise convergence lim

gk (uk ) =

M3k→∞

lim

uk ⊗ uk Φk (|uk |) = u ⊗ u

M3k→∞

2 almost everywhere in Q. The bound uk ⊗ uk Φk (|uk |) ≤ uk , see (4.41), then yields lim M3k→∞

gk (uk ) =

lim M3k→∞

uk ⊗ uk Φk (|uk |) = u ⊗ u in L5/3 (Q)3×3

due to the general version of Lebesgue’s theorem. We may therefore pass to the limit M 3 k → ∞ in the identity ZT −

uk (t), ∂t η(t) H(t) dt +

0

ZT 0

ZT = 0

A(uk (t)), η(t)

Vσ (t)

ZT dt +

gk (uk (t)), η(t) Vσ (t) dt

0

f (t), η(t) Vσ (t) dt + uk (0), η(0) H


107

for any η ∈ D with η(T ) = 0 to obtain ZT −

u(t), ∂t η(t)

ZT dt + H(t)

0


0

ZT =

ZT


0

f (t), η(t) Vσ (t) dt + u∗ , η(0) H .

0

However, by the method of construction, for every k ∈ N there holds uk (0) = u0 in Hσ (0). From this we eventually infer, using again the Leray–Helmholtz decomposition in H = ∞ (Ω )3 in H (0), that L2 (Ω0 )3 and the density of C0,σ 0 σ u∗ = u0 in Hσ (0). In summary we have proved Theorem 4.16 (Non-cylindrical Navier–Stokes: Existence) ∗ For any f ∈ L2 (I, V (t)) and any u0 ∈ Vσ (0), there exists at least one vector field u ∈ L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) such that ZT −

u(t), ∂t η(t)

ZT

H(t)

dt +

0

0

ZT =


ZT


0

f (t), η(t) Vσ (t) dt + u0 , η(0) H

0

holds for any η ∈ D =

∞ (Ω(t))3 , t ∈ I¯ with η(T ) = 0. ϕ ∈ C ∞ (Q)3 ϕ(t) ∈ C0,σ

Remark 4.17. The regularity assumption u0 ∈ Vσ (0) can be replaced by the more natural condition u0 ∈ Hσ (0). In fact, by density of the embedding Vσ (0) ,→ Hσ (0), any initial value u0 ∈ Hσ (0) may be approximated by a sequence (uk0 )k∈N ⊂ Vσ (0). Hence, we can simultaneously approximate the initial value and regularize the convective term in order to produce corresponding approximate solutions in L∞ (I, Hσ (t)) ∩ L2 (I, Vσ (t)) of the system thus obtained. Carrying out the compactness arguments from the previous sections one can actually show that the result of Theorem 4.16 prevails for the case u0 ∈ Hσ (0). Remark 4.18. As in the cylindrical case, uniqueness of weak solutions in the sense of Theorem 4.16 is an open problem in the case of non-cylindrical domains Q ⊂ R4 , too. However, the weak solution is unique in the case Q ⊂ R3 . In that case, the proof rests upon the fact that the weak solution itself is an admissible test function. For the details of the proof in the respective cylindrical case we refer to [BF13, Chapter V]. In Proposition 6.18 of Chapter 6, we will prove uniqueness of weak solutions of a variant of the p-Navier–Stokes equations. Similar arguments though, can be used to establish uniqueness of weak solutions of the Navier–Stokes equations (4.28) in a non-cylindrical domain Q ⊂ R3 .

108


5. The Aubin–Lions Lemma in Generalized Bochner Spaces 5.1. Basic Classical Version of the Aubin–Lions Lemma One of the most powerful tools in existence theory for nonlinear initial boundary value problems via compactness methods is the Aubin–Lions lemma. Ever since its appearance in [Aub63] and in Lions’ famous monograph [Lio69], it has been subject to numerous generalizations and applications. Further evidence for its perpetual importance is the fact that Simon’s survey article [Sim87] is still among the most cited papers within the nonlinear PDE community. Although the Aubin–Lions lemma is by now a standard result we state and sketch a proof of its basic version, the reason being that we want to use this proof as a starting point towards a non-cylindrical generalization. Lemma 5.1 (Aubin–Lions Lemma in Bochner Spaces) Let B0 , B and C denote Banach spaces, where B0 and C are reflexive, and set I := (0, T ). i

i

Furthermore, assume that B0 ,→,→ B ,→ C, where B ,→ C means that the embedding B ,→ C is injective. Then, the embedding du o n ∈ Lr (I, C) ,→ Lp (I, B) W 1,p,r (B0 , C) = u ∈ Lp (I, B0 ) dt is compact for 1 < p, r < ∞. Proof. Let (un )n∈N be a bounded sequence in W 1,p,r (B0 , C). It is sufficient to prove that a subsequence of (un )n∈N is a Cauchy sequence in Lp (I, C). In fact, due to Ehrling’s lemma, see Lemma 5.2 below, we know that for any > 0 there is a constant c > 0 such that

un − uk p + c un − uk p ≤ un − uk p L (I,B)

L (I,B0 )

L (I,C)

holds for any n, k ∈ N. Any un ∈ W 1,p,r (B0 , C) is absolutely continuous as a function with values in C and satisfies the identity 0

Zs

un (s) − un (s ) =

dun (t) dt dt

(5.1)

s0

for every s0 , s ∈ I¯ with s0 ≤ s. The last equation is to interpreted as an identity in C and, by means of Hölder’s inequality, yields the embedding ¯ C). W 1,p,r (B0 , C) ,→ C 0,1−1/r (I,

(5.2)

109

110

Chapter 5. The Aubin–Lions Lemma in Generalized Bochner Spaces

However, the Arzelà-Ascoli theorem, see [BF13, Theorem then implies that a subse II.3.1], p quence is relatively compact in L (I, C) provided that un (t) n ∈ N is relatively compact in C for any t in a dense subset G ⊂ I. The latter assertion in turn is a consequence of the continuous embeddings ¯ (B0 , C)1/s,s ) and (B0 , C)1/s,s ,→,→ C W 1,p,r (B0 , C) ,→ C(I,

(5.3)

for a suitable 1 < s < ∞, see [DR05, Theorem 33] or [BL76, Theorem 3.8.1]. Here, (B0 , C)1/s,s denotes the real interpolation space. Before we go on, we want to revise the proof of Ehrling’s lemma as well. Although this result is contained in many textbooks on PDEs and functional analysis, it seems as if [Alt06] is one of the few references that explicitly stresses the necessity of the injectivity of the embedding B ,→ C. Admittedly, this condition is satisfied in most applications. Nevertheless, in the next chapter we will encounter a particular example where the injectivity assumption is violated. The lack of injectivity though, causes serious problems as it thwarts the application of the classical Aubin–Lions lemma. Lemma 5.2 (Ehrling Lemma) Suppose B0 , B and C are three Banach spaces and that B0 is reflexive. If i

B0 ,→,→ B ,→ C, then, for every > 0 there exists a positive constant c such that for any u ∈ B0 there holds

u ≤ u + c u . B0 C B Proof. Assume that the claim is false. Then there exists 0 > 0 and a sequence (un )n∈N ⊂ B0 such that

un ≥ 0 un + n un . (5.4) B B0 C By homogeneity we may assume kun kB0 = 1 for every n ∈ N. The compact embedding B0 ,→,→ B then implies the strong convergence unk → u in B for some subsequence (unk )nk ∈N and some u ∈ B. On the other hand (5.4) yields for nk → ∞ that unk → 0 in C. i

Since B ,→ C we can infer u = 0 in B, thereby contradicting (5.4) since 0 > 0. This finishes the proof. Example 5.3. An instructive counterexample illustrating the injectivity assumption in Lemma 5.2 can be constructed as follows. Let I = [−2, 2] and J = [−1, 1] denote closed intervals. Arzelà-Ascoli indeed yields C 1 (I) ,→,→ C(I) but on the other hand, the continuous

5.2 Towards a Non-cylindrical Version of the Aubin–Lions Lemma

111

embedding C(I) ,→ L2 (J) is not injective, so that we cannot apply Ehrling’s lemma12 to the triple C 1 (I) ,→,→ C(I) ,→ L2 (J): In fact, for every > 0 and for any u ∈ C 1 (I) that vanishes identically on J we would obtain

u ≤ u C 1 (I) . C(I) However, the last inequality cannot hold in general.

5.2. Towards a Non-cylindrical Version of the Aubin–Lions Lemma The proof of the Aubin–Lions lemma demonstrates that uniform boundedness of the time derivatives is used to infer uniform equicontinuity. The interplay of (5.3), the Arzelà-Ascoli theorem and the Ehrling lemma then yields compactness. In the context of non-cylindrical domains or generalized Bochner spaces, respectively, the meaning of "uniform equicontinuity" has to be grasped suitably since the identity (5.1) does not make sense as an abstract equation if the spaces involved explicitly depend on time. The passage to the limit in approximate weak formulations is a typical application of the Aubin–Lions lemma. In this context, the Banach spaces in Lemma 5.1 are determined by the equation under consideration: The spaces B0 and B are typically tied to a priori estimates whereas the space C usually is the dual space of the "spatial part" of the test functions. These considerations motivate the functional setting in the following theorem.

5.2.1. Arzelà–Ascoli-type Argument In this section we assume without explicit mention, that the time-dependent Banach spaces that appear are spaces of functions or fields over time-dependent domains Ω(t). We also assume that these time-dependent domains constitute a non-cylindrical domain in the sense of the previous chapters. The generalized Bochner spaces that will appear are to be understood as generalized Bochner spaces over that non-cylindrical domain as introduced in Chapter 2. Theorem 5.4 (Arzelà–Ascoli-type Argument) ¯ let B0 (t), B(t) and B1 (t) denote reflexive Banach function spaces over Ω(t) For any t ∈ I, with the property ¯ B0 (t) ,→,→ B(t) ,→,→ B1 (t)0 , t ∈ I. Assume that there are isomorphisms ¯ Ft : B1 := B1 (0) → B1 (t), t ∈ I, that are uniformly bounded in the following sense: There exists a constant c0 which is independent of t ∈ I¯ such that for any η ∈ B1 there holds

Ft η ≤ c0 η B1 . B1 (t) 12

Notice that the reflexivity in Lemma 5.2 is only used to provide weak compactness of bounded sets, so that it can be substituted by the Arzelà–Ascoli theorem in our example.

112


Let h : R+ → R+ be continuous at 0 with h(0) = 0 and let (un )n∈N be a bounded sequence in Lp (I, B0 (t)), 1 ≤ p < ∞. Assuming that the evaluation [ un : I¯ → B1 (t)0 , t 7→ un (t), (5.5) t∈I¯

is well-defined for any n ∈ N, we define for t ∈ I¯ and η ∈ B1 the quantities

MB1 (t)0 (un , η, Ft ) := un (t), Ft η B1 (t) , NB1 (t)0 (un , Ft ) :=

sup kηkB1 ≤c−1 0

MB1 (t)0 (un , η, Ft ).

Suppose that sup

sup

n∈N kηkB ≤c−1 0 1

MB1 (t)0 (un , η, Ft ) − MB1 (s)0 (un , η, Fs ) ≤ h |t − s|

(5.6)

¯ Suppose further that there is a constant C > 0 such that holds for any t, s ∈ I.

sup un L∞ (I,B(t)) ≤ C.

(5.7)

Then there is an infinite subset M ⊂ N such that, as k, l ∈ M tend to infinity,

max NB1 (t)0 (uk − ul , Ft ) = max uk (t) − ul (t) B1 (t)0 → 0,

(5.8)

n∈N

t∈I¯

t∈I¯

¯ (uk (t))k∈M is a Cauchy sequence in B1 (t)0 . If there is a which means that for any t ∈ I, constant K > 0 such that (5.9)

sup NB1 (t)0 (un , Ft ) ≤ K

n∈N

is satisfied for almost every t ∈ I then there holds

lim NB1 (·)0 (uk − ul , F· ) Lp (I) = 0. M3k,l→∞

(5.10)

Remark 5.5. By definition, we have for f ∈ B1 (t)0 :

f f , η . = sup 0 t B1 (t) B1 (t) kη t kB1 (t) ≤1

Using the uniform boundedness of the isomorphisms Ft we see

f = sup f , η 0 t B1 (t) B1 (t) kη t kB1 (t) ≤1

=


f , Ft η

B1 (t)

=


MB1 (t)0 (f , η, Ft ) = NB1 (t)0 (f , Ft ).

The pointwise estimate (5.9) enables us to deduce the Cauchy property (5.10) from the assertion (5.8) by means of Lebesgue’s theorem on dominated convergence. On the other hand, condition (5.6) may be seen as the non-cylindrical analogue of (5.1) and (5.2). Therefore, we refer to (5.6) as the "generalized equicontinuity condition". Condition (5.7) in turn is a reformulation of (5.3). In applications however, this condition is guaranteed by a priori estimates as we will see later on.

5.2 Towards a Non-cylindrical Version of the Aubin–Lions Lemma

113

Proof. The proof of Theorem 5.4 proceeds in two steps. In the first step we are going to construct a subsequence (un )n∈M ⊂ (un )n∈N such that the sequence n o un (tk ) | n ∈ M converges for any tk out of a countable and dense subset G ⊂ I in the respective dual space B1 (tk )0 . This construction is a straightforward consequence of condition (5.7). In fact, we notice that by (5.7) there exists a countable and dense subset denoted by G := t1 , t2 , t3 , ... ⊂ I such that

sup max un (tk ) B(t) ≤ C.

n∈N tk ∈G

Exploiting this fact, we can now select a precompact subsequence. First, remember that for every t ∈ I¯ we have a compact embedding B(t) ,→,→ B1 (t)0 . Since

sup un (t1 ) B(t1 ) ≤ C n∈N

0 and 1 ) , there exists a subsequence indexed by N({t1 }) ⊂ N such that B(t1 ) ,→,→ B1 (t un (t1 ) | n ∈ N({t1 }) converges in B1 (t1 )0 . Similarly, for k = 2 there holds

sup un (t2 ) B(t2 ) ≤ C. n∈N({t1 })

The embedding B(t2 ) ,→ B1 (t2 )0 being indexed compact, there exists a further subsequence 0 by N({t1 , t2 }) ⊂ N({t1 }) such that un (t2 ) | n ∈ N({t1 , t2 }) converges in B1 (t2 ) . Repeating this procedure, we eventually get a diagonal subsequence M := N(G) with the property that n o un (tk ) | n ∈ M converges in B1 (tk )0 for any tk ∈ G. In the second step we can now combine the previous compactness result with the generalized continuity condition (5.6) in order to finish the proof of Theorem 5.4. Consider the sequence (un )n∈M constructed in the previous step. Due to the norm formula, see Remark 5.5, we have the identity

uk (t) − ul (t) 0 (uk − ul , η, Ft ) . 0 (uk − ul , Ft ) = M = N sup 0 B (t) B (t) 1 1 B1 (t) kηkB1 ≤c−1 0

¯ Let η ∈ B1 with kηkB1 ≤ c−1 0 and t ∈ I be arbitrary, then MB1 (t)0 (uk − ul , η, Ft ) =MB1 (t)0 (uk , η, Ft ) − MB1 (t)0 (ul , η, Ft ) =MB1 (t)0 (uk , η, Ft ) − MB1 (s)0 (uk , η, Fs ) + MB1 (s)0 (uk , η, Fs ) − MB1 (s)0 (ul , η, Fs ) + MB1 (s)0 (ul , η, Fs ) − MB1 (t)0 (ul , η, Ft ) =:∆k (t, s) + ∆k,l (s) + ∆l (s, t).

(5.11)

114


Owing to (5.6), the terms ∆k (t, s) and ∆l (s, t) can both be estimated by h(|t − s|). Now let > 0 be arbitrary. Since h is continuous at 0 with h(0) = 0, there is δ > 0 such that 0 ≤ τ ≤ δ implies h(τ ) < . Furthermore, as I¯ is compact there are finitely many s1 , s2 , ..., sN () ∈ G such that N ()

I⊂

[

(sm − δ, sm + δ).

m=1

In (5.11) we now choose s ∈ G := { s1 , ..., sN () } in such a way that t ∈ (s − δ, s + δ). The choice of δ then yields MB1 (t)0 (uk − ul , η, t) ≤ 2 + ∆k,l (s) . The sets G are finite for any > 0 and, by the method of construction, (un (s))n∈M is a Cauchy sequence for any s ∈ G ⊂ G. Hence, there exists n0 () such that |∆k,l (s)| ≤ holds uniformly in s for any s in the finite set G , if k, l ≥ n0 (). Altogether, this shows max NB1 (t)0 (uk − ul , Ft ) = max sup MB1 (t)0 (uk − ul , η, Ft ) ≤ 3 t∈I¯

t∈I¯ kηk ≤c−1 B1 0

for k, l ≥ n0 (). This proves (5.8). The assertion (5.10) then follows from Lebesgue’s theorem on dominated convergence as we already pointed out in Remark 5.5.

5.2.2. Ehrling-type Estimates In order to get compactness in a smaller space we need to invoke Ehrling’s lemma: If i

B0 ,→,→ B ,→ C, then for any > 0 there exists a constant c such that for any u, v ∈ B0 there holds

u − v ≤ u − v + c u − v . B B0 C However, it is not obvious whether c can be chosen independently of the time parameter if the Banach spaces B0 , B and C are to be replaced by time-dependent triples. Definition 5.6 (Uniform Ehrling Property) Suppose that the Banach spaces B0 (t), B(t) and C(t) satisfy i

B0 (t) ,→,→ B(t) ,→ C(t) for any t ∈ I¯ where B0 (t) t∈I¯ and C(t) t∈I¯ are additionally supposed to be reflexive. We say that the triples B0 (t), B(t), C(t) t∈I¯ have the uniform Ehrling property, if for any > 0 there exists a constant c > 0 such that for any t ∈ I¯ and for any α ∈ B0 (t) there holds

α ≤ α B0 (t) + c α C(t) . B(t)

115

5.3 The Non-cylindrical Aubin–Lions Compactness Lemma

The next lemma formulates a condition on the triples B0 (t), B(t), C(t) t∈I¯ which induces the validity of the uniform Ehrling property. In the context of problems on non-cylindrical domains, a condition of this type is fulfilled if time-dependent isomorphisms between the respective spaces can be controlled in a uniform way. At the end of this chapter, we will work out a tangible example of function spaces having the uniform Ehrling property. Lemma 5.7 Suppose that the Banach spaces B0 (t), B(t) and C(t) satisfy for any t ∈ I¯ i

B0 (t) ,→,→ B(t) ,→ C(t) and that B0 (t) t∈I¯ and C(t) t∈I¯ are reflexive. Furthermore, we assume that the triples B0 (t), B(t), C(t) satisfy the following "uniform squeezing property": For any sequence t∈I¯ S (αn )n∈N ⊂ tn ∈I¯ B0 (tn ), αn ∈ B0 (tn ),

sup αn B0 (tn ) ≤ c and

n∈N

lim αn C(tn ) = 0 imply

n→∞

lim αnk B(tn

nk →∞

at least for a subsequence nk → ∞. Then, the triples B0 (t), B(t), C(t) Ehrling property.

t∈I¯

k

)

=0

have the uniform

Proof. If the assertion is false, there exists 0 > 0 and a sequence (α ˜ n )n∈N ⊂ α ˜ n ∈ B0 (tn ), such that

S

tn ∈I¯ B0 (tn ),

α ˜ n B(tn ) > o α ˜ n B0 (tn ) + n α ˜ n C(tn ) . Considering the sequence (αn )n∈N :=

α ˜n kα ˜ n kB(tn ) n∈N ,

αn = 1, B(tn )

(5.12)

(5.13)

the latter inequality yields

sup αn B0 (tn ) ≤ 1/0 .

