Existence and Uniqueness Theorems for the Full

Existence and Uniqueness Theorems for the Full Three-Dimensional Ericksen-Leslie System Gregory A. Chechkin\ , Tudor S. Ratiu[ , Maxim S. Romanov\ , Vyacheslav N. Samokhin] October 21, 2016

\

Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia, [email protected], [email protected] [

Department of Mathematics Shanghai Jiao Tong University 800 Dongchuan Road, Minhang, Shanghai, 200240 China and Section de Mathématiques ´ Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland [email protected] ]

Moscow State University of Printing Arts 2A, Pryanishnikova ul., Moscow 127550, Russia [email protected]

Abstract In this paper we study the three dimensional Ericksen-Leslie equations for the nematodynamics of liquid crystals. We prove short time existence and uniqueness of strong solutions for the initial value problem for the periodic case and in bounded domains with both Dirichletand Neumann-type boundary conditions.

Keywords: liquid crystals, Ericksen-Leslie equations, nematodynamics, existence and uniqueness, director field, speed of propagation. MSC classification: 76A15, 35K61, 35A01, 35A02

1

1

Introduction

In this paper we consider the full Ericksen–Leslie system of equations, which models the hydrodynamics of nematic liquid crystals (see, for instance, [10]). Since the end of the 19th century, it is known that liquid crystals represent a new state of matter. They successfully combine the anisotropic properties of crystals and flow properties of fluids. There are two main classes of liquid crystals: thermotropic (formed by heated solids) and lyotropic (formed in mixtures of solids with solvents). In all types of liquid crystals, the orientation of the molecule is defined by a unit vector, called “director”. In calamitics (which has linear molecules or ellipsoids), the director is parallel to the generating axis of molecules, in discotic liquid crystals the director is perpendicular to discs. The thermotropic liquid crystals may be nematic, smectic, cholesteric (see Fig. 1), chiral, discotic, or in the blue phase. It is natural to study the behavior of nematic liquid crystals from the point of view of fluid dynamics, because of its similarity to the hydrodynamics of non-Newtonian fluids. Nematic liquid crystals can be calamitic or discotic (see Fig. 1). The nematodynamic theory for both cases, due to J. Ericksen and F. Leslie, was developed in the 1960’s [11], [12], [29], [30]; see also [13], [31]. For a brief discussion of the limitations of this theory and the indication of other existing models, we refer to §5. In this paper, we study the hydrodynamics of nematic liquid crystals. In previous work, we investigated plane periodic mesomorphic media [5] as well as non-periodic ([7] and [44]) and homogenization of micro inhomogeneous nematic liquid crystals (periodic in [45] and random in [8]).

Figure 1: The structure of horizontal flow of smectic (left), nematic calamitic and nematic discotic (center), cholesteric (right) liquid crystals

The subject of our research is the Ericksen-Leslie system describing the 2

dynamics of nematic liquid crystals ( ∂F · (∇n) + F + f, u˙ − µ∆u = −∇p − ∂x∂ j ∂n x

div u = 0,

j

¨ − 2qn + h = g + G, Jn

knk = 1,

(1)

where summation on repeated indices is understood (irrespective of their positions), the subscript xj on a real or vector valued function means partial derivative relative to the spatial coordinate xj (e.g., nxj := ∂x∂ j n), ∆ := div ∇ is the Laplacian, and ∂F ∂F · (∇n) := ∇nk ∂nxj ∂nk,xj (see also [10, formulas (3.90), (3.99), (3.100), (5.2)], [4, page 90], or [35]). Here, u is the Eulerian, or spatial, velocity vector field, n = (n1 , n2 , n3 ) is the director field, the constant µ > 0 is the viscosity coefficient, the constant J > 0 is the moment of inertia of the molecule, F(x, t) and G(x, t) are given ∂ external forces, and the overdot ˙ := ∂t +u·∇ is the material derivative. The terms f and g correspond to the dissipative part of the stress tensor and the dissipative part of intrinsic body force, respectively, and they depend on u, n, and their derivatives. The function F(n, ∇n) is the Oseen-Z¨ ocher-Frank free energy and is defined by F(n, ∇n) := Kn · curl n 1 + K1 (div n)2 + K2 (n · curl n)2 + K3 kn × curl nk2 , (2) 2 where K, K1 , K2 , K3 are real constants. The molecular field h is defined by ∂ ∂F ∂F − . h := ∂n ∂xj ∂nxj The pressure p is determined, as in usual ideal incompressible homogeneous hydrodynamics, by the condition div u = 0. The Lagrange multiplier 2q is determined by the condition knk = 1. Concretely, taking the inner product of the second equation in (1) with n yields ¨ + h − g − G) · n = −Jknk ˙ 2 + (h − g − G) · n, 2q = (J n

(3)

¨ · n = −knk ˙ 2. since n Since the liquid crystal is nematic, we necessarily have K = 0. Using the identities kn × curl nk2 = knk2 kcurl nk2 − (n · curl n)2 and knk = 1, the Oseen-Z¨ ocher-Frank free energy (2) takes the form: K1 K2 − K3 (div n)2 + (n · curl n)2 + 2 2 K1 K2 − K3 = (div n)2 + (n · curl n)2 + 2 2

F=

3

K3 kcurl nk2 2 K3 (nj,xi nj,xi − ni,xj nj,xi ). (4) 2

In this paper, we study the Ericksen-Leslie dynamics under the assumption K2 = K3 in the Oseen-Z¨ ocher-Frank free energy. In this case ∂F = K1 nk,xk δij + K2 (ni,xj − nj,xi ) ∂ni,xj

(5)

and the corresponding molecular field is h = (K2 − K1 )∇div n − K2 ∆n.

(6)

Remark 1.1. If we use the original formula (2) for F, instead of (4), , but a different molecular field h we obtain the same expression for ∂n∂F i,x j

with an additional term q 0 n, where q 0 is a scalar function depending on ∇n. However, the corresponding Ericksen-Leslie systems are equivalent, since this new term affects only the Lagrange multiplier q, not u or n. We study, from now on, only the Ericksen-Leslie equations without additional dissipation terms, i.e., both the director-caused dissipative part f of the stress tensor and the dissipative part g of the intrinsic body force vanish; hence we assume from now on that f = g = 0. Define the linear differential operator L by Lv := (K2 − K1 )∇(div v) − K2 ∆v.

(7)

We want to express ∂ ∂F ∂F ∂F − · (∇n) = − · (∇n) − · (∇nxj ) ∂xj ∂nxj ∂nxj x ∂nxj

(8)

j

in terms of L. We have ∂F (5) = K1 nk,xk xi + K2 (ni,xj xj − nj,xi xj ) ∂ni,xj x j

= K2 ∆ni − (K2 − K1 )(div n)xi = (K2 ∆n − (K2 − K1 )∇div n)i , that is, −

∂F ∂nxj

(7)

= (K2 − K1 )∇div n − K2 ∆n = Ln.

(9)

xj

Next, we prove that ∂F ∂F · (∇nxj ) := ∇nk,xj = ∇F. ∂nxj ∂nk,xj

(10)

By (4) (with K2 = K3 ), F is a quadratic form in ∇n, which immediately implies ∂F = 0, ∂n

2F =

∂F ∂2F ∂F · nxj =⇒ · n xj = . ∂nxj ∂nxj ∂ns,xl ∂ns,xl 4

(The second relation also follows directly from (4) and (5).) Consequently, ∂F ∂2F · nxj xi + nk,xj ns,xl xi ∂nxj ∂nk,xj ∂ns,xl ∂F ∂F ∂F · nxj xi + · nxj xi + (F)xi , = ns,xl xi = ∂nxj ∂ns,xl ∂nxj

(2F)xi =

proving (10). Thus, (8), (9), and (10), imply ∂ ∂F − · (∇n) = (Ln) · (∇n) − ∇F. ∂xj ∂nxj

(11)

Lemma 1.1. Suppose that (u, n, p) is a solution of the Ericksen-Leslie system (1) with f = g = 0 and K2 = K3 . Define the vector field ν := n × n˙ (first introduced in [16]). Then (u, ν, n) satisfy the following system u˙ − µ∆u = −∇p0 + (Ln) · (∇n) + F,

div u = 0,

(12)

J ν˙ = (Ln) × n + n × G,

(13)

n˙ = ν × n,

(14)

where p0 := p + ∇F, with initial conditions satisfying kn(x, 0)k = 1,

n(x, 0) ⊥ ν(x, 0).

(15)

Conversely, suppose that (u, ν, n, p0 ) is a solution of (12)–(15). Then ν = n × n˙ and (u, n, p) solves the Ericksen-Leslie system (1) with f = g = 0 and K2 = K3 , where p := p0 − ∇F. ˙ We Proof. Suppose that (u, n, p) is a solution of (1). Let ν := n × n. 0 show that (u, ν, n, p ) is a solution of (12)–(15). Since knk = 1 and n · ν = ˙ = 0, the initial conditions (15) hold. Taking the cross product of n · (n × n) ν := n × n˙ with n and using knk = 1, n · n˙ = 0 yields (14). Formulas (6) and (7) imply that Ln = h and hence ˙ · = Jn × n ¨ = 2qn × n − n × h + n × G J ν˙ = J(n × n) = (Ln) × n + n × G, which is equation (13). Identity (11) shows that the first equation in (1) implies (12). Thus, any solution of the Ericksen-Leslie system (1), with f = g = 0 and K2 = K3 , solves the system (12)–(15). Conversely, suppose that (u, ν, n, p0 ) is a solution of (12)–(15). Since (14)

(n · n)· = 2n˙ · n = 2(ν × n) · n = 0, we conclude that kn(x, 0)k = 1 implies kn(x, t)k = 1 for all time t for which the solution exists. In addition, by 5

(13) and (14), it follows that (n · ν)· = n˙ · ν + n · ν˙ = 0. Thus, if the initial conditions satisfy n(x, 0) ⊥ ν(x, 0) it follows that n(x, t) ⊥ ν(x, t) for all time t for which the solution exists. Therefore, taking the cross product of ˙ The first equation in (1), with f = g = 0 (14) with n, we get ν = n × n. and K2 = K3 , coincides with (12) by (11), where p := p0 − ∇F. It remains to show that the second equation in (1) holds, assuming that ¨ · n = −knk ˙ 2 , n · ν = 0, f = g = 0 and K2 = K3 . Since knk = 1, n · n˙ = 0, n ˙ and Ln = h (see (6) and (7)), we get ν = n × n, (14)

(14)

¨ = J(ν × n)· = J ν˙ × n + Jν × (ν × n) Jn (13)