(5.14)

n∈N

However, inequality (5.13) also shows limn→∞ kαn kC(tn ) = 0. Together with (5.14) this in turn implies limnk →∞ kαnk kB(tnk ) = 0 by the uniform squeezing property (5.12), thereby contradicting the fact that

1 = lim αn B(tn ) . n→∞

This yields the claim.

5.3. The Non-cylindrical Aubin–Lions Compactness Lemma Just like in the proof of the Aubin–Lions lemma, we can combine the Arzelà-Ascoli argument from Theorem 5.4 and the uniform interpolation inequality from Definition 5.6 to obtain a generalized version of the Aubin–Lions compactness lemma. This version is adapted to problems that are formulated in the framework of generalized Bochner spaces.

116


Lemma 5.8 (Aubin–Lions Lemma in Generalized Bochner Spaces) ¯ let B0 (t), B(t) and B1 (t) denote reflexive Banach function spaces over Ω(t) For any t ∈ I, with the properties i ¯ B0 (t) ,→,→ B(t) ,→,→ B1 (t)0 and B(t) ,→ B1 (t)0 , t ∈ I.

Suppose that the triples B0 (t), B(t), B1 (t)0 t∈I¯ have the uniform Ehrling property from Definition 5.6 and that there are uniformly bounded isomorphisms Ft : B1 = B1 (0) → B1 (t), ¯ Let furthermore (un )n∈N be a bounded sequence in Lp (I, B0 (t)), 1 ≤ p < ∞, that t ∈ I. satisfies conditions (5.5), (5.6) and (5.7) as well as condition (5.9). Then, the subsequence (un )n∈M constructed in Theorem 5.4 is a Cauchy sequence in Lp (I, B(t)). Proof. It follows from the uniform Ehrling property that for any > 0 there exists a constant c > 0 such that for any n, k ∈ M there holds

un − uk p ≤ un − uk Lp (I,B0 (t)) + c NB1 (·)0 (un − uk , F· ) Lp (I) . L (I,B(t)) Since N (un , ·) n∈M is a Cauchy sequence in Lp (I) by Theorem 5.4, the latter estimate implies that (un )n∈M is a Cauchy sequence in Lp (I, B(t)).

5.4. Uniform Ehrling Property of the Basic Function Spaces We now present an example that will be important for the application of the generalized Aubin–Lions lemma in the context of non-Newtonian fluids in non-cylindrical domains. Suppose that, for some open, bounded domain Ω0 ⊂ Rd , d ≥ 2, with Lipschitz boundary, Q=

[

{t} × Ω(t) :=

t∈I

[

{t} × ϕt (Ω0 )

t∈I

is a non-cylindrical domain, generated by the smooth flow (ϕt )t∈I¯ with smooth inverse (ψt )t∈I¯. On this non-cylindrical domain, we consider for any t ∈ I¯ the following reflexive Banach spaces we had already used in connection with the p-Stokes system: 1,p ∞ (Ω(t))d Vσ (t) := W0,σ (Ω(t))d = C0,σ ∞ (Ω(t))d Hσ (t) := C0,σ

k·kL2 (Ω(t))

k∇·kLp (Ω(t))

, 2
0 there exists a constant c > 0 such that for any t ∈ I¯ and for any u ∈ S t∈I¯ Vσ (t), u(t) ∈ Vσ (t), there holds

NHσ (t) (u(t), Pψt ) ≤ u(t) Vσ (t) + c NVσ (t)0 (u(t), Pψt ). Proof. Assuming the contrary, there exists 0 > 0 and sequences (tn )n∈N ⊂ I¯ and (un )n∈N ⊂ S tn ∈I¯ Vσ (tn ), un ∈ Vσ (tn ), such that

NHσ (tn ) (un , Pψtn ) ≥ 0 un Vσ (tn ) + n NVσ (tn )0 (un , Pψtn ). (5.15) 13

Note that Vσ (t), Hσ (t), Vσ (t)0 is a family of Gelfand triples so that we consequently identify the Hilbert space Hσ (t) with its dual space Hσ (t)0 .

118


Without loss of generality we may assume

un =1 Vσ (tn ) for any n ∈ N. By uniform boundedness of the isomorphisms Pϕtn : Vσ (tn ) → Vσ (0), the sequence (Un )n∈N ⊂ Vσ (0), Un := Pϕtn un , is uniformly bounded. By compactness of the embedding Vσ (0) ,→ Hσ (0), there exists a joint subsequence, again indexed by n ∈ N, U ∈ Vσ (0) and t ∈ I¯ such that ¯ tn → t in I, Un * U in Vσ (0), Un → U in Hσ (0), as n tends to infinity. Extending the fields (un )n∈N and (Un )n∈N by zero to Rd we will show that lim un = lim Pψtn Un = u := Pψt U in L2 (Rd )d .

n→∞

n→∞

(5.16)

∞ (Ω )d To see this, first note that for U ∈ Hσ (0) and any > 0 there exists U ∈ C0,σ 0 such that kU − U kL2 (Rd ) ≤ /3. Moreover, using the triangle inequality and the uniform boundedness of the isomorphism we can estimate

un − u 2 d = Pψ Un − Pψ U 2 d t tn L (R ) L (R )

≤ Pψtn (Un − U) L2 (Rd ) + Pψtn (U − U) L2 (Rd )

+ Pψt (U − U) L2 (Rd ) + Pψtn U − Pψt U L2 (Rd )

≤ c Un − U L2 (Rd ) + U − U L2 (Rd ) + Pψtn U − Pψt U L2 (Rd ) .

From this we infer kun − ukL2 (Rd ) ≤ if n is sufficiently large. In fact, Tϕtn uniformly converges to Tϕt and U ◦ ψtn uniformly converges to U ◦ ψt due to the smoothness of U and the smoothness of the flows. The transformation formula therefore implies

Pψ U − Pψ U 2 d ≤ /3 tn t L (R ) for large n. Now (5.16) will help us to prove lim NHσ (tn ) (un , Pψtn ) = NHσ (t) (u, Pψt ).

n→∞

In fact, for any η ∈ Hσ (0) with kηkHσ (0) ≤ c−1 0 we have MHσ (tn ) (un , η, Pψtn ) − MHσ (t) (u, η, Pψt ) = un , Pψtn η L2 (Ω(tn )) − u, Pψt η L2 (Ω(t))

≤ un − u L2 (Rd ) Pψtn η Hσ (tn ) + u W 1,p (Rd ) Pψtn η − Pψt η (W 1,p (Rd ))0 , so that (5.17) follows if we manage to prove that

lim Pψtn η − Pψt η (W 1,p (Rd ))0 = 0 n→∞

(5.17)

5.4 Uniform Ehrling Property of the Basic Function Spaces

119

holds uniformly in η ∈ Hσ (0) with kηkHσ (0) ≤ c−1 0 . If the latter assertion is false, there exists δ > 0 and a sequence (η n )n∈N ⊂ Hσ (0) with supn∈N kη n kHσ (0) ≤ c−1 0 such that

Pψ η n − Pψ η n 1,p d 0 > δ. tn t (W (R ))

(5.18)

However, we then find a further (not relabeled) subsequence (η n )n∈N and η ∈ Hσ (0) such that as n → ∞ η n * η in Hσ (0). Moreover, by the triangle inequality we also get

Pψ η n − Pψ η n 1,p d 0 ≤ Pψ η n − Pψ η 1,p d 0 + Pψ η − Pψ η n 1,p d 0 . tn t t t t t n n n (W (R )) (W (R )) (W (R )) For any g ∈ L2 (Rd )d the transformation formula shows Pψtn η n − Pψtn η, g L2 (Rd ) = η n − η, Pψ0 tn g L2 (Ω0 ) .

(5.19)

Approximating g by smooth functions similarly as above one can again show the strong convergence limn→∞ Pψ0 tn g = Pψ0 t g in L2 (Ω0 )d . Thus, from (5.19) we infer Pψtn η n − Pψtn η * 0 in L2 (Rd )d as n → ∞. The compact embedding W 1,p (Rd )d ,→,→ L2 (Rd )d in turn implies the compact 0 embedding L2 (Rd )d ,→,→ W 1,p (Rd )d by Schauder’s theorem above, so that actually

lim Pψtn η n − Pψtn η (W 1,p (Rd ))0 = 0. n→∞

Similar arguments also yield that

lim Pψtn η − Pψt η n (W 1,p (Rd ))0 = 0, n→∞

thereby contradicting (5.18). Going back to (5.15), essentially the same reasoning implies lim NVσ (tn )0 (un , Pψtn ) = NVσ (t)0 (u, Pψt ),

n→∞

but since the left-hand side of (5.15) is bounded we must have lim NVσ (tn )0 (un , Pψtn ) = 0.

n→∞

However, since the embedding Vσ (0) ,→ Hσ (0) is dense, we then obtain lim NHσ (tn ) (un , Pψtn ) = 0,

n→∞

which ultimately contradicts (5.15) since 0 > 0. This finishes the proof.

120


6. Shear Thickening Fluids in Non-cylindrical Domains 6.1. Motivation and Model In Chapter 4 we proved an existence result concerning Newtonian fluids in a time-dependent domain. There, the constitutive law relating the Cauchy stress tensor14 S to the rate of deformation tensor Du was given by S(Du) = 2 σ Du. In many modern applications though, such linear dependence can hardly be justified. This is why non-Newtonian fluid models have become increasingly important in recent years. For example, magma, lava and quicksand are just three examples for non-Newtonian fluids in geophysics. In chemical engineering, certain polymers and emulsions exhibit a non-Newtonian behavior as well. Another important field for non-Newtonian fluid models in modern life sciences is hemodynamics, that is, the study of blood flow in the human cardiovascular system. Apparently, blood is a highly heterogeneous mixture of platelets and red and white blood cells suspended in liquid plasma consisting of numerous components itself. While the plasma is generally considered a Newtonian fluid, the non-Newtonian character of blood as a whole is mainly due to particular mechanical properties of the red blood cells. Depicting the distinguished rheological properties of blood in adequate constitutive relations leads to highly nonlinear equations governing the dynamics of blood flow. We refer to the thorough introduction to hemodynamics in [FQV09] and the references therein. However, for a realistic modelling of blood flow through an artery it is indispensable to take also into account the coupling between the fluid and the artery caused by elastic effects on the artery wall. The coupling is established through boundary conditions for the respective systems of PDEs. In order to treat the resulting system one has to solve the fluid equations in time-dependent domains. As the solution of the fluid equations enters the dynamics of the artery through the coupled boundary conditions, the time-dependent domains themselves are unknowns. Such a problem is usually circumvented by applying a fixed-point procedure in order to decouple the vessel dynamics from the dynamics of the fluid. But still, one is forced to solve, as a subproblem, the nonlinear fluid equations on a non-cylindrical domain Q ⊂ R4 with moving boundary Γ. For convenience, we assume Q to be generated by the flow of a smooth, time-dependent and incompressible vector field v : R × R3 → R3 with compact support in R3 . The precise definition of Q and Γ is already given in (4.24) and (4.25) in Chapter 4. The smoothness assumptions on v are not very realistic from the mathematical point of view as the solutions of the structure equations are merely continuous in general. 14

The terms "Cauchy stress tensor" and "stress tensor" will be used synonymously.

121

122

Chapter 6. Shear Thickening Fluids in Non-cylindrical Domains

However, by regularizing the structure equations through adding appropriate damping terms one can gain a certain amount of smoothness, see for instance [CDEG05]. It is also possible to smooth the solution of the structure equation directly. For this approach we refer to [LR14]. Our setting may therefore serve as a first approximation for a subproblem of a fully coupled FSI problem for a non-Newtonian fluid. The problem takes the following form: We look for a fluid velocity field u and a pressure π such that ∂t u + u · ∇ u = f + div S(Du) − π idR3 in Q, div u = 0

in Q,

u=v

on Γ,

u(0) = u0

(6.1)

in Ω0 .

Note that the Dirichlet condition (6.1)3 means that the fluid particles on the boundary move according to the velocity of the boundary. We will come back to this point. As we already explained, the interpretation of blood as a non-Newtonian fluid amounts to a nonlinear dependence of S on Du. Since there is a vast diversity of constitutive models, we just consider two commonly used constitutive relations, also known as power-law ansatz models: p−2 S(Du) := µ0 δ + |Du| Du, (6.2) p−2 S(Du) := µ0 δ + |Du|2 2 Du. Here, µ0 > 0, δ ≥ 0 and 1 < p < ∞ are constants. Note that for p = 2, both relations in (6.2) reduce to the well-known Navier–Stokes model we treated in Chapter 4. However, the latter relations can also be seen as two examples of the even broader class of Cauchy stress tensors with (p, δ)-structure which we will now introduce. The definition follows the one given in [Růž13]. Definition 6.1 ((p, δ)-Structure) d×d We say that a stress tensor S ∈ C 0 (Rd×d sym , Rsym ) with S(0) = 0 has (p, δ)-structure if there exist 1 < p < ∞, δ ≥ 0 and constants c0 , c1 > 0 such that

p−2 S(A) − S(C) : A − C ≥ c0 δ + |C| + |A − C| |A − C|2 , p−2 |S(A) − S(C)| ≤ c1 δ + |C| + |A − C| |A − C|.

(6.3)

holds for any A, C ∈ Rd×d sym . The next lemma summarizes the fact that stress tensors having (p, δ)-structure induce monotone, coercive operators. Therefore, the physical models for this class of non-Newtonian fluids are accessible to the theory of monotone operators. Lemma 6.2 Any stress tensor having (p, δ)-structure induces a (strictly) monotone operator on suitable function spaces. Moreover, there exist constants c˜0 , c˜1 > 0 and non-negative functions g0 ∈

123

6.1 Motivation and Model 0

d×d L1 (Q) and g1 ∈ Lp (Q) such that we have for any A ∈ Rsym

S(A) : A ≥ c˜0 |A|p − g0 |S(A)| ≤ c˜1 |A|p−1 + g1 . Proof. The (strict) monotonicity is a direct consequence of (6.3)1 . If p ≥ 2, the coercivity of S follows from the fact that x 7→ xα is increasing for α > 0. Also, plugging C = 0 into (6.3)2 we recover the standard growth condition |S(A)| ≤ c1 |A|p−1 . Similar arguments also show the assertion for the case 1 < p < 2 We henceforth assume that our Cauchy stress tensor S has (p, δ)-structure for some δ ≥ 0 and an exponent p to be fixed below. We now come to define the functional setting and the weak formulation of problem (6.1). With regard to the non-homogeneous boundary condition (6.1)3 and the dependence of S on the symmetric gradient Du we set for any t ∈ I¯ 1,p Xσ (t) := Wσ,s (Ω(t))3 := Cσ∞ (Ω(t))3 1,p ∞ (Ω(t))3 Vσ (t) := W0,σ (Ω(t))3 = C0,σ

k·k

1,p Ws (Ω(t))

k∇·kLp (Ω(t))

H(t) := L2 (Ω(t))3 Yσ (t) := Cσ∞ (Ω(t))3

k·kH(t)

Here, the superscript s refers to the symmetric gradient. Thus, the norm on Xσ (t) is given by ku(t)kXσ (t) := ku(t)kLp (Ω(t)) + kDu(t)kLp (Ω(t)) . The fact that this quantity induces an equivalent norm on the Sobolev space W 1,p (Ω(t))3 is guaranteed by Korn’s second inequality which is valid if the domain has a sufficiently smooth boundary. The following result can be found in [Fuc94]. Lemma 6.3 (Korn Inequalities) If Ω ⊂ Rd is a bounded domain with Lipschitz boundary and if 1 < p < ∞, then there exists a constant c that only depends on Ω such that

u 1,p ≤ c u p + Du p W

(Ω)

L (Ω)

L (Ω)

holds for any u ∈ W 1,p (Ω)d . This inequality is referred to as Korn’s second inequality. Moreover, there is a constant c = c(Ω) such that for any u ∈ W01,p (Ω)d Korn’s (first) inequality holds, i.e.

∇u p ≤ c Du Lp (Ω) . L (Ω) The Cauchy stress tensor gives rise to a bounded and continuous operator S : Xσ (t) → Xσ (t)0 Z

S(u(t)), η(t) Xσ (t) := S(Du(t)) : Dη(t) dx. Ω(t)

(6.4)

124


With a slight abuse of notation, we extend S to a bounded and continuous mapping 0 S : Lp (I, Xσ (t)) → Lp (I, Xσ (t)) Z

S(u(t)), η(t) Xσ (t) dt. Su, η := I

Note that due to Korn’s inequality, kD · kLp (Ω(t)) is an equivalent norm on Vσ (t). Hence, the (p, δ)-structure of S implies that the restriction of S to Vσ (t) yields a bounded, continuous and coercive operator S : Vσ (t) → Vσ (t)0 . This restriction is again defined by (6.4)2 . As a consequence,S then induces a bounded, continuous and coercive operator S : Lp (I, Vσ (t)) 0 → 0 Lp (I, Vσ (t)) . Of course we can analogously define S : Lp (I, Xσ (t)) → Lp (I, Vσ (t)) and 0 S : Lp (I, Vσ (t)) → Lp (I, Xσ (t)) . The nonlinear convective term is represented by 0 G : L∞ (I, Yσ (t)) ∩ Lp (I, Xσ (t)) → Lp (I, Vσ (t)) Z Z Z

Gu, η := g(u(t)), η(t) Vσ (t) dt := − u(t) ⊗ u(t) : ∇η(t) dxdt. I

(6.5)

I Ω(t)

To make sure that G is well-defined and bounded one generally has to restrict the possible values of p. However, in the three-dimensional case we are interested in, it is sufficient to choose the lower bound p≥

11 . 5

(6.6)

We would like to refer to [MNRR96, Lemma 2.44] for a general discussion of integrability results concerning the convective term. The justification for our choice of the lower bound for the exponent p can also be found there. If the stress tensor has (p, δ)-structure for some exponent p > 2, the fluid is called shear thickening or dilatant. By definition, a shear thickening fluid increases its viscosity if shear stresses are applied. A popular example for shear thickening behavior is given by mixing corn starch and water in a certain ratio. The interested reader can find numerous recipes and videos about these mixtures on the internet. A more serious application are specially designed shock absorbers and brakes exploiting the mechanical properties of dilatant brake fluids. Although blood is a shear thinning fluid, we treat shear thickening fluids first because this model will be used to approximate the shear thinning case. The details are presented in the next chapter. Furthermore, the lower bound p ≥ 11 5 , which is necessary for the framework of monotone operators, restricts this theory to shear thickening fluids. Definition 6.4 (Weak Solution of 6.1) 0

Given f ∈ Lp (Q)3 and u0 ∈ Xσ (0) satisfying the compatibility condition u0 − v(0) ∈ Vσ (0),

6.2 Nonhomogeneous Boundary Conditions and a Pitfall in the Penalty Method 125

a vector field u ∈ L∞ (I, Yσ (t)) ∩ Lp (I, Xσ (t)) is called a weak solution of (6.1) if Z Z Z

− u(t), ∂t η(t) H(t) dt+ g(u(t)), η(t) Vσ (t) dt + Su(t), η(t) Vσ (t) dt I

I

I

Z =

f (t), η(t) Vσ (t) dt + u0 , η(0) H(0)

I

holds for any η in the space of test functions which is given by n o X = ϕ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) ϕ(T, ·) = 0 . Furthermore we require for almost every t ∈ I that u(t) − v(t) ∈ Vσ (t). Remark 6.5. For u ∈ Lp (I, Xσ (t)), the mapping trΓ : Lp (I, Xσ (t)) → L2 (I, L2 (Γ(t))3 ) trΓ u (t) := trΓ(t) u(t) := u(t)|Γ(t) is well-defined for almost every t ∈ I, owing to the continuous embedding Xσ (t) ,→ L2 (Γ(t))3 and the fact that p > 2.

6.2. Nonhomogeneous Boundary Conditions and a Pitfall in the Penalty Method 6.2.1. Lacking Energy Identity Before we come to the construction of weak solutions of (6.1) in the sense of Definition 6.4, we want to work out a number of characteristic difficulties connected to the nonhomogeneous Dirichlet condition (6.1)3 . In the course of these considerations we also hope to justify the choice of our definition of weak solutions. A severe drawback is the lack of an energy identity, even on a formal level. To make things clearer, let us assume that u is a sufficiently smooth solution of (6.1). Taking the scalar product of (6.1)1 and u and integrating the result over Ω(t) we obtain Z Z Z ∂t u(t) · u(t) dx + u(t) · ∇ u(t) · u(t) dx − div S(Du(t)) − π(t) idR3 · u(t) dx Ω(t)

Ω(t)

Ω(t)

Z f (t) · u(t) dx.

= Ω(t)

126


Applying the transport theorem, we may integrate by parts in the integral containing the time derivative to infer Z Z 1 ∂t u(t) · u(t) dx = ∂t |u(t)|2 dx 2 Ω(t)

Ω(t)

Z

1d = 2 dt

1 |u(t)| dx − 2 2

Ω(t)

Z

1d = 2 dt

Z

|u(t)|2 v(t) · n(t) da

∂Ω(t)

1 |u(t)| dx − 2 2

Ω(t)

Z

|v(t)|2 v(t) · n(t) da.

∂Ω(t)

Integrating by parts in the convective term yields Z Z Z u(t) · ∇ u(t) · u(t) dx + u(t) · ∇ u(t) · u(t) dx = − ∂Ω(t)

Ω(t)

Ω(t)

|u(t)|2 u(t) · n(t) da.

Due to the inhomogeneous boundary condition (6.1)3 we see that Z Z 1 u(t) · ∇ u(t) · u(t) dx = |v(t)|2 v(t) · n(t) da, 2 Ω(t)

∂Ω(t)

and as a result Z

Z ∂t u(t) · u(t) dx +

Ω(t)

1 d u(t) · ∇ u(t) · u(t) dx = 2 dt

Z

|u(t)|2 dx.

Ω(t)

Ω(t)

We note that in case of time-dependent domains, the convective term is essential to absorb the additional boundary integral induced by the transport theorem. The remaining integral is a typical "Gronwall term" that can easily be handled. Integrating by parts also in the term containing the stress tensor we get Z Z Z − div S(Du(t)) − π(t) idR3 · u(t) dx = S(Du(t)) : Du(t) dx + π(t) div u(t) dx Ω(t)

Ω(t)

Ω(t)

Z −

S(Du(t))n(t) − π(t)n(t) · u(t) da

∂Ω(t)

Due to the incompressibility constraint (6.1)2 the pressure integral in the last identity is equal to zero and therefore Z Z − div S(Du(t)) − π(t) idR3 · u(t) dx = S(Du(t)) : Du(t) dx Ω(t)

Ω(t)

Z − ∂Ω(t)

S(Du(t))n(t) − π(t)n(t) · u(t) da.


In conclusion, we obtain that any sufficiently smooth solution of the system (6.1) satisfies Z Z Z 1 d |u(t)|2 dx + S(Du(t)) : Du(t) dx = f (t) · u(t) dx (6.7) 2 dt Ω(t)

Ω(t)

Ω(t)

Z +

S(Du(t))n(t) − π(t)n(t) · u(t) da.

∂Ω(t)

However, the last identity reveals a severe discrepancy between the potential regularity of weak solutions of (6.1) and the regularity that is necessary to keep the integrands in (6.7) well-defined at all. On the one hand, the evaluation of S(Du(t)) and π(t) on ∂Ω(t) requires u(t) and π(t) to be at least in W 2,1 (Ω(t))3 and in W 1,1 (Ω(t)), respectively. On the other hand, even in the case of homogeneous Dirichlet boundary conditions, the left-hand side of (6.7) at most yields W 1,p -estimates for u(t). Moreover, there is no chance to get any estimate for the pressure from (6.7). As a consequence, we cannot expect to prove existence of weak solutions of (6.1) that satisfy an energy equality similar to (6.7). Note that this fact is in sharp contrast to the homogeneous Dirichlet case where weak solutions can additionally be characterized by energy (in)equalities. These energy (in)equalities are of course due to the fact that one may test the system with the solution itself without producing boundary integrals. Note also that Definition 6.4 circumvents these issues by using test functions that vanish on the moving boundary. From the physical point of view, the lack of the energy equality for solutions of (6.1) is obvious: There is no additional condition that balances the energy that is dissipated across the boundary of ∂Ω(t), which is why the integral Z S(Du(t))n(t) − π(t)n(t) · u(t) da ∂Ω(t)

does not cancel. Things are quite different in the context of genuine FSI problems. For example, imagine a shear thickening fluid that occupies the volume enclosed by an elastic, impermeable material. In that case, the nonhomogeneous Dirichlet condition (6.1)3 on the elastic interface is such that the fluid velocity matches the velocity of the structure. The definite formulation of this condition depends on several aspects and can hardly be cast in a unified form which is why our discussion remains informal. The first point to be noted is that the displacement of the ˆ : I × ∂Ω(0) → Rd is usually formulated on a fixed reference domain, whereas the structure η fluid velocity is defined in Ω(t), i.e. the domain enclosed by the deformed reference domain at time t. Therefore, these different domains first have to be related suitably. Moreover, the character of the displacement and especially its direction depend on the nature of the elastic material under consideration. For example, elastic shells behave differently than elastic plates. As the mathematical modelling of elastic structures is a highly complex matter in itself we refer to [Cia98], [Cia97], [Cia00] and [MH94] for a comprehensive exposition. If there is no interpenetration, the coupling between the fluid and the solid has to be complemented by a continuity condition on the stresses exerted to the separating interface by the fluid on the one hand and by the solid on the other hand. This point is important, since the two coupling conditions then typically yield global energy identities for FSI problems.

128


Note however, that another equation for the solid part of the interaction problem enters the model. Though, the coupled system does not become overdetermined as would be the case if we added another boundary condition for the stress tensor to (6.1). As we do not want to get into further technical details of these exceptionally complex systems, we draw attention to [LR14] and [Gra08b] and the references therein for deeper insight. A detailed introduction to FSI problems in the context of hemodynamics can be found in [FQV09]. The previous considerations about the system (6.1) give reason to assume that the construction of weak solutions in the sense of Definition 6.4 cannot be established without any further approximation that compensates for the lacking energy identity. An obvious approach to deal with the nonhomogeneous boundary condition (6.1)3 is given by a so-called fictitious domain approximation. This method was initiated by Peskin in [Pes73] in a numerical study of blood flow in and around heart valves. Later on, Glowinski and his coworkers refined it to study flow problems subject to different boundary conditions. See for instance [WPG10] and [DGP03]. A variant of this method has also been successfully applied by Angot, see [Ang10] and [Ang05] and the references therein, in connection with stationary flows in a bounded domain Ω ⊂ Rd around an obstacle ω ⊂⊂ Ω. Roughly speaking, it consists of adding an appropriate penalty term which enforces the right boundary values on the obstacle in the limit. Thereby, the new computational domain, i.e. the fictitious domain, is given by Ω rather than by Ω \ ω ¯ . If homogeneous boundary conditions are imposed on the boundary ∂Ω, the penalized system then allows for energy identities, at least on this level of approximation. Utilizing such a specific penalization procedure for the problem (6.1) we want to draw attention to a subtle pitfall in connection with the Aubin–Lions lemma that may present a major source of error.

6.2.2. Non-Applicability of the Classical Aubin–Lions Lemma Since the vector field v that generates the non-cylindrical domain is supposed to be smooth, incompressible and compactly supported, there exists a subset U ⊂ R3 such that Ω(t) ⊂⊂ U holds uniformly in t ∈ I. Without loss of generality we can assume that U is a sufficiently large ball in R3 . In order to define the penalized problems on the cylindrical domain I × U we introduce yet another set of function spaces. We set

∞ (U )3 H(U ) := L2 (U )3 , Hσ (U ) := C0,σ

k·kH(U )

,

1,p K(t) := L2 (U \ Ω(t))3 , Vσ (U ) := W0,σ (U )3 .

and equip these spaces with their canonical norms. Extending f by zero to I × U and u0 0 by v(0) to U , we obtain, with a slight abuse of notation, f ∈ Lp (I × U )3 and u0 ∈ Hσ (U ). Then, denoting by χQc the characteristic function of the complement of Q in I × U , the


following system represents the penalized form of (6.1): ∂t u + u · ∇ u + n χQc (u − v) = f + div S(Du) − π idR3

in I × U, in I × U,

div u = 0

on I × ∂U,

u=0

(6.8)

in U.

u(0) = u0 Definition 6.6

A vector field un ∈ L∞ (I, Hσ (U )) ∩ Lp (I, Vσ (U )) is a weak solution of the penalized problem (6.8) if15 Z Z Z

S(un (t)), η(t) Vσ (U ) dt g(un (t)), η(t) Vσ (U ) dt + − un (t), ∂t η(t) H(U ) dt + I

I

I

Z =

f (t), η(t) Vσ (U ) dt − n

I

holds for any η ∈

Z

(6.9) un (t) − v(t), η(t) K(t) dt + u0 , η(0) H(0)

I

H 1 (I, H(U ))

∩

Lp (I, V

σ (U ))

with η(T, ·) = 0.

Existence of solutions to (6.9) is obtained by means of the classical theory of monotone operators. For the sake of clarity though, we recall the most important steps in some detail. To begin with, the system is transformed into a finite dimensional dynamical system via Galerkin approximation. As it turns out, the right choice of the set of ansatz functions thereby plays a decisive role. The actual basis is formed by eigenvectors of the Stokes k,2 operator on the space W0,σ (U )3 ,→ Vσ (U ), where k ∈ N is sufficiently large. Apart from k,2 that, this basis can be chosen to be orthogonal in W0,σ (U )3 and orthonormal in H(U ). It follows that the projections onto finite dimensional spaces spanned by these vectors are k,2 (U )3 and in H(U ). uniformly bounded in W0,σ The local solvability of the resulting Galerkin system then follows from ODE theory. Global a priori estimates subsequently imply long-time existence of the Galerkin solutions. These estimates take the familiar form

(6.10) max uln (t) Hσ (U ) + uln Lp (I,Vσ (U )) ≤ K(n). t∈I¯

Here, l ∈ N denotes the approximation parameter of the Galerkin scheme whereas uln stands for the corresponding Galerkin solution to the penalization level n ∈ N. Note that the constant K(n) depends on n through the penalty term but is independent of l. The above estimates are obtained by testing the Galerkin system with the Galerkin solution itself. In doing so, we again use the fact that the convective term vanishes. Furthermore, one has to apply a Gronwall-type argument to treat the penalty term. Another important tool in this context is Korn’s first inequality for Sobolev fields that vanish on the boundary of U :

∇u p ≤ c Du Lp (U ) . L (U ) 15

The operators g and S denote the natural extensions to the cylindrical spaces of the operators we had introduced above.

130


The latter inequality enables us to obtain boundedness in Lp (I, Vσ (U )) from the estimate Z Z

p

p

l l S(un (t)), un (t) Vσ (U ) dt ≥ c Duln (t) Lp (U ) dt − K0 ≥ c˜ uln Lp (I,Vσ (U )) − K0 . I

I

Here, we also make use of the (p, δ)-structure of the Cauchy stress tensor, see Lemma 6.2. Exploiting the uniform boundedness of the projections together with the fact that they commute with the time derivative, see also (4.8), we furthermore obtain

dul sup n dt l∈N

(6.11)

0 ≤ c(n).

k,2 (U )3 ) Lp (I,W0,σ

Again, the constant c(n) depends on n and the data but is independent of l. By weak compactness, (6.10) yields a subsequence such that for l → ∞ uln * un in Lp (I, Vσ (U )), ∗

uln * un in L∞ (I, Hσ (U )), uln * un in L2 (I, Hσ (U )), uln (0) → u0 in Hσ (U ), 0

Suln * χn in Lp (I × U )3×3 . The last convergence is due to the fact that the stress tensor induces a bounded mapping 0 from Lp (I, Vσ (U )) to Lp (I × U )3×3 by virtue of its (p, δ)-structure. Moreover, we have uln → un in L2 (I × U )3 as l → ∞ by the classical Aubin–Lions lemma, see Lemma 5.1. It can be applied thanks to the estimates (6.10) and (6.11) and the embeddings 0 i k,2 Vσ (U ) ,→,→ Hσ (U ) ,→ W0,σ (U )3 . Note that this compactness result is necessary for the limiting process in the convective term. Thus, passing to the limit in the Galerkin scheme along a suitable subsequence, we obtain that the weak limit un ∈ L∞ (I, Hσ (U )) ∩ Lp (I, Vσ (U )) satisfies Z Z Z

− un (t), ∂t η(t) H(U ) dt + g(un (t)), η(t) Vσ (U ) dt + χn (t), η(t) Vσ (U ) dt I

I

Z =

f (t), η(t) Vσ (t) dt − n

I

I

Z

un (t) − v(t), η(t) K(t) dt + u0 , η(0) H(U ) ,

I

for any test function η vanishing at t = T . However, this identity also implies 0 dun p ∈ L (I, Vσ (U )) dt for any fixed n ∈ N. Since Vσ (U ) ,→ Hσ (U ) ,→ Vσ (U )0

(6.12)


forms a Gelfand triple, we know from Lemma 3.9 that every un is continuous as a function with values in Hσ (U ) and satisfies for every s ∈ I¯ the energy equality Zs Zs

2

1 1

un (s) 2

− u0 Hσ (U ) = f (t), un (t) Vσ (U ) dt − χn (t), un (t) Vσ (U ) dt H (U ) σ 2 2 0

0

Zs

un (t) − v(t), un (t) K(t) dt.

−n 0

The latter identity can now be used to obtain the classical pseudomonotonicity estimate

lim sup Suln , uln ≤ χn , un . l→∞

A subsequent application of Minty’s trick finishes the proof since we then have 0

χn = Sun in Lp (I × U )3×3 .

The fact that un solves the penalized problem can be expressed equivalently as 0 dun + Sun + Gun + Pn un = f in Lp (I, Vσ (U )) . (6.13) dt Here Pn refers to the operator induced by the penalty term. Restricting the set of admissible test functions in (6.13) allows us to pass from the auxiliary cylindrical problem to a noncylindrical approximate system. As a consequence we deduce the following corollary which provides a sequence of approximate solutions in the sense of Definition 6.4. However note that the specific boundary values of un on Γ do not enter as the test functions vanish on Γ anyway. For the same reason, the evaluation of the stress tensor on the boundary Γ does not contribute either. Corollary 6.7 The zero extension of η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) is an admissible test function in (6.13). Thus, every solution un ∈ L∞ (I, Hσ (U ))∩Lp (I, Vσ (U )) of the penalized system (6.9) satisfies Z Z Z

− un (t), ∂t η(t) H(t) dt+ g(un (t)), η(t) Vσ (t) dt + S(un (t)), η(t) Vσ (t) dt I

I

I

Z

=

(6.14)

f (t), η(t) Vσ (t) dt + u0 , η(0) H(0)

I

¯ for any η ∈ ∩ σ (t)) with η(T ) = 0. Furthermore, for every s ∈ I and 1 p every test function η ∈ H (I, H(t)) ∩ L (I, Vσ (t)) there holds Zs un (s), η(s) H(s) − u0 , η(0) H(0) − un (t), ∂t η(t) H(t) dt H 1 (I, H(t))

Lp (I, V

0

Zs = 0

f (t), η(t)

Vσ (t)

Zs dt − 0

g(un (t)), η(t) Vσ (t) dt −

Zs 0

S(un (t)), η(t) Vσ (t) dt.

132


The last assertion in Corollary 6.7 may be obtained by the following approximation argu0 p ment. By (6.12), any un has a generalized time derivative in L (I, Vσ (U )) . Thus, for any ¯ Vσ (U )) such that as → 0: n ∈ N, Lemma 3.9 yields a sequence (un )>0 ⊂ C ∞ (I, un → un dun dun → dt dt un → un

in Lp (I, Vσ (U )), 0 in Lp (I, Vσ (U )) , ¯ Hσ (U )). in C(I,

For any s ∈ I¯ and for any η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) ,→ Lp (I, Vσ (U )) we then pass to the limit → 0 in the identity

dun

, χ(0,s) η Lp (I,Vσ (U )) + ∂t η, χ(0,s) un Lp (I,Vσ (U )) = un (s), η(s) H(s) − un (0), η(0) H(0) . dt After routinely checking that un (0) = u0 we obtain the result. In order to pass to the limit n → ∞ one has to establish a priori estimates for the sequence of approximate solutions (un )n∈N ⊂ L∞ (I, Hσ (U )) ∩ Lp (I, Vσ (U )). However, for the penalization we chose, it does not make sense to use the approximate solutions as test functions. In view of the quadratic structure of the penalty term we will use un − v instead. Keep in mind that v was assumed to be smooth, compactly supported, incompressible and known in advance, so that v is an admissible test function and it can be treated as data. The next lemma summarizes the fact that the approximate solutions (un )n∈N satisfy essentially the same energy estimates as the Galerkin solutions. Lemma 6.8 There is a positive constant C that only depends on the data such that

max un (t) Hσ (U ) + un Lp (I,Vσ (U )) ≤ C t∈I¯

holds uniformly in n ∈ N. Proof. Similarly to Corollary 6.7 we use un − v as a test function in (6.13). Again, this step ¯ Vσ (U )) such that is justified by Lemma 3.9 which provides a sequence (un )>0 ⊂ C ∞ (I, un → un dun dun → dt dt un → un

in Lp (I, Vσ (U )), 0 in Lp (I, Vσ (U )) , ¯ Hσ (U )) in C(I,


for → 0. Thus, we can let tend to zero in

1

un (s) 2 + Hσ (U ) 2

Zs

du

n

dt

, (un − v)χ(0,s)

Sun (t), un (t) Vσ (U ) dt + n

0

Zs

Lp (I,Vσ (U ))

to obtain

un (t) − v(t) 2 dt K(t)

0

1 2 = u0 Hσ (U ) + un (s), v(s) Hσ (U ) − u0 , v(0) Hσ (U ) + 2

Zs

f (t), un (t) − v(t) Vσ (U ) dt

0

Zs +

g(un (t)), v(t) Vσ (U ) dt +

0

Zs

Sun (t), v(t) Vσ (U ) dt

0

after rearranging the terms. The (p, δ)-structure of S, Korn’s inequality, the smoothness of v and a straightforward application of Young’s inequality now imply the assertion of the preceding lemma. As we merely used the positivity of the penalty term in the above calculation, we likewise obtain the following crucial estimate. Corollary 6.9 There is a positive constant C only depending on the data such that Z

un (t) − v(t) 2 dt ≤ C/n. K(t)

(6.15)

I

Moreover, restricting the sequence (un )n∈N to the non-cylindrical domain Q, Lemma 6.8 further implies Corollary 6.10 There is a positive constant C that only depends on the data such that

max un (t) Yσ (t) + un Lp (I,Xσ (t)) ≤ C t∈I¯

(6.16)

holds uniformly in n ∈ N. The energy estimate (6.16) now allows us to pass to the limit in the first and the third integral on the left-hand side of the identity (6.14). Due to the nonlinearity of the operator induced by the Cauchy stress tensor S, up to now we merely know that (Sun )n∈N 0 weakly converges to some tensor field χ ∈ Lp (Q)3×3 that needs to be identified afterwards. The boundedness of (un )n∈N implies that a subsequence weakly converges to some u ∈ Lp (I, Xσ (t)). The estimate (6.15) in turn implies the strong convergence lim un = v in L2 (I, K(t))

n→∞

as well as the fact that a subsequence of (un )n∈N converges to v almost everywhere in Qc . From this it can be easily verified that u(t) − v(t) ∈ Vσ (t)