= ((Ln) × n + n × G) × n − Jkνk2 n

˙ 2n = −Ln + ((Ln) · n)n − (G · n)n + G − Jkn × nk ˙ 2 n + ((Ln) · n)n − (G · n)n − Ln + G = −Jknk ¨ + h − G) · n) n − h + G = ((J n (3)

= 2qn − h + G,

which is the second equation in (1). In the rest of the paper we work exclusively with the system (12), (13), (14) with initial conditions kn(x, 0)k = 1, n(x, 0) ⊥ ν(x, 0). There has been much work dedicated to the Ericksen–Leslie equations, mostly concerned with the case J = 0. Since the structure of this system is quite complicated, previous work tried to simplify it to get some reasonable results. For the incompressible model, Lin [33] introduced a simplification of the general Ericksen–Leslie system that keeps many of the mathematical difficulties of the original system by using a Ginzburg–Landau approximation to relax the nonlinear constraint. Namely, instead of the restriction knk = 1, the penalty term 12 (knk2 − 1)2 has been added to the free energy functional. The global existence of weak solutions was shown by Lin and Liu [34]. They also showed the existence of smooth solutions for this approximation. Shkoller [46] gave a very simple proof of local well-posedness for this coupled system using a contraction mapping argument. It was proved that this system is globally well-posed and has compact global attractors in 2D. Recently, Hong [21] and Lin, Liu, Wang [36] showed independently the global existence of weak solutions of an incompressible model in 2-dimensional space. Moreover, in [36], the regularity of solutions, except for a countable set of singularities whose projection on the time axis is a finite set, has been obtained (see also [22]). The results were extended to a more general case by Huang, Lin, Wang, see [23]. Wang [49] established global well-posedness for incompressible liquid crystals for rough initial data. The simplified system (the director evolution equation was replaced with n˙ = ∆n + |∇n|2 n) for

6

compressible media was considered by Huang, Wang, Wen in [24]; the existence and uniqueness of strong solutions were obtained. Hieber and Pr¨ uss (see [19], [20]) studied thermodynamical modeling of nematic liquid crystals working with a simplified system having zero molecular moment of inertia, and showed well-posedness of the initial-boundary problem in the Lp -setting. Feiereisl et al. (see [15]) proposed a new class of non-isothermal models for nematodynamics and showed that their model is consistent with thermodynamics. This system has first order time derivatives for the director field. They proved that the initial boundary value system has time-global weak solutions. See also [3, 50, 9, 25] for some new aspects of nematodynamics. In this article we focus on the system (12)–(14) with J 6= 0, as opposed to the papers mentioned above. The two-dimensional system with J 6= 0 was considered in [44]. We prove local-in-time existence and uniqueness of strong solutions for 3-dimensional periodic media (Theorems 2.1, 2.2, 2.4) as well as for the problem in a bounded domain with Dirichlet (Theorems 3.1, 3.3) and Neumann (Theorem 4.1) boundary conditions in appropriate natural spaces. Uniqueness holds in function spaces with weaker regularity. We also prove the finite speed of propagation of the director field disturbance in such media (Theorem 3.2, Corollary 3.1). Some results presented in this paper were announced in [6]. Notational conventions. Sum over repeated indices is always assumed, regardless of their positions. Sometimes, for emphasis, we also write the sum sign. Let Ω ⊂ Rn be a bounded domain, either with C 1 boundary ∂Ω or periodic (and hence ∂Ω = ∅). Let k ∈ N. For the Sobolev spaces Wpk (Ω) appearing in this paper, we use the conventions in [38, §1.1]; Wpk (Ω) are Banach spaces relative to the norm  

p Z k X

p

α ∂

 kwkpW k (Ω) =

∂x 1 · · · ∂xαnn w + kwk dx, p Ω 1 α1 +···+αn =k

where x := (x1 , . . . , xn ) ∈ Rn , dx denotes the Lebesgue measure on Rn , and α1 , . . . , αn are any natural numbers between 0 and k. These spaces are, in general, different from those defined by adding all partial derivatives up to and including order k (see [38, §1.1]). If ∂Ω is C 1 these two spaces coincide (which is the case in this paper). We prefer to work with the norm above since it simplifies certain computations.

2

Solution in periodic domains

Let QT := (0, T ) × T, where T = R3 /Z3 is the 3-dimensional flat torus. We study the system (12)–(14) in QT with initial conditions u(0, x) = u0 ,

ν(0, x) = ν0 , 7

n(0, x) = n0 .

(16)

Here u, ν, n are unknown vector fields, p0 is an unknown scalar function, and J, Ki , µ are fixed strictly positive numbers.

2.1

Notations, definitions, and the existence theorem

Throughout the paper we use the following notations: • f˙ :=

∂f ∂t

+ u · ∇f = ft + uj fxj is the material time derivative of f ;

• a bold letter b denotes a 3-dimensional vector b = (b1 , b2 , b3 ), or a vector field with values in R3 ; • standard summation convention is used P on repeated indices, independent on their position, e.g., ai bi := i ai bi ; R • L2 (T) := v : T → R3 | kvk22 := T kvk2 dx < ∞ ; R • (u, v) := T u · v dx is the inner product in L2 (T); • W2m (T) is the Sobolev space of functions on T having m distributional derivatives in L2 (T); • for any v ∈ W2m (T), m ∈ N, define kD

m

vk22

:=

X i1 +i2 +i3 =m

2

∂mv

i1 i2 i3 ;

∂x1 ∂x2 ∂x3 2

• Sol(T) := {v : T → R3 | v ∈ C ∞ (T), div v = 0}; • Sol(QT ) := {v ∈ C ∞ (QT ) | v(t, ·) ∈ Sol(T), ∀t ∈ (0, T )}; • Sol2 (T) • Sol2m (T)

is the closure of Sol(T) in the norm L2 (T); is the closure of Sol(T) in the norm W2m (T).

Definition 2.1. A quadruple (u, ν, n, ∇p0 ) is a strong solution of problem (12)–(16) in the domain QT if (i) u is a time-dependent vector field in L2 ((0, T ); Sol23 (T)), ut ∈ L2 (QT ); (ii) ν is a vector field in L∞ ((0, T ); W22 (T)), νt ∈ L∞ ((0, T ); L2 (T)); (iii) n is a vector field in L∞ ((0, T ); W23 (T)), nt ∈ L∞ ((0, T ); W21 (T)); (iv) ∇p0 ∈ L2 (QT ); (v) u, n, ν satisfy the initial conditions (16), i.e., (u, n, ν) → (u0 , n0 , ν0 ) in L2 (T) as t → 0; (vi) equations (12)–(14) hold almost everywhere. 8

The first goal of the paper is to prove local in time existence of strong solutions to the problem (12)–(16). Theorem 2.1. Let F = 0, G = 0. Suppose u0 ∈ Sol22 (T), ν0 ∈ ∈ W23 (T). Then there is a T > 0 such that the solution to problem (12)–(14), (16) (as given in Definition 2.1) does exist. The time of existence T can be estimated from below W22 (T), n0

T >

ku0 k2W 2 2

Const , + kν0 k2W 2 + kn0 k2W 3 + 1 2

2

where the constant depends only on the positive constants J, K1 , K2 = K3 , and µ. The proof of this theorem is given in the next subsections.

2.2

Properties of the operator L

We need to of the operator L defined in (7). Writing study some properties K1 −K2 1 L = −K2 K2 ∇(div v) + ∆v , we are led to the operator L : W2 (T) → W2−1 (T) defined for any α ∈ R by Lv : = α∇(div v) + ∆v   ∂ 2 v1 ∂ 2 v1 ∂ 2 v1 ∂ 2 v3 ∂ 2 v2 (α + 1) + α + + + α  ∂x1 ∂x2 ∂x1 ∂x3  ∂x21 ∂x22 ∂x23    2  2 2 2 2  ∂ v2 ∂ v3  ∂ v2 ∂ v2 ∂ v1   =  2 + (α + 1) 2 + +α +α ∂x1 ∂x2 ∂x2 ∂x3  ∂x2 ∂x23  ∂x1   2  2 2 2 2  ∂ v3 ∂ v3 ∂ v3 ∂ v1 ∂ v2  + + (α + 1) 2 + α +α ∂x1 ∂x3 ∂x2 ∂x3 ∂x21 ∂x22 ∂x3 =

3 X

Aij vxi xj ,

(17)

i,j=1 i≤j

where Aij are the  α+1 0 1 A11 =  0 0 0  1 0 22  A = 0 α+1 0 0

symmetric 3 × 3 matrices    0 0 α 0 0 , A12 = α 0 0 , 1 0 0 0    0 0 0 0 0 , A23 = 0 0 α , 1 0 α 0



A13

A33

0 = 0 α  1  = 0 0

 0 α 0 0 0 1  0 0 1 0 . 0 α+1

Proposition 2.1. For any α > −1, the operator L is a Vishik elP strong ij i liptic operator (see [48]) which means that the symmetric matrix i,j A ξ ξ j is positive definite for all ξ i ∈ R, ξ i 6= 0. 9

Proof. Using the concrete expressions of the symmetric matrices Aij given above, for any non-zero vectors ξ, x ∈ R3 , we have   X  Aij ξ i ξ j x · x = kxk2 kξk2 + α(x · ξ)2 . i,j

If α ≥ 0 and x 6= 0, this is bounded below by kxk2 kξk2 > 0. If α < 0, the Cauchy inequality shows that this expression is bounded below by (1 + α)kxk2 kξk2 > 0 if α > −1. e ∈ C ∞ (T) Proposition 2.2. For any v, v e ) = (Le (Lv, v v, v) = −

3 Z X i,j=1 T

αvxj j vexkk + vxkj vexkj dx.

(18)

Also, for any m ∈ N, m ≥ 1 Lm v = ((α + 1)m − 1)∆m−1 ∇(div v) + ∆m v

(19)

(−1)m (Lm v, v) ≥ CkDm vk22 ,

(20)

and where constant C does not depend on v. Proof. Identity (18) is a direct verification. Identity (19) is proved in [48, Theorem 1]. Since the operator L is a strongly elliptic operator with constant coefficients, we have the inequality (−1)m (Lm v, v) ≥ Cm (−1)m (∆m v, v) for each v ∈ C ∞ (T), m ∈ N, and some Cm > 0 depending only on m and α. The identity (−1)m (∆m v, v) = kDm vk22 concludes the proof of inequality (20). Define the bilinear form 3 Z X (18) hu, vi := (u, v) + αujxj vxkk + ukxj vxkj dx = (u, v) − (Lu, v). i,j=1 T

It is a scalar product in W21 (T) inducing the same topology. Indeed, due to Proposition 2.2 we have C −1 kvk2W 1 (T) ≤ hv, vi ≤ Ckvk2W 1 (T) 2