134


holds for almost every t ∈ I. The passage to the limit in the convective term represents the 0 major obstacle. Since G : L∞ (I, Yσ (t)) ∩ Lp (I, Xσ (t)) → Lp (I, Vσ (t)) is a bounded mapping, see (6.5), we know that (Gun )n∈N weakly converges in the dual space of Lp (I, Vσ (t)). In order to identify the weak limit we need a compactness result for the sequence of restrictions of (un )n∈N to the non-cylindrical space-time domain Q. Such compactness is usually guaranteed by the Aubin–Lions lemma. At this point though, one encounters severe difficulties as we will now explain. 0 p n Due to (6.12) we have du for any fixed n ∈ N. However, the penalty dt ∈ L (I, Vσ (U )) term impedes a uniform estimate for the time derivatives in this space so that we cannot apply the Aubin–Lions lemma globally on I × U . One is tempted to get around this problem by localizing the necessary estimates through restricting the set of test functions. So let us denote by J × B a cylindrical subdomain that is completely contained in Q. With the help of the identity Z Z Z

S(un (t)), η(t) Vσ (U ) dt g(un (t)), η(t) Vσ (U ) dt + − un (t), ∂t η(t) H(U ) dt + I

I

I

Z = I

f (t), η(t) Vσ (U ) dt − n

Z un (t) − v(t), η(t)

dt + u , η(0) 0 K(t) H(0)

I

which is valid for any η ∈ H 1 (I, H(U )) ∩ Lp (I, Vσ (U )) with η(T, ·) = 0, one easily verp (J, V (B)) 0 , where V (B) := W 1,p (B)3 . n ifies that du is uniformly bounded in L σ σ 0,σ dt n∈N 2 3 Strong convergence of (un )n∈N in L (J × B) allegedly follows from the seemingly compact embedding dϕ n o 0 0 p ϕ ∈ L (I, Vσ (U )) ∈ Lp (J, Vσ (B)) ∼ = Lp (J, Vσ (B)0 ) ,→,→ L2 (J, Yσ (B)), (6.17) dt where Yσ (B) denotes the closure of Cσ∞ (B)3 in L2 (B)3 . The subtle point is that the classical Aubin–Lions lemma cannot be applied in the present case as we now try to elaborate. To begin with, recall from the discussion of Chapter 5, that the Aubin–Lions lemma crucially relies on the interpolation result supplied by the Ehrling lemma. The Ehrling lemma in the form of Lemma 5.2 though, not only requires the embedding Vσ (U ) ,→ Yσ (B) to be compact, which is still fine, but it also requires continuity and injectivity of the embedding Yσ (B) ,→ Vσ (B)0 . At this point we also want to refer to Example 5.3: Based on the lack of injectivity, we had constructed a counterexample to the assertion of the Ehrling Lemma there. However, although the embedding Yσ (B) ,→ Vσ (B)0 is still continuous, the necessary injectivity can neither be guaranteed in the present example. In order to substantiate this point, suppose u ∈ Yσ (B) is such that u = 0 in Vσ (B)0 , that is, Z

u, ϕ Vσ (B) = u · ϕ dx = 0 B


for every ϕ ∈ Vσ (B). By virtue Rham’s A.11, there exists a R theorem, of de see Theorem p p 2 unique function π ∈ L0 (B) = h ∈ L (B) B h dx = 0 ,→ L (B) such that the identity u = ∇π holds in the sense of distributions on B. Since C0∞ (B)3 is dense in W01,2 (B)3 this identity 0 also holds in W01,2 (B)3 . As a consequence, an easy application of Nečas’ theorem on negative norms, see Theorem A.12, shows that actually π ∈ W01,2 (B). Therefore, π is the unique weak solution in W 1,2 (B) of the Neumann problem ∆ π = 0 in B ∂π = γn (u) on ∂B, ∂n

(6.18)

where γn (u) ∈ H −1/2 (∂B) denotes the trace in the direction of the unit outward normal on ∂B 16 . Note problem (6.18) is well-posed: By virtue of the inclusion Yσ (B) ⊂ that the 2 3 2 Hdiv (B) := u ∈ L (B) div u ∈ L (B) we know from Theorem A.13 that γn (u) is

well-defined in H −1/2 (∂B) and that γn (u), 1 W 1/2,2 (∂B) = 0. Following [BF13], for any u ∈ Hdiv (B) and any ϕ ∈ W 1,2 (B) there holds Z Z

u · ∇ϕ dx + ϕ div u dx = γn (u), ϕ|∂B W 1/2,2 (∂B) . B

B

Approximating u ∈ Yσ (B) by a sequence (u )>0 ⊂ Cσ∞ (B)3 in L2 (B)3 , the last identity applied to π ∈ W 1,2 (B) yields Z Z 2 u dx = ∇π 2 dx ≥ 0. B

B

However, the following argument shows that the latter inequality is strict in general. For simplicity we assume that the ball B ⊂ R3 is strictly contained in U ⊂ R3 which was supposed to be a large ball, too. Consider a solution h of ∆ h = 0 in B h = b on ∂B, for a smooth function b which is chosen in such way that ∇h does not vanish identically in B. Such a function h certainly exists. The restriction of ∇h to B is smooth, incompressible, lies in Yσ (B) and satisfies ∇h = 0 in Vσ (B)0 . Repeating the above arguments we see that h is the unique weak solution of the corresponding Neumann problem (6.18) with boundary ∂h values γν (∇h) = ∂n . However, by the method of construction there holds Z 2 ∇h dx > 0 B 16

At this point we want to refer to Theorem A.9 and to Theorem A.13 for a brief summary of the notation and the results we now apply.

136


which eventually implies that the continuous embedding Yσ (B) ,→ Vσ (B)0 is not injective. The loss of injectivity in the Ehrling lemma can therefore be seen as a manifestation of the strongly non-local character of the incompressibility constraint. In other words, by restricting the approximate solutions to small cylinders, we loose control over the gradient parts of the solutions. Note also that this problem is not specifically due to the time-dependency of the domain but rather to the incompressibility constraint. That is to say, the same problem occurs if the penalty method is used to treat the unsteady incompressible p-Navier–Stokes system with inhomogeneous Dirichlet boundary conditions, even on a cylindrical domain. As a consequence, it might be said that the penalty method is an inadequate approach to deal with unsteady incompressible problems, even though it works fine for stationary ones. Further evidence for the relevance of these issues can be found in a series of papers of Feireisl and his coworkers. They recently studied compressible Navier–Stokes models on non-cylindrical domains and the behavior of rigid bodies immersed in compressible fluids. In order to obtain existence results they devised compactness methods based on a thorough analysis of the so-called Lighthill acoustic analogy. This analysis allows them to control the gradient parts of (approximate) velocity fields. As we do not pursue these ideas we want to refer the interested reader to [FKN+ 13], [FF12], [FKKS13] and [Fei11] for further details regarding the latter approach.

6.3. Existence of Weak Solutions As we have tried to explain, we cannot derive the classical energy identity for the potential solution u of (6.1). Furthermore, we refrain from applying a penalty method like the one we had introduced above since it causes serious difficulties with regard to the Aubin–Lions lemma. Since the deformation of the domain is predicted by the prescribed smooth vector field v, we may choose a different, in a way natural approach. We will transform (6.1) into a problem subject to zero Dirichlet conditions on the spatial boundary of the non-cylindrical domain. The resulting problem then allows for energy estimates, weak compactness and monotonicity methods. In order to avoid local compactness arguments we shall choose a global approach based on the non-cylindrical Aubin–Lions lemma from Chapter 5. Setting w := u − v, we see that u solves (6.1) if and only if w solves ∂t w + (w + v) · ∇ (w + v) = f − ∂t v + div S(Dw + Dv) − π idR3

in Q,

div w = 0

in Q,

w=0

on Γ,

w(0) = w0 := u0 − v(0)

(6.19)

in Ω0 .

Definition 6.11 (Weak Solution: Shifted Problem) Assume that the time-dependent vector field v generating the non-cylindrical domain Q is 0 smooth and incompressible. Suppose further that f ∈ Lp (Q)3 and w0 ∈ Vσ (0). A vector field

137

6.3 Existence of Weak Solutions

w ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) is a weak solution of (6.19) if Z

w(t), ∂t η(t) H(t) dt+

−

Z

gv (w(t)), η(t) Vσ (t) dt +

Sv (w(t)), η(t) Vσ (t) dt

I

I

I

Z

Z =

¯f (t), η(t) dt + w0 , η(0) H(0) Vσ (t)

I 0 holds for any η ∈ X. Here, ¯f stands for f − ∂t v ∈ Lp (Q)3 and the shifted operators gv and Sv are defined by Z Z

gv (w), η Lp (I,Vσ (t)) := gv (w(t)), η(t) Vσ (t) dt := g(w(t) + v(t)), η(t) Vσ (t) dt,

Sv (w), η Lp (I,Vσ (t)) :=

ZI

Sv (w(t)), η(t) Vσ (t) dt :=

I

ZI

S(w(t) + v(t)), η(t) Vσ (t) dt,

I

respectively. Remark 6.12. Obviously, w is a weak solution of (6.19) in the sense of the latter definition if and only if u = w + v is a weak solution of (6.1) in the sense of Definition 6.4. In order to construct solutions in the sense of Definition 6.11 we will first linearize and truncate the shifted convective term similarly as in the second part of Chapter 4. Note that for any w ∈ L2 (Q)3 and η ∈ Lp (I, Vσ (t)) there holds Z

gv (w(t)), η(t) Vσ (t) dt = −

I

Z

Z w ⊗ w : ∇η dx dt −

w ⊗ v : ∇η dx dt

Q

Q

Z

Z

−

v ⊗ w : ∇η dx dt − Q

v ⊗ v : ∇η dx dt Q

For ξ ∈ L2 (Q)3 we then define Kk : L2 (Q)3 → L∞ (Q)3 Kk (ξ) := ξ Φk (|ξ|), where Φk : R+ → [0, 2k] denotes the cut-off function we had introduced previous to (4.32). For any k ∈ N and any ξ ∈ L2 (Q)3 we define linearized and truncated convective terms by Z

gvk,ξ (w(t)), η(t) Vσ (t) dt

Z := −

Z Kk (ξ) ⊗ ξ : ∇η dx dt −

Q

I

Q

Z −

Z v ⊗ w : ∇η dx dt −

Q

Kk (ξ) ⊗ v : ∇η dx dt

v ⊗ v : ∇η dx dt. Q

138


Lemma 6.13 (Linearized, Truncated, Shifted Problem: Existence and Uniqueness) Let the assumptions of Definition (6.11) be satisfied. Then, for any k ∈ N and for any ξ ∈ L2 (Q)3 there exists a unique vector field w ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) such that Z Z Z

k,ξ Sv (w(t)), η(t) Vσ (t) dt gv (w(t)), η(t) Vσ (t) dt + w(t), ∂t η(t) H(t) dt+ − I

I

I

Z =

¯f (t), η(t) dt + w0 , η(0) H(0) Vσ (t)

I

holds for any η ∈ X. The solution satisfies the a priori estimate

w ∞ + w Lp (I,Vσ (t)) ≤ c(k). L (I,Hσ (t))

(6.20)

¯ L2 (R3 )3 ) and the pointwise in time evaluation The zero extension of w lies in C(I, [ w : I¯ → H(s), s 7→ w(s), s∈I¯

is well-defined. Moreover, for any s ∈ I¯ and any η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) there holds w(s), η(s) H(s) − w0 , η(0) H(0) Zs =

w(t), ∂t η(t)

Zs dt + H(t)

0

¯f (t), η(t) dt − Vσ (t)

0

Zs −

Zs 0

gvk,ξ (w(t)), η(t) Vσ (t) dt

(6.21)

Sv (w(t)), η(t) Vσ (t) dt

0

Proof. The existence of at least one weak solution can be proved by means of a Galerkin approximation built on time-dependent ansatz fields as in Chapter 3 and we just discuss the most important steps here. We can still exploit the (p, δ)-structure of S by noticing that for any Galerkin solution wn there holds

Sv (wn (t)), wn (t) Vσ (t)

= Sv (wn (t)), wn (t) + v(t) Xσ (t) − Sv (wn (t)), v(t) Xσ (t)

p−1

p ≥ c Dwn (t) + Dv(t) Lp (Ω(t)) − Dwn (t) + Dv(t) Lp (Ω(t)) Dv(t) Lp (Ω(t)) − g(t) for some g ∈ L1 (I). The last estimate follows from Lemma 6.2. From the above estimate and a straightforward application of Young’s inequality with one can easily derive the uniform a priori estimate

max wn (t) Hσ (t) + wn Lp (I,Vσ (t)) ≤ c(k) (6.22) t∈I¯

for the sequence of Galerkin solutions (wn )n∈N . Notice that the constant c(k) depends on the truncation parameter k, on the norms of f and w0 and on various norms of v. Using (6.22)

139


we may then justify the passage to the limit in the Galerkin system. Note that gvk,ξ (wn ) is linear in wn and therefore we do not require strong compactness at this stage to pass to the limit in the linearized, truncated convective term. As Sv induces a bounded operator 0 Sv : Lp (I, Vσ (t)) → Lp (Q)3×3 we infer that a subsequence of Sv (wn ) n∈N weakly converges 0 to some χ ∈ Lp (Q)3×3 . The identification of χ rests upon the generalized integration by parts formula from Theorem 3.20 and Minty’s monotonicity trick. At this point we only want to remark that the monotonicity of S carries over to Sv , since for any t ∈ I¯ and any w(t), eta(t) ∈ Vσ (t) there holds

Sv (w(t)) − Sv (η(t)), w(t) − η(t) Vσ (t)

= S(w(t) + v(t)) − S(η(t) + v(t)), w(t) + v(t) − (η(t) + v(t)) Vσ (t) (6.23)

p

p ≥ Dw(t) − Dη(t) p = w(t) − η(t) ≥ 0 L (Ω(t))

Vσ (t)

It follows that χ = Sv (w), where w ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) denotes the weak limit ¯ L2 (R3 )3 ) and that it of the sequence of Galerkin solutions. The fact that w lies in C(I, actually satisfies the identity (6.21) follows from Theorem 3.20, too. We eventually prove uniqueness of the solution thus constructed. To this end, let wi ∈ ¯ L2 (R3 )3 ), i = 1, 2, denote weak solutions corresponding L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) ∩ C(I, 0 to w0i ∈ Vσ (0), f i ∈ Lp (Q)3 , i = 1, 2, and the same linearization ξ ∈ L2 (Q)3 at the same truncation level k ∈ N. Applying Theorem 3.20, we obtain for any s ∈ I¯ that

1

w1 (s) − w2 (s) 2 + H(s) 2

Zs

Sv (w1 (t)) − Sv (w2 (t)), w1 (t) − w2 (t) Vσ (t) dt

0

2 1 = w01 − w02 H(0) + 2

Zs

f 1 (t) − f 2 (t), w1 (t) − w2 (t)

Vσ (t)

dt

(6.24)

0

Zs −

k,ξ gv (w1 (t)) − gvk,ξ (w2 (t)), w1 (t) − w2 (t) Vσ (t) dt.

0

However, since Zs −

gvk,ξ (w1 (t)) − gvk,ξ (w2 (t)), w1 (t) − w2 (t) Vσ (t) dt

0

Zs Z =

1 v ⊗ (w1 − w2 ) : ∇(w1 − w2 ) dx dt = − 2

0 Ω(t)

Zs Z

2 div v w1 − w2 dx dt = 0,

0 Ω(t)

it easily follows from (6.23) and (6.24) that the solution continuously depends on the data, which in turn implies uniqueness of the weak solution.

6.3.1. Compactness of the Solution Operator Having established the existence and uniqueness of solutions to the linearized and truncated problems, our next goal is again to show that for any k ∈ N, the solution operator Lk ,

140


mapping ξ to the unique solution w, satisfies the requirements of Schauder’s fixed point theorem. First off, the a priori estimate (6.20) shows that Lk maps the closed, convex set n o Bk := ξ ∈ L2 (Q)3 ξ L2 (Q) ≤ c(k) into itself. In order to infer compactness of Lk we will employ the generalized Aubin–Lions lemma in the next Proposition 6.14 (Compactness of Lk ) For any k ∈ N, the solution operator Lk : Bk → Bk , ξ 7→ Lk (ξ) =: w is continuous and compact. Proof. Consider a bounded sequence of linearizations (ξ n )n∈N ⊂ Bk and denote by wn = Lk (ξ n ) the unique weak solutions of the corresponding linearized and truncated problems. Our goal is to prove relative compactness of (wn )n∈N in L2 (Q)3 . By the method of construction, any wn satisfies for any s ∈ I¯ and for any η in the space of test functions X wn (s), η(s) H(s) − w0 , η(0) H(0) +

Zs

Sv (wn (t)), η(t) Vσ (t) dt

0

Zs =

wn (t), ∂t η(t) H(t) dt +

0

Zs

¯f (t), η(t) dt − Vσ (t)

0

Zs

(6.25)

k,ξn gv (wn (t)), η(t) Vσ (t) dt.

0

We will now exploit the identity (6.25) and the uniform a priori estimate

wn ∞ + wn Lp (I,Vσ (t)) ≤ c(k), L (I,Hσ (t))

(6.26)

which easily follows from (6.25), in order to check the validity of the assumptions of Theorem 5.4, i.e. we essentially need to show the existence of a continuous function h : R+ → R+ with h(0) = 0 and 0 ˆ ˆ 0 0 0 sup sup M (w , η , P ) − M (w , η , P ) (6.27) Vσ (s) n n ψs ψs0 ≤ c(k) h |s − s | . Vσ (s ) n∈N kˆ η kVσ (0) ≤c−1 0

Here, Pψ· : Vσ (0) → Vσ (·) denotes the isomorphism induced by the Piola transform. We will come back to this point below. Note that the estimate

sup wn L∞ (I,Hσ (t)) ≤ c(k) n∈N

already corresponds to condition (5.7) in Theorem 5.4 and that our function spaces are linked by a chain of compact embeddings i

¯ Vσ (t) ,→,→ Hσ (t) ,→,→ Vσ (t)0 , Hσ (t) ,→ Vσ (t)0 , t ∈ I,

(6.28)

see also the discussion previous to Proposition 5.10 in Chapter 5. To begin with, the boundedness of (wn )n∈N in L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) implies the existence of a subsequence

141


(wn )n∈M1 ⊂ (wn )n∈N and of w ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) such that for almost every s ∈ I we get wn (s) * w(s) in Hσ (s) for M1 3 n → ∞. Here, the identification of the weak limit on almost every time slice is established using also the implications of Lemma 3.6 and Corollary 3.8. On the other hand, having established (6.27), Theorem 5.4 yields a further subsequence (wn )n∈M ⊂ (wn )n∈M1 such that for any s ∈ I¯ the sequence (wn (s))n∈M is a Cauchy sequence in Vσ (s)0 . It follows ˜ that for any s ∈ I¯ there exists w(s) ∈ Vσ (s)0 such that lim M3n→∞

˜ wn (s) = w(s) in Vσ (s)0 .

i ¯ eventually implies The embedding Hσ (s) ,→ Vσ (s)0 , s ∈ I,

lim M3n→∞

wn (s) = w(s) in Vσ (s)0

at least for almost every s ∈ I. Furthermore, we have shown in Proposition 5.10 that the triples Vσ (t), Hσ (t), Vσ (t)0 t∈I¯ have the uniform Ehrling property, i.e. for any > 0 there exists a constant c such that

NHσ (t) (wn (t) − w(t), Pψt ) ≤ wn (t) − w(t) Vσ (t) + c NVσ (t)0 (wn (t) − w(t), Pψt ) at least holds for almost every t ∈ I. Taking the p-th power of the latter inequality and integrating with respect to t we obtain Z Z

NH (t) (wn (t) − w(t), Pψ ) p dt = wn (t) − w(t) p dt t σ Hσ (t) I Z

NV (t)0 (wn (t) − w(t), Pψ ) p dt. ≤ c wn Lp (I,Vσ (t)) + w Lp (I,Vσ (t)) + c t σ

I

I

Using also Lebesgue’s theorem on dominated convergence, the right-hand side can be made arbitrarily small. Thus, we infer that (wn )n∈M strongly converges to w in Lp (I, Hσ (t)) ,→ L2 (Q)3 , thereby proving that the sequence (wn )n∈N is relatively compact in L2 (Q)3 . To finish the compactness part of the proof we still have to establish (6.27). Choosing a test function η ∈ X with η(0) = 0, we get from (6.25) that for any wn and for any such test function there holds Z − wn (t), ∂t η(t) H(t) dt I

Z = I

¯f (t), η(t) dt − Vσ (t)

Z

k,ξ gv n (wn (t)), η(t) Vσ (t) dt

I

Z −

Sv (wn (t)), η(t) Vσ (t) dt.

I

∗ By Definition 2.12, this means dt wn ∈ Lp (I, Vσ (t)) . Combining the last identity with the a priori estimate (6.26) and the (p, δ)-structure of S, we can even deduce the uniform bound

0 ≤ C(k) sup dt wn p (6.29) n∈N

L (I,Vσ (t))

142


from the fact that the set η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) η(0) = η(T ) = 0 is dense in Lp (I, Vσ (t)). The identity (6.25) can now be recast as follows: For every s ∈ I¯ and for every η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) there holds

wn (s), η(s) H(s) − w0 , η(0) H(0) =

Zs

wn (t), ∂t η(t) H(t) dt +

0

Zs

dt wn (t), η(t)

0

Vσ (t)

dt.

(6.30)

The latter identity can be viewed as a substitute for the fundamental theorem of calculus, see (5.1), which does not make sense in generalized Bochner spaces. As a consequence, (6.30) will allow us to recover the validity of the generalized equicontinuity condition (6.27) for the sequence of solutions (wn )n∈N ⊂ Lp (I, Vσ (t)). At first, we construct special test functions. The flow of the smooth vector field v naturally 1,p 1,p induces isomorphisms between Vσ (0) = W0,σ (Ω0 )3 and Vσ (t) = W0,σ (Ω(t))3 by means of the Piola transform Pψt . Note that in the present case, the Piola transform coincides with the pull-back of vector fields as v was assumed to be incompressible. Thanks to the smoothness of v, these isomorphisms are uniformly bounded in the sense that there is a uniform constant c0 such that

ˆ

Pψ η (6.31) t ˆ Vσ (t) ≤ c0 η Vσ (0) ˆ ∈ Vσ (0). For an arbitrary η ˆ ∈ Vσ (0) we now set holds for any t ∈ I¯ and any η ˆ ˆ (t) := Pψt η Pψ η ˆ ∈ Vσ (t) and Pψ η ˆ ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)). Here, the fact that ∂t (Pψ η ˆ) to obtain Pψt η 2 lies in L (I, H(t)) follows from the chain rule and the assumption p ≥ 11/5 > 2 whereas the other assertions follow from (6.31) and the fact that Vσ (t) ⊂ H(t). Now let s0 , s ∈ I¯ with s0 < s be arbitrary. Comparing the identity (6.30) with the test ˆ at the times s0 and s we see function Pψ η ˆ , Pψs ) − MVσ (s0 )0 (wn , η ˆ , Pψs0 ) MVσ (s)0 (wn , η

0 ˆ V (s) − wn (s ), Pψs0 η ˆ V (s0 ) = wn (s), Pψs η ˆ H(s) − wn (s0 ), Pψs0 η ˆ H(s0 ) = wn (s), Pψs η σ σ Zs ˆ wn (t), ∂t (Pψt η

= s0

Zs dt + H(t)

ˆ dt wn (t), Pψt η

Vσ (t)

dt

s0

But now, Hölder’s inequality, (6.26) and (6.29) show that we can find a continuous function h with h(0) = 0 such that ˆ , Pψs ) − MVσ (s0 )0 (wn , η ˆ , Pψs0 ) ≤ c(k) h |s0 − s| η ˆ V (0) . (6.32) MVσ (s)0 (wn , η σ To see this, we only take care of the integral containing the generalized time derivative dt wn . The remaining integral may easily be treated by exploiting the smoothness of the flows, the chain rule and the Poincaré inequality.

143


Using Hölder’s inequality we estimate Zs

ˆ dt wn (t), Pψt η Vσ (t) dt ≤ dt wn

1/p Zs

p

Pψ η 0 t ˆ Vσ (t) dt

Lp (I,Vσ (t))

s0

s0

1/p ˆ V (0) , ≤ C(k) s0 − s η σ where we also used (6.29) and (6.31) for the second inequality. Eventually, (6.32) therefore implies ˆ , Pψs ) − MVσ (s0 )0 (wn , η ˆ , Pψs0 ) ≤ C(k) h |s0 − s| sup sup MVσ (s)0 (wn , η n∈N kˆ η kVσ (0) ≤ c−1 0

for a suitably chosen function h : R+ → R which is continuous at 0 and satisfies h(0) = 0. But this is just the generalized equicontinuity condition (6.27) which finally yields the compactness assertion. We now come to the continuity of Lk . Let (ξ n )n∈N ⊂ Bk denote a sequence of linearizations that strongly converges to some ξ ∈ Bk . Since the truncation operator Kk is continuous, the strong convergence of (ξ n )n∈N implies lim Kk (ξ n ) = Kk (ξ) = ξ Φk (|ξ|) in L∞ (Q)3 .

n→∞

Denoting by wn = Lk ξ n the corresponding weak solutions, the a priori estimate (6.26) and a subsequent application of the generalized Aubin–Lions compactness lemma yield a subsequence (wn )n∈M ⊂ (wn )n∈N such that wn * w in Lp (I, Vσ (t)), wn → w in L2 (I, Hσ (t)), 0

Sv (wn ) * χ in Lp (Q)3×3 , as M 3 n → ∞. Passing to the limit along this subsequence in the identity Z Z Z

k,ξn

− wn (t), ∂t η(t) H(t) dt+ gv (wn (t)), η(t) Vσ (t) dt + Sv (wn (t)), η(t) Vσ (t) dt I

I

I

Z =

¯f (t), η(t) dt + w0 , η(0) H(0) , Vσ (t)

I

we obtain that Z Z Z

k,ξ

− w(t), ∂t η(t) H(t) dt+ gv (w(t)), η(t) Vσ (t) dt + χ(t), η(t) Vσ (t) dt I

I

I

Z =

¯f (t), η(t) dt + w0 , η(0) H(0) Vσ (t)

I

holds for any test function η ∈ X. But now the remaining identification 0

χ = Sv (w) in Lp (Q)3×3

144


can easily be established by using Theorem 3.20 and Minty’s trick. As the same reasoning applies to any converging subsequence of (wn )n∈N we can actually infer that limn→∞ wn = limn→∞ Lk ξ n = Lk ξ = w which amounts to the continuity of the solution operator Lk for any fixed k ∈ N. Having established the continuity and the compactness of Lk , Schauder’s fixed point theorem now asserts the existence of wk ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) ,→ L2 (Q)3 with wk = Lk wk . The properties of the fixed points are summarized in the following lemma. Lemma 6.15 (Truncated, Shifted Problem: Existence and Uniqueness) Let the assumptions of Definition (6.11) be satisfied. For any k ∈ N there exists a unique wk ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) such that Z Z Z

k,w k Sv (wk (t)), η(t) Vσ (t) dt − wk (t), ∂t η(t) H(t) dt+ gv (wk (t)), η(t) Vσ (t) dt + I

I

I

Z =

¯f (t), η(t) dt + w , η(0) 0 Vσ (t) H(0)

I

(6.33)

holds for any η ∈ X. The sequence (wk )k∈N satisfies the uniform estimate

wk ∞ + wk Lp (I,Vσ (t)) ≤ C. L (I,Hσ (t))

(6.34)

¯ L2 (R3 )3 ) and the pointwise in time evaluation The zero extension of any wk lies in C(I, [ H(s), s 7→ wk (s), wk : I¯ → s∈I¯

is well-defined for every k ∈ N. Moreover, for any s ∈ I¯ and any η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) there holds Zs Zs

k,w k ¯f (t), η(t) wk (s), η(s) H(s) − w0 , η(0) H(0) = (w (t)), η(t) dt − g dt k v Vσ (t) Vσ (t) 0

0

Zs +

wk (t), ∂t η(t)

Zs

H(t)

dt −

0

Sv (wk (t)), η(t) Vσ (t) dt.

0

(6.35)

Proof. We will only illustrate the uniform estimate (6.34) because the remaining assertions are essentially due to the fixed point property of the fields (wk )k∈N . Now, using also Theorem 3.20, the estimate (6.34) basically follows from testing (6.35) with the solution wk itself. In doing so, we particularly need to control the truncated convective term Z Z Z

k,w k gv (wk (t)), wk (t) Vσ (t) dt = − Kk (wk ) ⊗ wk : ∇wk dx dt − Kk (wk ) ⊗ v : ∇wk dx dt I

Q

Q

Z −

Z v ⊗ wk : ∇wk dx dt −

Q

v ⊗ v : ∇wk dx dt. Q

145


By Lemma 4.14, the first integral on the right-hand side of the latter identity equals zero. R Also, Q v ⊗ wk : ∇wk dx dt = 0 as may easily be obtained from integration by parts and the fact that v is incompressible. The trivial pointwise estimate Kk (wk ) ≤ wk and Hölder’s inequality imply Z Z

Kk (wk ) ⊗ v : ∇wk dx dt + v ⊗ v : ∇wk dx dt ≤ c(v) wk Lp (I,Vσ (t)) . Q

Q

However, the quantity kwk kLp (I,Vσ (t)) can now be routinely absorbed in the Cauchy stress tensor by using the (p, δ)-structure of S, the triangle inequality and Young’s inequality.

6.3.2. Passage to the Limit The passage to the limit k → ∞ in the identity Z Z Z

k,w

− wk (t), ∂t η(t) H(t) dt+ gv k (wk (t)), η(t) Vσ (t) dt + Sv (wk (t)), η(t) Vσ (t) dt I

I

I

Z =

¯f (t), η(t) dt + w , η(0) , 0 Vσ (t) H(0)

I

which, owing to Lemma 6.15, holds for any test function η ∈ X, will eventually prove existence of a weak solution of the transformed system (6.19) in the sense of Definition 6.11. Note that due to the uniform estimate (6.34) there exists w ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) and a subsequence of (wk )k∈N which weakly converges to w in Lp (I, Vσ (t)). With the help of the non-cylindrical Aubin–Lions lemma we can prove that a further subsequence (wk )k∈M ⊂ (wk )k∈N actually converges to w strongly in L2 (Q)3 and almost everywhere in Q. As the proof of the latter assertions proceeds similarly as the compactness part of Proposition 6.14 we just illustrate the most important steps. First off, the uniform energy estimate (6.34) and the identity (6.33) now imply uniform boundedness of the generalized time derivatives (dt wk )k∈N , i.e. there exists a constant C > 0 such that

0 ≤ C. sup dt wk p k∈N

L (I,Vσ (t))

This information can be used subsequently to infer the generalized equicontinuity of the sequence (wk )k∈N in the form ˆ , Pψs ) − MVσ (s0 )0 (wk , η ˆ , Pψs0 ) ≤ c h |s0 − s| , sup sup MVσ (s)0 (wk , η k∈N kˆ η kVσ (0) ≤ c−1 0

for a constant c only depending on the data and a continuous function h with h(0) = 0. Furthermore, we can make use of the estimate

sup wk L∞ (I,Hσ (t)) ≤ C k∈N

146


and the embeddings (6.28) to complete the assumptions of Theorem 5.4. Using again the uniform Ehrling property of the triples Vσ (t), Hσ (t), Vσ (t)0 t∈I¯, see Proposition 5.10, the noncylindrical Aubin–Lions lemma eventually shows that a subsequence (wk )k∈M ⊂ (wk )k∈N strongly converges to w in L2 (Q)3 . Without loss of generality we can even assume that this subsequence converges to w almost everywhere in Q. From this it is not hard to see that for any η ∈ X there holds Z Z

k,w k gv (w(t)), η(t) Vσ (t) dt. gv (wk (t)), η(t) Vσ (t) dt = lim M3k→∞

I

I

Combining the energy estimate (6.34) with the growth properties of Sv , there exists χ ∈ 0 Lp (Q)3×3 such that 0

Sv (wk ) * χ in Lp (Q)3×3 , for M 3 k → ∞. However, applying Theorem 3.20 and Minty’s trick to the weak limit w ∈ L∞ (I, Hσ (t) ∩ Lp (I, Vσ (t)), we can again identify Sv (w) with χ and prove the usual continuity properties of the weak limit. Theorem 6.16 (Shifted Problem: Existence) Assume that the time-dependent vector field v generating the non-cylindrical domain Q is 0 smooth and incompressible and suppose that ¯f ∈ Lp (Q)3 and w0 ∈ Vσ (0). Then there is at least one vector field w ∈ L∞ (I, Hσ (t)) ∩ Lp (I, Vσ (t)) such that Z Z Z

− w(t), ∂t η(t) H(t) dt+ gv (w(t)), η(t) Vσ (t) dt + Sv (w(t)), η(t) Vσ (t) dt I

I

I

Z =

¯f (t), η(t) dt + w0 , η(0) H(0) Vσ (t)

I

holds for any η ∈ X. The solution satisfies the energy estimate

w ∞ + w Lp (I,Vσ (t)) ≤ C. L (I,Hσ (t)) ¯ L2 (R3 )3 ) and the pointwise in time evaluation The zero extension of w lies in C(I, [ w : I¯ → H(s), s 7→ w(s), s∈I¯

is well-defined. Moreover, for any s ∈ I¯ and any η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) we have the identity w(s), η(s) H(s) − w0 , η(0) H(0) =

Zs w(t), ∂t η(t) 0

Zs − 0

Zs dt + H(t)

¯f (t), η(t) dt Vσ (t)

0

gv (w(t)), η(t) Vσ (t) dt −

Zs 0

Sv (w(t)), η(t) Vσ (t) dt.

147

6.4 Uniqueness

As an immediate consequence of the latter theorem we obtain the following existence result for the p-Navier–Stokes system with inhomogeneous Dirichlet boundary conditions. Theorem 6.17 (Problem 6.1: Existence) 0

For any f ∈ Lp (Q)3 and any u0 ∈ Xσ (0) satisfying the compatibility condition u0 ·n = v(0)·n there exists at least one weak solution u ∈ L∞ (I, Yσ (t)) ∩ Lp (I, Xσ (t)) of (6.1) in the sense that Z Z Z

− u(t), ∂t η(t) H(t) dt+ g(u(t)), η(t) Vσ (t) dt + S(u(t)), η(t) Vσ (t) dt I

I

I

Z =

(6.36)

f (t), η(t) Vσ (t) dt + u0 , η(0) H(0)

I

holds for any η ∈ X and that for almost every t ∈ I there holds u(t) − v(t) ∈ Vσ (t). ¯ L2 (R3 )3 ) and the pointwise in time evaluation The extension of u by v to R3 lies in C(I, [ u : I¯ → H(s), s 7→ u(s), s∈I¯

is well-defined. Moreover, for any s ∈ I¯ and any η ∈ H 1 (I, H(t)) ∩ Lp (I, Vσ (t)) we have the identity u(s), η(s) H(s) − u0 , η(0) H(0) =

Zs

u(t), ∂t η(t) H(t) dt +

0

Zs − 0

Zs

f (t), η(t) Vσ (t) dt

0

g(u(t)), η(t)

Vσ (t)

Zs dt −

S(u(t)), η(t)

Vσ (t)

dt

0

6.4. Uniqueness In contrast to the Navier–Stokes equation, uniqueness of weak solutions to systems with (p, δ)-structure can be proved without restrictive assumptions on the dimension or the data. However, still p has to be sufficiently large and δ has to be strictly positive. Proposition 6.18 (Uniqueness) If the stress tensor S has (p, δ)-structure for 5/2 ≤ p < ∞ and 0 < δ, then the weak solution we had obtained in Theorem 6.17 is unique within the natural energy class L∞ (I, Yσ (t)) ∩ Lp (I, Xσ (t)). Proof. The proof follows the ideas from [MNRR96, Theorem 4.29] where an analogue result is proved in the cylindrical setting. Moreover, we restrict our attention to the case p ∈ [5/2, 3)

148


because the other case can be treated similarly. Suppose that u1 and u2 are two weak solutions of (6.1) in the sense of Theorem 6.17 corresponding

to the same data (f , u0 ) ∈ 0 Lp (Q)3 × Xσ (0). It follows from (6.36) that the expression dt (u1 − u2 ), u1 − u2 Lp (I,Vσ (t)) is well-defined. Hence, Theorem 3.20 yields

1

u1 (s) − u2 (s) 2 + H(s) 2

Zs

S(u1 (t)) − S(u2 (t)), u1 (t) − u2 (t)

Vσ (t)

dt

0

Zs =−

(6.37)

g(u1 (t)) − g(u2 (t)), u1 (t) − u2 (t) Vσ (t) dt.

0

The convective term may be rewritten as follows: Zs −

g(u1 (t)) − g(u2 (t)), u1 (t) − u2 (t) Vσ (t) dt

0

Zs Z = 0 Ω(t) Zs Z

=

u2 ⊗ u2 − u1 ⊗ u1 : ∇ u1 − u2 dx dt

u2 − u1 ⊗ u1 : ∇ u1 − u2 dx dt

0 Ω(t) Zs Z

−

u2 ⊗ u1 − u2 : ∇ u1 − u2 dx dt.

0 Ω(t)

Since u1 − u2 is solenoidal and vanishes on Γ(t) for almost every t ∈ I, integration by parts implies that Zs −

g(u1 (t)) − g(u2 (t)), u1 (t) − u2 (t) V (t) dt

0

Zs Z

u1 − u2 ⊗ u1 : ∇ u1 − u2 dx dt

= 0 Ω(t) Zs

=−

g(u1 (t) − u2 (t)), u1 (t)

Xσ (t)

Zs Z dt ≤

0

u1 − u2 2 |∇u1 | dx dt.

0 Ω(t)

With the help of Hölder’s inequality we estimate Zs Z 0 Ω(t)

u1 − u2 2 ∇u1 dx dt ≤

Zs

u1 (t) − u2 (t) 2 2p p−1 L

0

(Ω(t))

∇u1 (t) p dt. L (Ω(t))

149

6.4 Uniqueness

Applying the interpolation inequality

u1 (t) − u2 (t)

2p L p−1 (Ω(t))

2p−3

3 2p ≤ c u1 (t) − u2 (t) H(t) ∇u1 (t) − ∇u2 (t) L2p2 (Ω(t)) ,

we can further estimate Zs Z

u1 − u2 2 ∇u1 dx dt

0 Ω(t)

Zs ≤ c

3

2p−3

∇u1 (t) − ∇u2 (t) p 2

u1 (t) − u2 (t) p ∇u1 (t) p dt. H(t) L (Ω(t)) L (Ω(t))

0

But now Young’s inequality with yields Zs Z

u1 − u2 2 ∇u1 dx dt ≤ c()

0 Ω(t)

Zs

2p

u1 (t) − u2 (t) 2 ∇u1 (t) 2p−3 Lp (Ω(t)) dt H(t)

0

Zs +

(6.38)

∇u1 (t) − ∇u2 (t) 2 2 dt. L (Ω(t))

0

On the other hand, our assumptions on the stress tensor S imply that Zs

S(u1 (t)) − S(u2 (t)), u1 (t) − u2 (t) Vσ (t) dt ≥ δ p−2

Zs

Du1 (t) − Du2 (t) 2 2 dt L (Ω(t))

0

0

Using also Korn’s inequality for fields with vanishing traces, we can absorb the second term on the right-hand side of (6.38) in the left-hand side of (6.37) by choosing sufficiently small. Thus, we eventually infer

u1 (s) − u2 (s) 2 ≤ c H(s)

Zs

2p

u1 (t) − u2 (t) 2 ∇u1 (t) 2p−3 Lp (Ω(t)) dt H(t)

0

However, for p ≥ 5/2 there holds

2p 1

∇u1 (·) 2p−3 Lp (Ω(·)) ∈ L (I) so that ultimately, Gronwall’s inequality yields the asserted uniqueness.