2

for any v ∈ W22 (T) and, since C ∞ (T) is dense in W21 (T), these inequalities hold for any v ∈ W21 (T). 10

The operator I − L : W21 (T) → W2−1 (T) is invertible. Indeed, due to the Riesz Representation Theorem, for any f ∈ W2−1 (T), there exists w ∈ W21 (T), such that hw, vi = f (v)

∀v ∈ W21 (T)

and the mapping f 7→ w is a continuous linear operator (I−L)−1 : W2−1 (T) → W21 (T). (See also [48, §4] for the more general case of inversion of nonsymmetric operators.) Proposition 2.3. Define Q := (I − L)−1 AB : W21 (T) → W21 (T), where A : L2 (T) ,→ W2−1 (T) and B : W21 (T) ,→ L2 (T) are the standard compact embeddings given by the inclusion map. Then the following statements hold. (i) Q is a linear positive definite compact operator. (ii) Consider the spectral problem hu, vi = λ(u, v), ∀v ∈ W21 (T) ⇐⇒ λ−1 hu, vi = hQu, vi, ∀v ∈ W21 (T). Then there exists an orthogonal basis of eigenvectors uk ∈ W21 (T) of the operator Q. (iii) If uk is an eigenfunction of Q, then uk ∈ W22 (T) and for some λ0k ∈ R we have L(uk ) = λ0k uk almost everywhere. Similar results for domains with smooth boundary are presented in Section 3.

2.3

Galerkin-type approximation and norm estimates

We begin the proof with a classical approximation method. First of all, we select sequences ofSsubspaces E 1 ⊂ E 2 ⊂ . . . , F 1 ⊂SF 2 ⊂ . . . , and G1 ⊂ G2 ⊂ . . . such that E k is dense in Sol12 (T) and F k, k∈N k∈N S k G are dense in W21 (T). k∈N

Denote by E k and F k the linear span of the first k eigenfunctions of ∆ in Sol2 (T) and in L2 (T), respectively. The subspaces Gk are eigenspaces of operator L defined in (17). The constant α > −1 will be determined later. Proposition 2.4. Let (u0k , ν0,k , n0,k ) ∈ E k × F k × Gk be an approximation of the initial data (u0 , ν0 , n0 ) for fixed k. Then, for some T0 > 0, there exists a solution (uk , νk , nk ) ∈ C 0 ((0, T0 ); E k × F k × Gk ) of the problem (uk,t , ω) = −(ulk uk,xl , ω) + µ(∆uk , ω) + (Lnk · ∇nk , ω), 11

(21)

J(νk,t , ζ) + (Julk νk,xl , ζ) − (Lnk × nk , ζ) = 0, (nk,t , ψ) +

(ulk nk,xl , ψ)

− (νk × nk , ψ) = 0,

(uk , νk , nk )|t=0 = (u0k , ν0k , n0k ), where the identities above hold for all ω ∈

Ek,

ζ, ψ ∈

(22) (23) (24)

F k.

The system (21)–(24) can be regarded as a Cauchy problem for the d ordinary differential equation dt X = f (X) in 3k-dimensional space with continuous right-hand side. Due to the Cauchy-Peano theorem, there exists some small T0 > 0 such that this problem has a solution for t ∈ (0, T0 ). Remark 2.1. If u30k = 0, then for any t < T0 , we have u3k (t) = 0. The next step is to get a uniform estimate on (uk , νk , nk ) in some appropriate norm to prove the convergence of (uk , νk , nk ) to the solution of (12)–(16). Lemma 2.1. There exist T > 0, C > 0, and C1 > 0 depending only on the initial data and the positive constants J, K1 , K2 = K3 , µ > 0 appearing in the system (12)–(14), such that kuk kL2 ((0,T );W23 ) , kνk kL∞ ((0,T );W22 ) , knk kL∞ ((0,T );W23 ) ≤ C, kuk,t kL2 (QT ) , kνk,t kL2 (QT ) , k∇nk,t kL2 (QT ) ≤ C, and T is estimated from below by T >

C1 . ku0 k2W 2 + kν0 k2W 2 + kn0 k2W 3 + 1 2

2

2

Proof. In this proof, we omit the index k in (uk , νk , nk ), in order to simplify notation. Also, summation on repeated indices is understood, regardless of their position. We begin by proving the first set of inequalities. Let the approximate solution be defined on some interval [0, T ]. In equation (22), set ζ = ∆2 ν and integrate over the interval (0, T ) using Green’s identities. Since ∆2 ν ∈ F k , this substitution is allowed. We have Z T Z J k∆νk2 dx = − J(∆uj νxj + 2ujxk νxj xk )∆ν 2 T 0 QT + (Ln × n + 2Lnxl × nxl + Ln × ∆n) · ∆ν dxdt. (25) Due to (19), integrating (23) with ψ = −K2 L3 n, yields Z T K2 − L3 n · ndx 2 T 0 Z = −K2 (∆ν × n) · ∆(β∇div n + ∆n) + ∆uj nxj + 2ujxi nxj xi + uj ∆nxj QT + 2νxj × nxj + ν × ∆n x · (β∇div n + ∆n)xk dxdt, (26) k

12

1)3

where β := (α + − 1. Set α := K2 (β∇div n + ∆n) = −Ln.