150


7. Shear Thinning Fluids in Non-cylindrical Domains In the foregoing chapters we transferred some of the cornerstones of the classical theory of monotone operators to the framework of generalized Bochner spaces. In doing so, we had to make assumptions that appear to be somehow artificial in comparison with the classical results on cylindrical domains. This point is especially evident if we take a look at the proof that led to the results of Proposition 3.13, i.e. the integration by parts formula and the welldefinedness of time traces. There, we had to assume the lower bound p ≥ 2 for the growth parameter of the monotone nonlinearity and also higher regularity for the initial value u0 . Both these restrictions are necessary for the properties of the approximation in Proposition 3.10 and Proposition 3.12 and both restrictions are connected to the Piola transform. The analogue results in the setting of Bochner spaces require the natural lower bound p > 1. Also, the natural regularity for the initial value is u0 ∈ H, where H denotes the Hilbert space in the underlying Gelfand triple. In applications though, different mathematical reasons for a lower bound appear, as we saw in the last chapter: The presence of the convective term gave rise to the lower bound p ≥ 1 + 2d/(d + 2) in order to fit the physical model into the mathematical framework of (pseudo)monotone operators. However, applied fluid dynamics advocates the need for abandoning purely mathematical restrictions, or, to be more specific: It is necessary to understand stress tensors having (p, δ)structure for 1 < p < 2. In fact, the borderline case p = 2 divides the class of power-law models for non-Newtonian fluids, see definition 6.1, into shear thickening fluids (p > 2) and shear thinning fluids (1 < p < 2). By definition, shear thinning fluids are characterized by their ability to decrease resistance to flow with an increasing rate of shear stress. Everybody who has ever squeezed too much ketchup out of a ketchup bottle by accident knows this effect. However, blood once more is a by far more important example of a shear thinning fluid. Under low shear rates, red blood cells tend to form three-dimensional aggregates known as rouleaux. For blood at rest these rouleaux appear solid-like and resist flow until a certain level of stress is applied. This level is known as yield stress. When applied stresses increase, the rouleaux break up into smaller pieces thereby significantly lowering the effective viscosity of blood. For an introduction to blood rheology we again refer to [FQV09, Chapter 6]. In order to build realistic models for the human cardiovascular system it is therefore of utmost importance to understand also the mathematical properties of shear thinning fluids, especially in time-dependent domains. In the last decades though, there has been a growing interest in mathematical tools that allow for small values of p, thereby extending the earlier results of the standard monotone operator theory. In this context, the classical theory of monotone operators is merely used to establish existence of some sequence of approximate solutions (un )n∈N and their basic

151

152

Chapter 7. Shear Thinning Fluids in Non-cylindrical Domains

properties. One of the first results in this direction is due to Landes and Mustonen, see also [LM87]. There, an L∞ -approximation (ukn )n,k∈N is constructed which strongly converges to the potential weak solution u, has better integrability properties than (un )n∈N and agrees with the original sequence on a large set. Within the existence theory for non-Newtonian fluids, these ideas were significantly sharpened by Frehse, Málek and Steinhauer in [FMS03] and by Diening, Málek and Steinhauer in [DMS08]. In the latter works, the authors developped the so-called Lipschitz truncation method which provides an explicit approximation in W 1,∞ with the additional property that the set of non-coincidence is controlled in terms of the Hardy-Littlewood maximal function. Using this technique, they were able to simplify existence results for stationary models of non-Newtonian fluids. Especially, the existence theory for shear thinning fluids can be extended down to the lower bound p > 6/5 in three space dimensions. Note also that this lower bound is optimal as the convective term can no longer be treated as a compact perturbation for 1 < p ≤ 6/5. However, further improvement of the Lipschitz approximation method was recently made by Breit, Diening and Schwarzacher. In [BDS13] they were able to construct a refined Lipschitz truncation that is tailor-made for unsteady problems in the context of incompressible non-Newtonian fluids. More precisely, their method even provides solenoidal Lipschitz approximations of fields belonging to the natural energy space Lp (I, Wσ1,p (Ω)d ). We will use this technique in order to prove existence of weak solutions of the problem ∂t u + u · ∇ u = f + div S(Du) − π idR3 in Q, div u = 0

in Q,

u=v

on Γ,

(7.1)

in Ω0 .

u(0) = u0

for any stress tensor S having (p, δ)-structure down to the critical lower bound p > 6/5. Moreover, we can weaken our earlier assumptions on the regularity of u0 . As in the previous chapter, Q denotes the non-cylindrical domain generated by the flow of the time-dependent, smooth, solenoidal vector field v with compact spatial support in U ⊂ R3 . The nonhomogeneous boundary condition (7.1)3 again takes into account the fact that the viscous fluid adheres to the moving boundary. The results of the present chapter will be formulated in terms of the following function spaces: For any t ∈ I¯ and for exponents 1 < r < ∞ to be specified below we set 1,r ∞ (Ω(t))3 Vr (t) := W0,σ (Ω(t))3 = C0,σ 1,r Xr (t) := Wσ,s (Ω(t))3 := Cσ∞ (Ω(t))3 2

k·k

,

1,r Ws (Ω(t))3

, (7.2)

3

H(t) := L (Ω(t)) , ∞ (Ω(t))3 Hσ (t) := C0,σ

Yσ (t) := Cσ∞ (Ω(t))3

k∇·kLr (Ω(t))

k·kH(t)

k·kH(t)

.

The existence result we want to prove is the following:

153

7.1 Existence of Approximate Solutions

Theorem 7.1 (Problem 7.1: Existence) 0

Given 6/5 < p < ∞, f ∈ Lp (Q)3 and u0 ∈ Yσ (0) satisfying the compatibility condition u0 − v(0) ∈ Hσ (0), there exists a weak solution u ∈ L∞ (I, Yσ (t)) ∩ Lp (I, Xp (t)) of (7.1) in the sense that Z Z Z

− u(t), ∂t η(t) H(t) dt+ g(u(t)), η(t) Vq (t) dt + S(u(t)), η(t) Vq (t) dt I

I

I

Z =

f (t), η(t)

Vq (t)

dt + u0 , η(0)

H(0)

I

holds for any η ∈ X, where X := for almost every t ∈ I there holds

ϕ ∈ H 1 (I, H(t)) ∩ Lq (I, Vq (t)) ϕ(T ) = 0 . Moreover, u(t) − v(t) ∈ Vp (t)

Remark 7.2. The exponents p and q are related by17 q ≥ max (5p/6)0 , 5/2 . This choice of q is motivated by the following facts. The embedding L∞ (I, L2 (Ω(t))3 ) ∩ Lp (I, W 1,p (Ω(t))3 ) ,→ Lp5/3 (Ω(t))3 , see Corollary 4.4, shows that u ⊗ u ∈ L5p/6 (Ω(t))3×3 . This in turn implies u ⊗ u : ∇η ∈ L1 (Q) for any u ∈ L∞ (I, L2 (Ω(t))3 ) ∩ Lp (I, W 1,p (Ω(t))3 ) and any test function η ∈ Lq (I, W 1,q (Ω(t))3 ). Moreover, the lower bound q ≥ 5/2 will not only guarantee existence but also uniqueness of approximate solutions as we will see.

7.1. Existence of Approximate Solutions As we had already explained, the first step consists of constructing a suitable sequence of approximate solutions that will ultimately converge to a weak solution of (7.1). Suppose that S is a stress tensor with (p, δ)-structure in the sense of Definition 6.1 for 6/5 < p < 11/5. Since the theory we have developed so far does not directly apply in the case 6/5 < p ≤ 2, we approximate S by another tensor that compensates the lack of integrability and simultaneously preserves the monotonicity and coercivity properties of S. The easiest way to achieve that is to add a second Cauchy stress tensor A having (q, κ)structure for the exponent q we had introduced in the last remark and some κ ∈ (0, 1). That is, we set for n ∈ N Sn (Du) := S(Du) + An (Du) := S(Du) +

1 A(Du). n

In order to fit the approximate system into the framework of the previous chapter we additionally have to regularize the initial value. Since u0 ∈ Yσ (0) is assumed to satisfy the compatibility condition (u0 − v(0)) · n = 0, the field w0 := u0 − v(0) lies in Hσ (0). Hence, there exists a sequence (w0n )n∈N ⊂ Vq (0) such that limn→∞ w0n = w0 in L2 (Ω0 )3 . Setting un0 := w0n + v(0), we can formulate the following existence result for approximate solutions. 17

We will restrict ourselves to the physically relevant three-dimensional case.

154


Proposition 7.3 (Existence and Uniqueness of Approximate Solutions I) For every n ∈ N there exists a unique vector field un ∈ L∞ (I, Yσ (t)) ∩ Lq (I, Xq (t)) such that Z Z Z

n

S (un (t)), η(t) Vq (t) dt g(un (t)), η(t) Vq (t) dt + un (t), ∂t η(t) H(t) dt+ − I

I

I

Z

=

(7.3)

f (t), η(t) Vq (t) dt + un0 , η(0) H(0)

I

holds for every η ∈ X. Moreover, the nonhomogeneous boundary condition un (t) − v(t) ∈ Vq (t) is satisfied for every n ∈ N and for almost every t ∈ I. Proof. The existence of at least one weak solution having all of the above properties follows from Theorem 6.17 of the previous chapter. Furthermore, uniqueness of the approximate solutions is guaranteed by Proposition 6.18. We would just like to point out that the additional stress tensor An dominates S owing to the choice of q. However, since S is monotone, the proof of Proposition 6.18 can be carried over without any essential difficulties. Recall that by the method of construction, the solutions (un )n∈N from Proposition 7.3 are of the form un = wn + v, where the fields wn are the unique weak solutions of the corresponding shifted systems subject to homogeneous boundary conditions on Γ. We now reformulate Proposition 7.3 in terms of the sequence (wn )n∈N as the shifted problems easily yield uniform energy estimates. Of course, the weak solution u we have announced in Theorem 7.1 will then also be of the form u = w + v, where w is the weak limit of the sequence (wn )n∈N . Proposition 7.4 (Existence and Uniqueness of Approximate Solutions II) For every n ∈ N there exists a unique vector field wn ∈ L∞ (I, Hσ (t)) ∩ Lq (I, Vq (t)) such that Z Z Z

n − wn (t), ∂t η(t) H(t) dt+ gv (wn (t)), η(t) Vq (t) dt + Sv (wn (t)), η(t) Vq (t) dt I

I

I

Z =

¯f (t), η(t) dt + w0n , η(0) H(0) Vq (t)

I

(7.4)

0 holds for every η ∈ X. Here, ¯f ∈ Lp (Q)3 again denotes f − ∂t v and the shifted operators gv ¯ L2 (R3 )3 ) and Snv are defined as in Definition 6.11. The zero extension of any wn lies in C(I, and the pointwise in time evaluation [ wn : I¯ → H(s), s 7→ wn (s),

s∈I¯

155


is well-defined for every n ∈ N. Moreover, for any wn , any s ∈ I¯ and any η ∈ H 1 (I, H(t)) ∩ Lq (I, Vq (t)) there holds

wn (s), η(s) H(s) − w0 , η(0) H =

Zs

¯f (t), η(t) dt − Vq (t)

0

Zs

gv (wn (t)), η(t) Vq (t) dt

0

Zs +

wn (t), ∂t η(t)

Zs dt − H(t)

0

n Sv (wn (t)), η(t) Vq (t) dt.

0

(7.5)

Proof. These statements follow directly from Theorem 6.16.

Our next goal is to derive uniform energy estimates for the sequence (wn )n∈N to justify the limiting process in (7.4). Since (7.4) and (7.3) are equivalent, this will allow us the passage to the limit in (7.3), too. Lemma 7.5 The sequence of approximate solutions (wn )n∈N satisfies the uniform estimate

1

wn p

wn ∞

Dwn q q + + ≤ C. L (I,Vp (t)) L (I,Hσ (t)) L (Q) n

(7.6)

Proof. As always, we start by using wn as a test function in (7.5). Again, this point can be justified by Theorem 3.20: For every n ∈ N there exists a sequence (wn )>0 ⊂ H 1 (I, H(t)) ∩ Lq (I, Vq (t)) such that wn → wn in Lq (I, Vq (t)), 0 ∂t wn * dt wn in Lq (I, Vq (t)) ,

as → 0. For any s ∈ I¯ we may then pass to the limit in the duality ∂t wn , χ[0,s] wn Lq (I,Vq (t)) to infer the energy identity

1 1

wn (s) 2

w0n 2 − = H(s) H(0) 2 2

Zs

¯f (t), wn (t) dt − Vq (t)

0

Zs

Snv (wn (t)), wn (t) Vq (t) dt

0

Zs − 0

gv (wn (t)), wn (t) Vq (t) dt. (7.7)

156


The definition of Sn , the (p, δ)-structure of S and the (q, κ)-structure of A then show Zs

n Sv (wn (t)), wn (t) + v(t) Vq (t) dt

0

Zs =

Snv (wn (t)), wn (t)

+ v(t)

Zs

Vq (t)

dt +

0

n Av (wn (t)), wn (t) + v(t) Vq (t) dt

(7.8)

0

Zs ≥ c

1

Dwn (t) + Dv(t) p p dt + L (Ω(t)) n

Zs

Dwn (t) + Dv(t) q q dt − K0 . L (Ω(t))

0

0

On the other hand there holds18 Zs

Snv (wn (t)), v(t) Xq (t) dt ≤

0

Zs

Dv(t) p

Dwn (t) + Dv(t) p−1 dt p L (Ω(t)) L (Ω(t))

0

Zs +

g1 (t) p0

Dv(t) p dt L (Ω(t)) L (Ω(t))

0

1 + n

Zs

(7.9)

Dwn (t) + Dv(t) q−1

Dv(t) q dt Lq (Ω(t)) L (Ω(t))

0

Zs +

Dv(t) q

g˜1 (t) q0 dt. L (Ω(t)) L (Ω(t))

0

Due to the smoothness of the vector field v we know that there exists a positive constant c = c(v) such that for r ∈ {p, q} there holds

max Dv(t) Lr (Ω(t)) ≤ c(v). t∈I¯

Therefore, we can apply Hölder’s inequality to obtain an upper bound for right-hand side of (7.9) of the form p−1 Zs Zs q−1

p

1 p q

Dwn (t) + Dv(t) p

Dwn (t) + Dv(t) q q C +C dt + C , dt L (Ω(t)) L (Ω(t)) n 0

0

where the constant C > 0 depends on c(v), kg1 kLp0 (Q) , k˜ g1 kLq0 (Q) and the exponents p, q but not on n ∈ N. With the help of Young’s inequality with , the quantity Zs p−1

p

Dwn (t) + Dv(t) p p C dt L (Ω(t)) 0 18

The functions g1 and g˜1 are connected to the (p, δ)-structure of S and the (q, κ)-structure of A by Lemma 6.2.

157


can be absorbed in the first term on the right-hand side of (7.8). Concerning the remaining term we estimate, again using Young’s inequality, Zs q−1

1 q

Dwn (t) + Dv(t) q q dt C L (Ω(t)) n 0

q−1 1 1/q 1 1/q0 Zs

q q

= Dwn (t) + Dv(t) Lq (Ω(t)) dt C n n 0

1 Zs

Dwn (t) + Dv(t) q q dt + C() ≤ L (Ω(t)) n 0

so that we can absorb this term on the right-hand side of (7.9) by choosing > 0 sufficiently small, too. These considerations allow us to treat the term Zs Zs

n

n Sv (wn (t)), wn (t) Vq (t) dt = Sv (wn (t)), wn (t) + v(t) − v(t) Vq (t) dt 0

0

Zs

=

Snv (wn (t)), wn (t)

+ v(t)

Xq (t)

Zs dt −

0

Snv (wn (t)), v(t) Xq (t) dt

0

which appears on the right-hand side of the energy identity (7.7). Simplifying and rearranging the resulting inequality we obtain Zs Zs

p

2

1

Dwn (t) + Dv(t) p

Dwn (t) + Dv(t) q q

wn (s) dt + + dt L (Ω(t)) L (Ω(t)) Hσ (s) n 0

˜+ ≤ K

Zs

0

¯f (t), wn (t) dt + Vq (t)

0

Zs

gv (wn (t)), wn (t) Vq (t) dt

0

(7.10)

˜ denotes a finite constant that particularly depends on various (bounded) norms of Here, K v. In order to estimate the shifted convective term we notice that Zs

gv (wn (t)), wn (t) Vq (t) dt 0

Zs Z

Zs Z

wn ⊗ v : ∇wn dx dt

wn ⊗ wn : ∇wn dx dt +

= 0 Ω(t) Zs Z

0 Ω(t) Zs Z

v ⊗ wn : ∇wn dx dt +

+

0 Ω(t)

0 Ω(t)

Zs

Z

= 0 Ω(t)

wn ⊗ v : ∇wn dx dt +

v ⊗ v : ∇wn dx dt

Zs

Z

0 Ω(t)

v ⊗ v : ∇wn dx dt.

158


From this identity it is not hard to see that the shifted convective term can be absorbed in the left-hand side of (7.10), too. A subsequent Gronwall-type argument and the triangle inequality then yield the desired estimate. From the estimate (7.6) we deduce that the additional stress tensors (An )n∈N vanish in the limit. In fact, for any test function η ∈ Lq (I, Vq (t)) there holds Z Z

n 1

Av (wn (t)), η(t) Vq (t) dt = Av (wn (t)), η(t) Vq (t) dt n I I (7.11)

q−1

1 ˜ ≤ K + Dwn Lq (Q) Dη Lq (Q) . n However, the a priori estimate (7.6) yields the uniform estimate 1

Dwn q q sup ≤ K L (Q) n∈N n which in turn implies

1 1/q 0 1

Dwn q−1 q (Q) ≤ K L n n1/q for any n ∈ N. Hence, the right-hand side of (7.11) tends to zero for every fixed test function as n → ∞. As an immediate consequence we even see that for n → ∞ Z

−1

1

n ADv (Dwn ) q0 A (Dw ) : F ≤ sup dx dt n Dv L (Q) n kFkLq (Q) ≤1 Q (7.12)

q−1 1 ˜ ≤ K + Dwn Lq (Q) → 0. n Now, just like in the previous chapters, the uniform a priori estimate (7.6) is not completely sufficient for the limiting process n → ∞ in (7.4): The passage to the limit in the shifted convective term requires strong convergence of a subsequence of (wn )n∈N . The non-cylindrical Aubin–Lions lemma and the compactness principle of Landes and Mustonen provide two possible methods to achieve the necessary compactness. We are now going to discuss the most significant points in both approaches but we refrain from going into details since both methods have already been worked out at large in previous chapters. The a priori estimate (7.6) yields two assumptions of Theorem 4.1 since L∞ (I, Hσ (t)) ∩ Lp (I, Vp (t)) ,→ L∞ (I, L1 (Ω(t))3 ) ∩ Lp (I, W01,p (Ω(t))3 ). As a consequence, we only need to check that for almost every t ∈ I, a subsequence (wn (t))n∈M1 weakly converges to w(t) in H(t) = L2 (Ω(t))3 ,→ L1 (Ω(t))3 . To this end we first construct special test functions by using the isomorphisms Pψt : Vq (0) → Vq (t) supplied by the Piola transform. See Lemma 3.7 for the details. Using these test functions in the identity (7.5), the weak compactness properties of the sequence (wn )n∈N , (7.12) and the localization procedure from Lemma 3.6 and Corollary 3.8, we can eventually show wn (t) * w(t) in Hσ (t)

159


for almost every t ∈ I and a suitable subsequence (wn )n∈M1 . From this auxiliary result we can subsequently infer the weak convergence wn (t) * w(t) in L2 (Ω(t))3 for almost every t ∈ I thanks to the Leray–Helmholtz decomposition ⊥

L2 (Ω(t))3 = Hσ (t) ⊕ Hσ (t)⊥ . Invoking the properties of the Piola transform once more, the compactness principle of Landes and Mustonen yields a further subsequence (wn )n∈M ⊂ (wn )n∈M1 that strongly converges to w in Lp (Q)3 . Here, w of course denotes the weak limit of the sequence (wn )n∈M1 ⊂ L∞ (I, Hσ (t)) ∩ Lp (I, Vp (t)). Without loss of generality we may also assume that lim

wn = w

M3n→∞

holds almost everywhere in Q. We can alternatively apply the non-cylindrical version of the Aubin–Lions lemma by checking the assumptions of Theorem 5.4 and Definition 5.6 for the current functional setting. The most important point in Theorem 5.4 is the generalized equicontinuity condition (5.6) which resembles the uniform Hölder continuity of (wn )n∈N . The latter is a consequence of the available regularity of the generalized time derivatives (dt wn )n∈N . More precisely, the approximate solutions (wn )n∈N satisfy Z Z Z

n − wn (t), ∂t η(t) H(t) dt+ gv (wn (t)), η(t) Vq (t) dt + Sv (wn (t)), η(t) Vq (t) dt I

I

I

Z =

¯f (t), η(t) dt + w0n , η(0) H(0) Vq (t)

I

for every test function η ∈ turn implies

ϕ ∈ H 1 (I, H(t)) ∩ Lq (I, Vq (t)) ϕ(T ) = 0 . This identity in

sup dt wn n∈N

Lq (I,Vq (t))

0 ≤ C.