q 3

K1 K2

− 1, then β =

K1 K2

− 1, and

Remark 2.2. In the one constant approximation K1 = K2 = K3 we have α = β = 0. To estimate the second derivatives of u, we set ω = ∆2 u in (21) and integrate over (0, T ): Z Z T 1 k∇(∆u)k2 dxdt k∆uk2 dx + µ 2 T 0 QT Z = −(∆uj uxj + 2ujxk uxj xk ) · ∆u − (Ln · nxi )xk ∆uixk dxdt. (27) QT

Here we use the identity k∇(∆u)k2 := ∆uxk · ∆uxk . The sum of (25)–(27) gives Z Z T 1 2 2 3 Jk∆νk + k∆uk − K2 L n · n dx + µk∇(∆u)k2 dxdt 2 T 0 Qt Z ≤− ujxk (2Jνxj xk · ∆ν − 2uxj xk · ∆u + Lnxk · ∆nxj − 2nxi xj xk · Lnxi ) Qt

+ ∆uj (Jνxj · ∆ν + uxj · ∆u − nxj xk · Lnxk ) − ujxk xi nxj xi · Lnxk − ∆ujxk nxj · Lnxk + (Lnxk · nxj + nxj xk · Ln)∆ujxk − (∆Ln × n + 2∆nxk × nxk + Ln × ∆n)) · ∆ν − (∆ν × n) · ∆Ln − K(2νxj xk × nxj + 2νxj × nxj xk + νxk × ∆n + ν × ∆nxk ) · Lnxk dxdt Z T ≤C esssup k∇u(t)k kν(t)k2W 2 + ku(t)k2W 2 + kn(t)k2W 3 2 2 2 0 + ku(t)kW42 + kν(t)kW41 kν(t)kW41 + ku(t)kW41 + kn(t)kW42 kν(t)kW22 + ku(t)kW22 + kn(t)kW23 + ku(t)kW23 kn(t)k2W 2 + esssup k∇n(t)k ||ν(t)||W22 kn(t)kW23 4 2 + esssup kν(t)k kn(t)kW 3 dt. (28) 2

This inequality allows us to estimate derivatives of the unknown functions. To estimate their L2 -norm we suppose (ω, ζ, ψ) = (u, ν, n) and repeat

13

the procedure above. We have Z T Z TZ Z T 1 2 2 k∇u(t)kL2 k∇n(t)k2L4 dt, k∇u(t)k dxdt ≤ C ku(t)k dx + µ 2 T 0 0 0 T Z T Z T 1 kν(t)kL2 kn(t)kL4 kD2 n(t)kL4 dt, kν(t)k2 dx ≤ C 2 T 0 0 Z Z T T (29) 2 kn(t)k2 dx ≤ C kn(t)k2L4 kν(t)kL2 dt. T

0

0

Since for any periodic f ∈ W21 ([0, 1]3 ) we have kf kLp ≤ C(k∇f kL2 + kf kL2 )

for p < 6

(30)

max kgk ≤ C(k∇gkLp + kgkL2 )

for p > 3,

(31)

we conclude from (20), (28), and (29) that Z t I (t) ≤ I (0) + C4 I (t)2 dt,

as t < T,

(32)

and for any periodic g ∈ Wp1 ([0, 1]3 ),

0

where I (t) :=

ku(t)k2W 2 2

+

kν(t)k2W 2 2

+

kn(t)k2W 3 2

Z + 0

t

k∇(∆u)k2L2 dt + 1,

(33)

and the constant C4 > 0 depends only on the positive constants J, K1 , K2 = K3 , and µ. To continue the proof of the lemma, we need the following simple proposition, a kind of Gronwall–Bellman inequality (see [5] for the proof). Lemma 2.2. Let Y be a measurable function on R. Suppose that for almost all t we have Z t 0 ≤ Y (t) ≤ Y (0) + k Y (s)2 ds. 0

Then Y (t) ≤

Y (0) . 1 − ktY (0)

Thus, by Lemma 2.2, inequality (32) implies that for any t ∈ [0, T ], we have I (0) I (t) ≤ . (34) 1 − tC4 I (0) Let us h show that the approximate solution can be continued on the whole 1 interval 0, C4 I (0) =: [0, T1 ). If the triple (u, ν, n) is defined on the interval 14

[0, T ) and T < T1 , then the norm of the solution is uniformly bounded on [0, T ). Due to the extension theorem for ordinary differential equations (see, e.g., [18]), there exists a limit of the solution at t = T and the solution (u, ν, n) can be continued on some interval (T, T + δ). This means that the interval I := {t < T1 | the approximate solution is defined on [0, t)} is open in [0, T1 ). On the other hand, I is closed in [0, T1 ) and hence I = [0, T1 ). Thus the approximate solution exists on the interval [0, T1 ] and for all t < T := T1 /2 it satisfies the inequality I (t) ≤ C5 ,

(35)

where C5 depends only on K1 , k2 = K3 , µ, J, and I (0). This proves the first set of three inequalities in the statement of the lemma and gives the lower bound for T . Using inequality (35), we can now estimate the time-derivative ut . Set ω = ut in (21) and rewrite the resulting identity as Z Z t 1 2 kut k dxdt + µ k∇uk2 dx 2 T 0 QT2 Z −uj uxj · ut + Ln · nxj ujt dxdt. = QT2

Since uj uxj and ∆n · nxj are uniformly bounded in L2 (Qt ) (which follows from (35) and the Gagliardo-Nirenberg inequality; see, e.g., [37, Section 6.1,Th.1.13]), we conclude Z Z t 2 kut k + µ k∇uk2 dx ≤ kuj uxj k22 + kLn · ∇nk22 ≤ C6 . Qt

0

T

The same type of inequalities can be obtained in a similar fashion for νt and ∇nt . This proves the second set of inequalities in the statement of the lemma 2.1. Now we can complete the proof of Theorem 2.1. Lemma 2.1 provides the existence of measurable functions u, ν, n and a subsequence of (uk , νk , nk ) such that uk * u

weakly in L2 ((0, T ); Sol22 (T)),

weakly in Sol2 (QT ),

∗

*-weakly in L∞ ((0, T ); W22 (T)),

∗

*-weakly in L∞ ((0, T ); W23 (T)),

νk * ν nk * n uk,t * ut ,

uk,t * ut

νk,t * νt ,

∇nk,t * ∇nt

weakly in L2 (QT ).

Therefore, due to the standard embedding theorems, uk → u strongly in Sol2 (QT ), 15

νk → ν

strongly in L2 (QT ),

and Fix ω ∈

S

∇nk → ∇n strongly in L2 (QT ). S S C 1 ((0, T ); E k ), ζ(t) ∈ C 1 ((0, T ); F k ), ψ(t) ∈ C 1 ((0, T ); Gk ) k

k

k

and integrate (21)–(23) over (0, T ). Passing to the limit as k → ∞, we have Z   (u˙ − µ∆u − (Ln)i ∇ni ) · ωdxdt = 0,    QT  Z   (J ν˙ · ζ − (Ln × n) · ζ) dxdt = 0, (36)  QT   Z     (n˙ · ψ − (ν × n) · ψ) dxdt = 0.  QT

Since

S

C 1 ((0, T ); E k ) is dense in L2 ((0, T ); Sol2 (T)) and

S

C 1 ((0, T ); F k )

k

k

is dense in L2 (QT ), these equations imply (12)–(14). If ∇p0 is the Hodge projection of u˙ − µ∆u − ∆n∇n ∈ L2 (QT ) on the orthogonal complement of L2 ((0, T ); Sol2 (T)), it follows that ∇p0 ∈ L2 (QT ). Finally, we verify the initial conditions (16). Fix φ ∈ Sol2 (T) and consider the family of functions fk (t) = (uk (t), φ)L2 (T) . Since fk0 (t) = (uk,t (t), φ)L2 (T) → (ut , φ)L2 (T) , it follows that fk tends to (u, φ)L2 (T) in C(0, T ). Thus, u(t, ·) tends to limk u0k = u0 weakly in L2 (T). Also, ut ∈ L2 (Qt ), consequently the function t 7→ u(t, x) is continuous in the L2 (T)-norm with respect to t. So u0 is both a weak and a strong limit. The limits of the other variables to their respective initial conditions are checked in the same way. This proves Theorem 2.1.

2.4

Uniqueness

Theorem 2.2. Under the hypotheses of Theorem 2.1, let (u1 , ν1 , n1 , p01 ) and (u2 , ν2 , n2 , p02 ) be solutions of the problem (12)–(16) in the domain QT . Then, for some 0 < T0 ≤ T (u2 , ν2 , n2 , ∇p02 ) = (u1 , ν1 , n1 , ∇p01 ) almost everywhere in QT0 . Proof. First of all, every solution of the problem satisfies identities (36) for all ω ∈ W21 (QT ), ζ ∈ W21 (Qt ), ψ ∈ L2 ((0, T ); W22 ). Let w := u1 − u2 , f := ν1 − ν2 , g := n1 − n2 , and set (ω, ζ, ψ) = (w, f , Lg) in (36). With this substitution, for any τ < T , the identities (36) yield Z wt · w + ui2 wxi · w + wi u1,xi · w + µk∇wk2 Qτ

− Ln1 · gxj wj − Lg · n2,xj wj dxdt = 0, 16

(37)

Z

J(ft · f + ui2 fxi · f + wi ν1,xi · f )

Qτ

−(Lg × n2 + Ln1 × g) · f ) dxdt = 0, Z

(38)

gt · Lg + u1 i gxi · Lg + wi n2,xi · Lg

Qτ

−(ν1 × g + f × n2 ) · Lg) dxdt = 0.

(39)

Adding the identities (37)–(39) and moving some terms to the right hand side, we get Z Z 1 2 2 (kw(τ )k + Jkf (τ )k + |Lg(τ ) · g(τ )| dx + µ k∇w(t)k2 dxdt 2 T Qτ Z = − wj ui1,xj wi + Ln1 · gxj wj + Jwi ν1,xi · f + (Ln1 × g) · f Qτ

+ (K2 − K1 )(∇ui1 · gxi − div (ν1 × g)) div g − K2 ui1,xk gxi · gxk − K2 (ν1,xk × g) · gxk dxdt. At the same time, Z Z 1 2 kg(τ )k dx = (f × n2 ) · g − wi n2,xi g dx. 2 T T

(40)

(41)

Due to the embedding theorems, the Hölder inequalities, and (20) we get Z (kw(τ )k2 + Jkf (τ )k2 + kg(τ )k2 + C1 k∇g(τ )k2 dx T Z h 1 +µ k∇w(t)k2 dxdt ≤ C2 τ 2 esssup kn2 (t)kW21 (T) + kLn1 (t)kL3 (T) t Qτ i + k∇ν1 (t)kL2 (T) + k∇u1 (t)kL2 ((0,T );L∞ (T)) esssup kw(t)k2L2 (T) t 2 2 2 + kf (t)kL2 (T) + k∇g(t)kL2 (T) + kwkW 1 (Qτ ) . 2