A straightforward adaption of the arguments we had used in Proposition 6.14 provides the generalized equicontinuity of the solutions in the form ˆ , Pψs ) − MVq (s0 )0 (wn , η ˆ , Pψs0 ) ≤ C h |s0 − s| , sup sup MVq (s)0 (wn , η n∈N kˆ η kVq (0) ≤ c−1 0

for a function h : R+ → R+ which is continuous at 0. In order to conclude with Lemma 5.8 we need to check that the triples 0 Vp (t), Hσ (t), Vq (t) , t∈I¯

160


satisfy19 for any t ∈ I¯ i

Vp (t) ,→,→ Hσ (t) ,→,→ Vq (t)0 and Hσ (t) ,→ Vq (t)0

(7.13)

and that they have the uniform Ehrling property, see Definition 5.6. However, we can repeat Proposition 5.10 and the relevant preceding arguments almost literally to prove (7.13). Thus, by Lemma 5.8, the sequence (wn )n∈N is relatively compact in Lp (I, Hσ (t)). Since the sequence is additionally bounded in L∞ (I, Hσ (t)) we deduce the relative compactness of (wn )n∈N in L2 (Q)3 from the estimate Z

wn (t) − w(t) 2

wn − w 2 2 dt = Hσ (t) L (Q) ZI =

wn (t) − w(t) p

wn (t) − w(t) 2−p dt Hσ (t) Hσ (t)

I

p ≤ C wn − w Lp (I,Hσ (t)) → 0 for M 3 n → ∞. In the end, a subsequence of (wn )n∈N converges almost everywhere in Q to w. We summarize the compactness result for the convective term in the following lemma. Lemma 7.6 There exists a subsequence (wn )n∈M ⊂ (wn )n∈N such that Z Z

lim gv (wn (t)), η(t) Vq (t) dt = gv (w(t)), η(t) Vq (t) dt M3n→∞

I

is satisfied for every η ∈

(7.14)

I

Lq (I, V

q (t)).

lim M3n→∞

Moreover, for sufficiently small r > 1 there holds

wn ⊗ wn = w ⊗ w in Lr (Q)3×3 .

Proof. The above compactness argument implies that (wn ⊗ wn )n∈M strongly converges to w ⊗ w almost everywhere in Q. By (7.6) the sequence (wn )n∈M is uniformly bounded in L∞ (I, Hσ (t))∩Lp (I, Vp (t)) and this space is continuously embedded in L5p/3 (Q)3 . Passing to another subsequence if necessary, we can assume without loss of generality that the sequence (wn ⊗ wn )n∈M weakly converges in L5p/6 (Q)3×3 , too. However, by Lemma A.1 these two limits coincide so that 7.14 follows. The strong convergence in Lr (Q)3×3 for small r > 1 is guaranteed by our compactness arguments, too. The a priori estimate (7.6) and the previous lemma now justify the passage to the limit in the approximate system (7.4). Since un = wn + v, we can equally pass to the limit M 3 n → ∞ in (7.3). We consequently obtain that for any test function η ∈ X there holds Z Z Z

− u(t), ∂t η(t) H(t) dt + g(u(t)),η(t) Vq (t) dt + χ(t), η(t) Vq (t) dt I

I

I

Z = I

19 Recall that p > 6/5 and q ≥ max (5p/6)0 , 5/2 .

f (t), η(t) Vq (t) dt + u0 , η(0) H(0) .

(7.15)

7.2 Identification via Solenoidal Lipschitz Truncation

161

0 Here, χ ∈ Lp (Q)3×3 denotes the weak limit of the sequence S(Dun ) n∈N which is uniformly 0 bounded in Lp (Q)3×3 thanks to the a priori estimate and the (p, δ)-structure of S.

7.2. Identification via Solenoidal Lipschitz Truncation 0

We still have to identify the weak limit χ ∈ Lp (Q)3×3 . Notice that we cannot proceed as usually by using Theorem 3.20 and Minty’s trick: The limit equation ∗ (7.15) shows that the generalized time derivative of u ∈ Lp (I, Xp (t)) lies in Lq (I, Vq (t)) , or to put it equiva∗ lently, the generalized time derivative of w ∈ Lp (I, Vp (t)) lies in Lq (I, Vq (t)) and not in ∗ Lp (I, Vp (t)) . As a consequence, we see that in the limit, we dropped out of the functional setting of Theorem 3.20. However, at this point we can alternatively apply the implications of the solenoidal Lipschitz truncation method developed by Breit, Diening and Schwarzacher. Taking the difference of the equation for some approximate solution un and the limit equation (7.15) we see that Z Z

− un (t) − u(t), ∂t η(t) H(t) dt = χ(t) − S(un (t)), η(t) Vq (t) dt I

I

Z +

g(u(t)) − g(un (t)) − An (un (t)), η(t) Vq (t) dt

I

holds for every η ∈ C0∞ (Q)3 with div η = 0. Setting hn := un − u and Hn := H1n + H2n with H1n : = χ − S(Dun ), H2n : = u ⊗ u − un ⊗ un −

1 A(Dun ) n

the above identity may equivalently be formulated as Z Z − hn · ∂t η dx dt = Hn : ∇η dx dt Q

Q

for every vector field η ∈ C0∞ (Q)3 with div η = 0. Now let Q0 := J × B, where J ⊂ I is an interval and B is a ball in R3 , be any cylindrical domain such that Q0 ⊂⊂ Q. The weak convergence results we had established up to now then imply hn * 0 in Lp (J, Wσ1,p (B)3 ) hn → 0 in Lr (Q0 )3 , ∗

hn * 0 in L∞ (J, Yσ (B)) 0

H1n * 0 in Lp (Q0 )3×3 . for M 3 n → ∞ and some sufficiently small r > 1. Here, the definition of the spaces Wσ1,p (B)3 and Yσ (B) follows the one given in (7.2). Moreover, together with (7.12), the compactness result from Lemma 7.6 yields lim M3n→∞

H2n = 0 in Lr (Q0 )3×3 ,

162


for some small r > 1. This setting serves as a starting point for the identification of χ by means of the solenoidal Lipschitz truncation method, which essentially boils down to the following two results. Theorem 7.7 (Solenoidal Lipschitz Truncation) Suppose 1 < p < ∞ and 1 < r < p, p0 . Let hn and Hn satisfy Z Z − hn · ∂t η dx dt = Hn : ∇η dx dt Q0

Q0

for every n ∈ N and for any η ∈ C0∞ (Q0 )3 with div η = 0. Assume that (hn )n∈N is uniformly bounded in L∞ (J, Lr (B)3 ) and that hn * 0 in Lp (J, Wσ1,p (B)3 ), hn → 0 in Lr (Q0 )3 as n → ∞. Moreover, assume that Hn = H1n + H2n is such that for n → ∞ 0

H1n * 0 in Lp (Q0 )3×3 , H2n → 0 in Lr (Q0 )3×3 . k

k+1

Then there is a double sequence (λn,k )n,k∈N ⊂ R+ , 22 ≤ λn,k ≤ 22 , and (hn,k )n,k∈N := αn,k αn,k 2−p , and open sets On,k := Oλn,k that satisfy the hλn,k ⊂ L1 (Q0 )3 , αn,k := λn,k n,k∈N n,k∈N following properties for any k ≥ k0 : 1,s 1 • hn,k ∈ Ls 41 J, W0,σ ( 6 B)3 for any 1 < s < ∞ and supp hn,k ⊂ 16 Q0 .20 • hn,k = hn almost everywhere in

• ∇hn,k L∞ ( 1 Q0 ) ≤ c λn.k .

1 8

Q0 \ On,k .

4

• hn,k → 0 in Ls ( 14 Q0 )3 , 1 ≤ s < ∞, for n → ∞ and any fixed k. • ∇hn,k * 0 in Ls ( 14 Q0 )3×3 , 1 ≤ s < ∞, for n → ∞ and any fixed k. • lim sup λpn,k On,k ≤ c 2−k . n→∞

R • lim sup Q0 Hn : ∇hn,k dx dt ≤ c λpn,k On,k . n→∞

Proof. The proof of this highly sophisticated result is given in [BDS13]. The sequence (hn,k )n,k∈N is called solenoidal Lipschitz approximation (of (hn )n∈N ). The first property shows that the approximation preserves the incompressibility of hn whereas the second assertion reflects the fact that the approximation coincides with the original sequence on 20

For α > 0 we denote by αQ the cylinder Q scaled by α with respect to its center. The same convention is used for intervals and balls

163


a large set. Here, the sets On,k can be characterized in terms of level sets of the HardyLittlewood maximal function applied to another auxiliary approximate sequence. This twostage approximation procedure is necessary due to the incompressibility constraint which is handled by a very careful and subtle application of the curl-operator in a first step. The continuity of the maximal function and the compactness of the sequence (hn )n∈N provide control over the set of non-coincidence On,k . The results of Theorem 7.7 are not restricted to the three-dimensional case. However, as the whole construction heavily relies on the properties of the curl-operator, the higher dimensional case must be formulated using differential forms. We cite yet another result from [BDS13] which turns out to be useful below. Corollary 7.8 Let the assumptions of Theorem 7.7 be satisfied. Then there exists ξ ∈ C0∞ ( 16 Q0 ), χ 1 Q0 ≤ 0

8

ξ ≤ χ 1 Q0 and a constant c > 0 that only depends on ξ such that for any K ∈ Lp ( 16 Q0 )3×3 6 there holds Z lim sup H1n + K : ∇hn ξ χO{ dx dt ≤ c 2−k/p . n,k

n→∞

Q0

In order to identify χ it is sufficient to prove that for some 0 < θ < 1 lim sup M3n→∞

Z

θ S(Du) − S(Dun ) : D(u − un ) dx dt

1 Q 8 0

≤ lim sup M3n→∞

Z

(7.16) θ S(Du) − S(Dun ) : D(u − un ) ξ dx dt = 0

Q0

In fact, by monotonicity of the integrand we may then conclude lim

S(Du) − S(Dun ) : D(u − un ) = 0

M3n→∞

almost everywhere in 18 Q0 . From the lemma of Dal-Maso and Murat, see Lemma 3.18, we infer lim

Dun = Du

M3n→∞

almost everywhere in 18 Q0 . The continuity of the stress tensor S then finally implies 0

S(Du) = χ in Lp (Q)3×3 since Q0 was arbitrary. Let us now show (7.16). While the first inequality follows from the properties of the cut-off function ξ, we will use Theorem 7.7 and Corollary 7.8 to establish

164


convergence of the second integral. We have Z θ S(Du) − S(Dun ) : D(u − un ) ξ dx dt Q0

=

Z

θ S(Du) − S(Dun ) : D(u − un ) ξχOn,k dx dt

Q0

Z

+

θ S(Du) − S(Dun ) : D(u − un ) ξχO{ dx dt. n,k

Q0

Using Hölder’s inequality, the first term on the right-hand side is bounded by θ Z S(Du) − S(Dun ) : D(u − un ) dx dt |On,k |1−θ . c Q0

By the sixth point in Theorem 7.7, the a priori estimates for (un )n∈M and the (p, δ)-structure of S we obtain Z θ lim sup S(Du) − S(Dun ) : D(u − un ) ξχOn,k dx dt ≤ c 2(θ−1)k . M3n→∞

Q0

On the other hand Z

θ S(Du) − S(Dun ) : D(u − un ) ξχO{ dx dt n,k

Q0

Z

=

θ S(Du) − S(Dun ) : D(u − un ) ξ ξ 1−θ dx dt

{ On,k

≤ c

Z

θ S(Du) − S(Dun ) : D(u − un ) ξ χO{ dx dt n,k

Q0

by Hölder’s inequality. Setting K := S(Du) − χ we get S(Du) − S(Dun ) = H1n + K. Therefore, Corollary 7.8 yields Z θ ˜ lim sup S(Du) − S(Dun ) : D(un − u) ξχO{ dx dt ≤ c 2−kθ , M3n→∞

n,k

Q0

θ˜ := θ/p, so that eventually Z θ ˜ lim sup S(Du) − S(Dun ) : D(u − un ) ξ dx dt ≤ c 2(θ−1)k + c 2−kθ . M3n→∞

Q0

Since k can be chosen arbitrary large, the assertion follows. Gathering the results we obtained from the solenoidal Lipschitz truncation method, we especially deduce

165


Corollary 7.9 Almost everywhere in the non-cylindrical domain Q there holds lim

Dun = Du.

M3n→∞

Thus, passing to the limit M 3 n → ∞ in (7.3), we found a vector field u ∈ L∞ (I, Yσ (t)) ∩ Lp (I, Xp (t)) that satisfies for any η ∈ X the identity Z Z Z

S(u(t)), η(t) Vq (t) dt g(u(t)), η(t) Vq (t) dt + u(t), ∂t η(t) H(t) dt+ − I

I

I

Z =

f (t), η(t)

Vq (t)

dt + u0 , η(0)

H(0)

.

I

Since u = w + v for w ∈ Lp (I, Vp (t)), we immediately deduce that the nonhomogeneous boundary condition u(t) − v(t) ∈ Vp (t) is valid for almost every t ∈ I. That is is to say, u is a weak solution of the p-Navier–Stokes system (7.1) in the sense of Theorem 7.1.

166


8. Outlook At the end we want to illustrate some potential applications of the current work and also highlight some further research directions. As for many unsteady problems in the cylindrical setting, the monotone operator theory we extended to the non-cylindrical one can be used first and foremost in order to provide existence of solutions within the natural functional setting of the problem at hand. In this context, the canonical solutions are weak solutions, having the advantage of existing for any finite time. However, the drawback of weak solutions for many advanced applications is that they do not possess higher regularity properties a priori. Regularity theory for unsteady nonlinear problems though, whether this be higher integrability or higher differentiability, still is a highly active field. An obvious investigation would therefore consist of transfering higher regularity results which are yet available for problems on cylindrical domains. Within the existence theory for nonlinear unsteady problems, techniques from maximal Lp -regularity theory have become increasingly popular in the last three decades. The reason therefore lies in the fact that this theory yields strong solutions that (in some sense) inherit their intrinsic smoothness from the data and the structure of the equation as a (non)autonomous Cauchy problem. However, these solutions usually only exist for short time. To the best of our knowledge, there are no results using maximal regularity techniques directly on non-cylindrical domains, but there are a few results in this context where the non-cylindrical problem is transformed into an auxiliary cylindrical one in a first step. Consequently, one could try to combine the amenities of both approaches to further improve intrinsic non-cylindrical techniques. Going back to the definition of our non-cylindrical domains, the smoothness assumptions on the generating vector field admittedly seem to be quite restrictive. Although these assumptions can surely be weakened, a certain degree of smoothness is still necessary in order to benefit from the Piola transform and its properties. Our approach is however restricted to rather smooth deformations of the initial domain. As a consequence, topological changes along the deformation process, such as the appearance of cracks or holes, cannot be taken into account, although this might be desirable in applications. But still, some of our methods can be used for example in the context of shape optimization problems under PDE constraints. Here, both the solution of the PDE and the vector field that determines the shape or the evolution of the domain enter certain cost functionals that are to be minimized subject to suitable optimality conditions. On the one hand one is then interested in how the solution of the PDE depends on the domain or its deformation. On the other hand the above mentioned minimization problems typically demand for compactness methods within the framework of generalized Bochner spaces over non-cylindrical domains. Our generalization of the compactness principle of Landes and Mustonen and the non-cylindrical version of the Aubin-Lions lemma can certainly be implemented there. As we had already mentioned in previous chapters, a natural field of application for our techniques is given by FSI problems. In that context, one could possibly use some of our

167

168

Chapter 8. Outlook

results in order to provide theoretical existence results in connection with the ALE (Arbitrary Lagrangian-Eulerian) method which lies at the heart of both theoretical and numerical approaches to FSI problems. One of the most challenging problems here are error estimates between continuous solutions on the non-cylindrical domain and their discrete counterparts which are usually obtained from numerical schemes. Such error estimates are indispensable for bridging the gap between theoretical and numerical approaches. Furthermore, they are of utmost importance for a deeper understanding of numerical experiments.

A. Appendix A.1. Lebesgue Spaces For a measurable subset Ω ⊂ Rd , d ∈ N, we denote by Lp (Ω), 1 ≤ p ≤ ∞, the space of (classes of) Lebesgue-measurable functions u : Ω → R, endowed with the norm 1/p Z

u p dx

u p , 1 ≤ p < ∞, := L (Ω) Ω

u ∞ := esssup u(x) . L (Ω) x∈Ω

If 1 ≤ p < ∞ the spaces Lp (Ω) are separable and if 1 < p < ∞ they are reflexive. In 0 0 p the reflexive case, the dual space L (Ω) is isometrically isomorphic to Lp (Ω), where 1/p + 1/p0 = 1. By Lp (Ω)l , l ∈ N, we denote the space of Lp fields u : Ω → Rl . These spaces share the above properties of Lp spaces of functions. An important fact we often use is the following lemma, the proof of which can be found in [GGZ74, Lemma 1.19]. Lemma A.1 Suppose that (un )n∈N is a sequence in Lp (Ω), 1 ≤ p < ∞ such that (un )n∈N converges to u weakly in Lp (Ω) and almost everywhere in Ω to v as n → ∞. Then there holds u = v in Lp (Ω). In order to formulate Vitali’s theorem we introduce another notion of convergence for a sequence of measurable functions (un )n∈N : Ω → R. We say that the sequence converges to u locally in measure, if for every > 0 and every measurable subset M ⊂ Ω there holds lim x ∈ Ω |un (x) − u(x)| ≥ ∩ M = 0. n→∞

The sequence converges to u in measure, if for every > 0 there holds lim x ∈ Ω |un (x) − u(x)| ≥ = 0. n→∞

Theorem A.2 (Vitali) Let Ω ⊂ Rd , d ∈ N, be measurable and 1 ≤ p < ∞. For (un )n ∈ N, u ∈ Lp (Ω) the following assertions are equivalent: • lim un = u in Lp (Ω). n→∞

•

– (un )n∈N converges to u locally in measure

169

170

Chapter A. Appendix

– For any > 0 there exists a measurable subset M ⊂ Ω with finite Lebesgue measure such that Z |u|p dx ≤ . sup n∈N

Mc

– For any > 0 there is a δ > 0 such that for every measurable subset M ⊂ Ω the following implication holds: Z M ≤ δ ⇒ sup |u|p dx ≤ . n∈N M

Proof. The proof can be found in [Els05]. A sequence with the properties 2)b) and 2)c) is called uniformly integrable in the p-th mean. If Ω has finite Lebesgue measure one can abandon condition 2)b) by choosing M = Ω. Moreover, if (un )n∈N converges to u almost everywhere in Ω then (un )n∈N converges to u (locally) in measure. A useful implication of Vitali’s theorem is the following compactness result. Corollary A.3 let Ω ⊂ Rd , d ∈ N, be measurable with finite Lebesgue measure. Suppose that (un )n∈N is uniformly bounded in Lp (Ω), 1 < p ≤ ∞, and converges to u in measure. Then, for any 1 ≤ q < p there holds lim un = u in Lq (Ω).

n→∞

Proof. The proof easily follows from Vitali’s theorem by noticing that since |Ω| < ∞, uniform boundedness in Lp (Ω), 1 < p ≤ ∞, implies uniform integrability in the q-th mean for 1 ≤ q < p. Proposition A.4 (Transformation Formula) Let Ω ⊂ Rd , d ∈ N, be open and let ϕ : Ω → Ω0 := ϕ(Ω) ⊂ Rd be a diffeomorphism. Then, u ∈ L1 (Ω0 ) if and only if Jϕ u ◦ ϕ ∈ L1 (Ω) and the following transformation formula for integrals holds true: Z Z u dx = Jϕ u ◦ ϕ dx. Ω0

Ω

Here, Jϕ denotes the Jacobian determinant of ϕ. Proof. This result follows from [Lee03, Proposition 14.2]. We also introduce the Hardy-Littlewood maximal operator which is one of the most powerful tools in harmonic analysis.

171

A.1 Lebesgue Spaces

Definition A.5 (Hardy-Littlewood Maximal Function) For u ∈ L1loc (Rd ) the (centered) Hardy-Littlewood maximal function at x0 ∈ Rd is defined by Z

1 M u(x0 ) := sup r>0 |Br (x0 )|

|u| dx. Br (x0 )

Here, Br (x0 ) stands for a ball of radius r > 0 centered at x0 . The operator u 7→ M u is called Hardy-Littlewood maximal operator. The most important facts concerning the centered maximal operator a summarized in the following theorem. For a proof we would like to refer to [Ste93], [Ste70] or [Gra08a]. Theorem A.6 For u ∈ L1loc (Rd ) the function M u : Rd → [0, ∞] is lower semi-continuous and |u| ≤ M u holds almost everywhere. If u ∈ Lp (Rd ), 1 ≤ p ≤ ∞, then the function M u is finite almost everywhere. For any u ∈ Lp (Rd ), 1 < p ≤ ∞ there holds

M u p d ≤ c u p d . L (R ) L (R ) The constant c only depends on p and d and blows up as p → 1. In connection with parabolic problems one is led to distinguish between the temporal and the spatial variable. Following this distinction we introduce yet two more maximal operators acting on functions defined on Rd+1 . Given u ∈ L1loc (Rd+1 ) we know that up to null sets we have u(t, ·) ∈ L1loc (Rd ) and u(·, x) ∈ L1loc (R). Then we define Z

1 M u(t, x0 ) := sup r>0 |Br (x0 )| s

|u(t, x)| dx, Br (x0 )

1 M u(t0 , x) := sup ρ>0 |Iρ (t0 )| τ

Z |u(t, x)| dt, Iρ (t0 )

where Iρ (t0 ) := [t0 − ρ, t0 + ρ]. Now, for u ∈ Lp (Rd+1 ), 1 < p ≤ ∞, Theorem A.6 yields kM s u(t, ·)kLp (Rd ) ≤ cku(t, ·)kLp (Rd ) for almost every t ∈ R. Integrating with respect to t we obtain

s

M u p d+1 ≤ c u p d+1 . L (R ) L (R ) Similar reasoning applied to M τ yields

τ

M u p d+1 ≤ c u p d+1 . L (R ) L (R ) The next theorem is a direct consequence of the last two estimates

172

Chapter A. Appendix

Theorem A.7 The operator M † := M τ ◦ M s is bounded as a mapping from Lp (Rd+1 ) to Lp (Rd+1 ), 1 < p ≤ ∞. Remark A.8. The results about the maximal operator are not restricted to the case where the supremum of centered averages is considered. For example, the boundedness of the ˜ τ , where, setting I˜ρ (t0 ) := temporal maximal operator M τ implies the boundedness of M [t0 , t0 + ρ], Z 1 τ ˜ |u(t, x)| dt. M u(t0 , x) := sup ρ>0 |I˜ρ (t0 )| I˜ρ (t0 )

A.2. Sobolev Spaces We also list some facts about Sobolev spaces that are used throughout this work. For a detailed introduction and further results about this topic we refer to [AF03] and [BF13]. For ease of notation, we formulate most of the results for the case of functions, although they remain valid for fields and although we mostly use them for fields. For any open set Ω ⊂ Rd , d ∈ N, we denote by W k,p (Ω), k ∈ N, the Sobolev space of real-valued Lp (Ω) functions whose distributional derivatives up to order k lie in Lp (Ω). For any k ∈ N and 1 ≤ p < ∞, the Sobolev spaces W k,p (Ω) are separable Banach spaces under the norm 1/p X

u k,p

∂ α u p p := . W (Ω) L (Ω) |α|≤k

For 1 < p < ∞ the spaces W k,p (Ω) are reflexive. By W k,p (Ω)l , l ∈ N, we denote the Sobolev space of vector fields u : Ω → Rl . These spaces share the above properties of Sobolev spaces of functions. The space W0k,p (Ω) is defined as the closure of C0∞ (Ω) in W k,p (Ω). Theorem A.9 Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary. Then, the mapping u 7→ u|∂Ω which is well-defined on C 1 (Ω), can be continuously extended to γ : W 1,p (Ω) → γ W 1,p (Ω) =: W 1−1/p,p (∂Ω) ⊂ Lp (∂Ω) u 7→ u|∂Ω . The mapping γ is called trace operator. The space W01,p (Ω), 1 ≤ p < ∞ can be characterized as the kernel of the trace operator. Let U ⊂ Rd be an open, bounded set such that Ω ⊂ U . Then there exists a linear extension operator E : W 1,p (Ω) → W 1,p (U ) u 7→ E(u), such that kE(u)kW 1,p (U ) ≤ c kukW 1,p (Ω) and such that E(u) = u almost everywhere in Ω. The constant c only depends on p, Ω and U .

173

A.2 Sobolev Spaces

Proof. For the proof we refer to [AF03]. Theorem A.10 (Chain Rule for Sobolev Functions) Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary and let ϕ : Ω → Ω0 := ϕ(Ω) ⊂ Rd be a smooth diffeomorphism with bounded derivatives up to order k ∈ N. Then ϕ extends to an isomorphism ϕ∗ : W k,p (Ω0 ) → W k,p (Ω), u 7→ ϕ∗ u := u ◦ ϕ. There exists constants c1 , c2 such that

c1 u W k,p (Ω0 ) ≤

for any u ∈ W k,p (Ω0 ) there holds

∗

u k,p 0 .

ϕ u k,p ≤ c 2 W (Ω ) W (Ω)

If ψ : Ω0 → Ω denotes the inverse of ϕ, then the inverse of ϕ∗ is given by ψ ∗ . Proof. The proof is given in [BF13, Theorem III.2.13]. 0 0 One can show that for any F ∈ W0k,p (Ω) , there exist fα ∈ Lp (Ω), |α| ≤ k, such that for any u ∈ W0k,p (Ω) there holds X Z

X F, u W k,p (Ω) = (−1)|α| ∂ α fα , u W k,p (Ω) := fα ∂ α u dx. 0

0

|α|≤k

|α|≤k Ω

Theorem A.11 (de Rham) 0 Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary. Suppose F ∈ W01,p (Ω)d , 1 < p < ∞, satisfies

F, ϕ W 1,p (Ω)d = 0 0

0

∞ d p for R any ϕ ∈ C0 (Ω) with div ϕ = 0. Then there exists a unique function π ∈ L (Ω) with Ω π dx = 0 such that F = ∇π holds in the sense of distributions.

Proof. The proof can be found in [Gal94, Theorem III.3.1 and Theorem III.5.2]. Theorem A.12 (Nečas) Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary. Suppose π ∈ W01,2 (Ω) 0 and ∂i π ∈ W01,2 (Ω) for every i = 1, ..., d. Then there is a constant c > 0 such that

2

π 2

π 2 ≤ c L (Ω)

0 +

W01,2 (Ω)

d X

2

∂i π i=1

0

0 .

W01,2 (Ω)

Proof. This is [BF13, Theorem IV.1.1.] We gather some results about classical spaces that arise in the mathematical theory of the Navier-Stokes equation

174

Chapter A. Appendix

Theorem A.13 Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary. Define n o Hdiv (Ω) := u ∈ L2 (Ω)d div u ∈ L2 (Ω) which is a Banach space under the canonical norm

u H

div

(Ω)

1/2

2

2 . := u L2 (Ω) + div u L2 (Ω)

The space C0∞ (Ω)d is dense in Hdiv (Ω). The mapping u 7→ u · n which is well-defined on C 1 (Ω)d extends to a linear, continuous operator γn : Hdiv (Ω) → γn Hdiv (Ω) =: H −1/2 (∂Ω). The space Hdiv ,0 (Ω), defined as the closure of C0∞ (Ω)d in Hdiv (Ω), the kernel coincides with ∞ (Ω)d := ϕ ∈ C ∞ (Ω)d div ϕ = 0 of γn . Let furthermore Hσ stand for the closure of C0,σ 0 in L2 (Ω)d . Then Hσ can be characterized as n o Hσ = u ∈ L2 (Ω)d div u = 0, γn (u) = 0 and Hσ⊥ , the orthogonal complement of Hσ in L2 (Ω)d , is given by n o ⊥ 2 d Hσ = u ∈ L (Ω) u = ∇π, π ∈ W 1,2 (Ω) . The orthogonal decomposition ⊥

L2 (Ω)d = Hσ ⊕ Hσ⊥

(A.1)

is called Leray-Helmholtz decomposition. Proof. Proofs of the assertions are given in [BF13, Theorem III.2.42, Theorem III.2.43, Theorem III.2.45, Theorem IV.3.5]. We now cite a classic embedding result for Sobolev spaces. The proof can be found in any textbook on this topic. Theorem A.14 Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary. If 1 ≤ p < d, there is a continuous embedding W 1,p (Ω) ,→ Lq (Ω)

(A.2)

dp for any 1 ≤ q ≤ p∗ := d−p . The constant of embedding only depends on d, p and Ω. If ∗ 1 ≤ q < p , the embedding (A.2) is compact.

Another useful result that is used throughout this work is the so-called Poincaré inequality.

175

A.3 Bochner Spaces

Proposition A.15 Let Ω ⊂ Rd , d ∈ N, be a bounded domain with Lipschitz boundary. For any u ∈ W01,p (Ω), 1 ≤ p < ∞, there holds

∇u p .

u p ≤ c L (Ω) L (Ω) The constant c only depends on d, p and Ω. Thus, k∇ · kLp (Ω) induces an equivalent norm on W01,p (Ω).

A.3. Bochner Spaces Definition A.16 Let I ⊂ R be Lebesgue measurable and let Z be a Banach space. A function u : I → Z is called simple if there exist zi ∈ Z, i = 1, ..., n, n ∈ N, and Lebesgue measurable sets Bi ⊂ I, i = 1, ..., n, with |Bi | < ∞, such that u=

n X

χBi zi .

i=1

The Bochner integral of a simple function u is defined as Z u(t) dt :=

n X

|Bi |zi ∈ Z.

i=1

I

Definition A.17 A function u : I → Z is called Bochner measurable if there is a sequence of step functions (un )n∈N : I → Z such that for almost every t ∈ I there holds

lim un (t) − u(t) Z = 0.

n→∞

Lemma A.18

If u : I → Z is Bochner measurable then u(·) Z : I → R is Lebesgue measurable. Hence, a Bochner measurable function u : I → Z is called Bochner integrable if Z

u(t) − un (t) dt = 0. lim Z n→∞

I

In that case, the Bochner integral of u is given by Z Z u(t) dt := lim un (t) dt. n→∞

I

176

Chapter A. Appendix

Theorem A.19 (Pettis) Let Z be a separable Banach space. A function u : I → Z is Bochner measurable if and only if F, u(·) Z : I → R is Lebesgue measurable for any F ∈ Z 0 . Proof. For a proof we refer to [Yos80]. Definition A.20 The space Lr (I, Z), 1 ≤ r < ∞, is defined as the set of Bochner measurable functions u : I → Z such that 1/r Z

u(t) r dt

u r < ∞. := Z

L (I,Z)

I

The space L∞ (I, Z) consists of Bochner measurable functions u : I → Z such that

u ∞ := esssup u(t) Z < ∞. L (I,Z) t∈I

Theorem A.21 The Bochner spaces Lr (I, Z), 1 ≤ r ≤ ∞ are Banach spaces. Proof. A proof can be found in [GGZ74]. Proposition A.22 Identifying u ∈ Lr (I, Z), 1 ≤ r < ∞, with its zero extension in time, the convolution with ρ ∈ C0∞ (R) Z Z u ∗ ρ (s) := u(t)ρ(s − t) dt = u(s − t)ρ(t) dt R

R

exists for any t ∈ R. There holds u ∗ ρ ∈ C ∞ (R, Z) and ∂tα (u ∗ ρ) = u ∗ (∂tα ρ) for any α ∈ N. The convolution defines a bounded mapping ∗ : Lr (I, Z) × C0∞ (R) → Lr (I, Z) with

u r

ρ 1 .

u ∗ ρ r ≤ L (R) L (I,Z) L (I,Z) For h ∈ R, the translation τh : u → τh u := u(· + h) defines R a bounded mapping τh : r r ∞ L (I, Z) → L (I, Z). Now suppose ρ ∈ C0 (R) is such that R ρ(t) dt = 1. Defining a sequence of mollifiers ρ (t) := −1 ρ(t/), > 0, for any u ∈ Lr (I, Z), 1 ≤ r < ∞, there holds lim u ∗ ρ = u in Lr (I, Z),

→0

¯ Z) is dense in Lr (I, Z), 1 ≤ r < ∞. which means that C ∞ (I, Proof. For a detailed proof of the latter proposition we refer to [Dro01].

177

A.4 Monotone Operators

Theorem A.23 Let Z be a separable and reflexive Banach space and suppose 1 < r < ∞. Let 1 < r0 < ∞ denote the dual exponent, i.e. 1/r + 1/r0 = 1, then there is an isometric isomorphism 0 0 I : Lr (I, Z 0 ) → Lr (I, Z) Z

f (t), v(t) Z dt, v ∈ Lr (I, Z). I(f ), v Lr (I,Z) := I

Moreover, if Z is separable and reflexive, so are the spaces Lr (I, Z), 1 < r < ∞. Proof. The proof is given in [GGZ74]. The results remain valid for Banach spaces Z having the Radon-Nikodym property.

A.4. Monotone Operators Definition A.24 Let X be a real, reflexive Banach space. An operator A : X → X 0 is called • monotone, if for any u, v ∈ X there holds

Au − Av, u − v X ≥ 0. • hemicontinuous, if the mapping

7→ A(u + v), w X is continuous on [0, 1] for any u, v, w ∈ X. • coercive, if

Au, u X → ∞. lim kukX kukX →∞ If X is additionally separable, then the main theorem on monotone operators states that a monotone, hemicontinuous and coercive operator A : X → X 0 is surjective. The proof is usually based on a Galerkin scheme and weak compactness methods. However, the identification of weak limits is established by using the following lemma which is referred to as Minty’s trick. Lemma A.25 (Minty’s Trick) Let X be a reflexive Banach space and suppose that A : X → X 0 is monotone, hemicontinuous and coercive. Then A is maximal monotone. That is, if u ∈ X and b ∈ X 0 are such that for any v ∈ X there holds

b − Av, u − v X ≥ 0, (A.3) then b = Au.

178

Chapter A. Appendix

A.5. Auxiliary Results Lemma A.26 (Gronwall) Let u ∈ L∞ (I) be non-negative and suppose there are g ∈ L1 (I) and u0 ∈ R such that for almost every s ∈ I there holds Zs u(s) ≤ u0 +

g(t)u(t) dt. 0

Then, for almost every s ∈ I there holds u(s) ≤ u0 exp

Zs

g(t) dt .

0

Proof. The proof is given in [BF13, Lemma II.4.10]. Proposition A.27 Isomorphisms preserve reflexivity. That is, if P : X → Y is a linear isomorphism between Banach spaces X, Y , then Y is reflexive if and only if X is reflexive. Proof. By Kakutani’s characterization of reflexivity, see [Bre11, Theorem 3.17], X is reflexive if and only if the unit ball B1X is compact in the weak topology τ (X, X ∗ ) on X. Since P is linear and continuous, it is continuous as a mapping P : X, τ (X, X ∗ ) → Y, τ (Y, Y ∗ ) . (A.4) Moreover, there is r > 0 such that B1Y ⊂ P(BrX ). X being reflexive, BrX is weakly compact by Kakutani’s theorem and an easy scaling argument. Thus, by the continuity of the mapping in (A.4) we deduce that P(BrX ) is weakly compact in Y . Now, B1Y is strongly closed and convex, thus, by Mazur’s lemma, it is weakly closed. Since B1Y is contained in the weakly compact set P(BrX ) it is weakly compact, too. The other direction follows along the lines by using P −1 . Theorem A.28 (Schauder Fixed-Point Theorem) Le B denote a non-void, closed, bounded and convex subset of a Banach space X. Then any compact mapping L : B ⊂ X → B has a fixed point. Proof. We refer to [Růž04, Folgerung 2.53]. In the next theorem we define the notion of a Gelfand triple. The presentation follows the one in [Alt12], the same results can be found in [GGZ74] and in [Rou13]. Theorem A.29 (Gelfand Triples) Let V be a Banach space and let H be a Hilbert space. Suppose that the continuous embedding

179

A.5 Auxiliary Results

I : V → H is dense. By means of the Riesz isomorphism RH : H → H 0 ,

RH u, v H := u, v H , u, v ∈ H we can identify H with its dual space H 0 . Denoting by I ∗ : H 0 → V 0 the adjoint of the embedding I : V → H, we define J : H → V 0, J u := I ∗ (RH u). As a consequence we obtain a sequence of continuous mappings I

J

V −→ H −→ V 0 .

(A.5)

In this case, the triple V, H, V 0 is called a Gelfand triple or evolution triple and we also write V ,→ H ∼ = H 0 ,→ V 0 , thereby suppressing the specific mappings in (A.5) if there is no danger of confusion. The following properties are easily derived from the above definitions: • For every v ∈ V there holds kI(v)kH ≤ c kvkV . • The mapping J : H → V 0 has the representation

J u, v V = u, I(v) H , u ∈ H, v ∈ V • The mapping J˜ := J ◦ I = I ∗ ◦ RH ◦ I : V → V 0 has the representation

J˜v1 , v2 V = I(v1 ), I(v2 ) H , v1 , v2 ∈ V • J is injective if and only if the embedding I : V → H is dense.

180

Chapter A. Appendix

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