Taking the τ -esssup of the left hand side and comparing it with the second factor on the right hand side, shows that for τ sufficiently small we have (w, f , ∇g) = (0, 0, 0). Consequently, (u1 , ν1 ) = (u2 , ν2 ). However, g˙ = −wi n2,xi + ν1 × g + f × n2 . d Since f = 0 and w = 0, we have dt g(t) = ν1 (t) × g(t) with initial condition g(0) = 0. This implies that g = 0, i.e., n1 = n2 . Since ∇p0i is the projection of u˙ − µ∆u − ∆n∇n, we have ∇p01 =∇p02 . Theorem 2.2 is proved. The hypotheses of Theorem 2.2 can be weakened. We introduce the notion of a weak solution. 17

Definition 2.2. The triple (u, ν, n) is a weak solution of (12)–(16) if • u ∈ L2 ((0, T ); W21 (T)) ∩ C((0, T ); L2 (T)), • ν ∈ C((0, T ); L2 (T)), n ∈ C((0, T ); W21 (T)) ∩ L4 ((0, T ); W41 (T)), • for all (ω, ζ, ψ) ∈ Sol21 (QT ) × W21 (QT ) × W21 (QT ), the triple (u, ν, n) satisfies the initial conditions and the identities Z T Z   −u · ω˙ + µuxi · ωxi u · ω +   0  Q T T     j k i  · n + (K − K )n n − K n · ω  xi 2 1 xj xi 2 xk xk dxdt = 0,   Z Z  T  −Jν · ζ˙ − (K2 nxk × n) · ζxk ν · ζ + (42) 0  Q T T       − (K − K )div n(curl n · ζ − n · curl ζ) dxdt = 0, 2 1    Z Z    T  n · ψ + −n · ψ˙ − (ν × n) · ψ dxdt = 0.  0

T

QT

It is easy to check that any strong solution is a weak solution. We need to show that, with some additional assumptions, the weak solution is strong. Theorem 2.3. Suppose (u1 , ν1 , n1 , p01 ) is a strong solution satisfying 1 (T)), ν ∈ L ((0, T ); W 1 (T)), n ∈ L ((0, T ); W 2 (T)), u1 ∈ Lδ ((0, T ); W∞ 1 1 2δ 2δ 3 3 where δ > 1. Let (u2 , ν2 , n2 , p02 ) be a weak solution. Suppose there exist sequences of smooth vector fields νh , nh such that  kν2 − νh kL2 (QT ) → 0, kn2 − nh kL4 ((0,T );W41 (T)) → 0,    kn2 − nh k2L4 ((0,T );W 1 (T)) kνh kL2 ((0,T );W21 (T)) → 0, (43) 4    kn k kν − ν k → 0, 2 h L2 ((0,T );W2 (T))

Z I3 :=

2

h L2 (QT )

(K2 − K1 )div(g − gh )(−n2 · curl fh ))

QT

+ (K2 ((gxk − ghxk ) × n2 ) · fhxk − Lgh · ((f − fh ) × n2 ) + Lgh · (n2 − nh )xj wj + (f − fh )ui2 fhxi + (g − gh )xi ui1 Lgh dxdt → 0

(44)

as h → 0, where (fh , gh ) := (ν1 − νh , n1 − nh ). Then, for some 0 < T0 ≤ T we have (u2 , ν2 , n2 ) = (u1 , ν1 , n1 ) almost everywhere in QT0 . Proof. Since w, f , g are not smooth enough, we need to modify the last proof. First, suppose that u2 , νh , and nh have time-derivatives in L2 (QT ) and νht → ν2t ,

nht → n2t , 18

as

h → 0.

Let (ω, ζ, ψ) in (36), (42) be equal to (w, fh , Lgh ) = (u1 −u2 , ν1 −νh , L(n1 − nh )). From the identities (42) and (36) we have I1 = I2 + I3 + o(1) where I1 and I2 are the left hand and right hand sides of (40), I3 is defined by (44), and o(1) stands for the terms tending to zero due to the conditions (43). Since I3 → 0 by hypothesis, we have (40), so we can repeat the procedure in the proof of the last theorem. If the weak solution does not have a regular time-derivative, we can use the method of time-homogenization from [28, Ch.III,§2, page 648] with minor changes. For any δ > 0 and any function f (t), we define fδ by the formula Z t+δ

fδ (t) = δ −1

f (τ )dτ. t

It is easy to show that uδ satisfies the identity Z Z uδt ·ω = (ui u)δ ·ωxi −uδxi ·ω+((K2 n2xk ·n2xi +(K2 −K1 )nj2xj nk2xi )δ ·ωxi k ). QT

QT

Similar identities hold for ν and n. Setting (ω, ζ, ψ) = (wδ , (fh )δ , (Lgh )δ ), and taking into consideration that if fh is an approximation of f in the sense of (43), (fh )δ is an approximation of fδ , we obtain (40), which proves the theorem.

2.5

Liquid crystal in the presence of external forces

Theorem 2.1 can be easily extended if F, G 6= 0. Theorem 2.4. Suppose u0 ∈ Sol22 (T), ν0 ∈ W22 (T), n0 ∈ W23 (T), and F ∈ L2 ((0, T ); W21 (T)), G ∈ L1 ((0, T ); W22 (T)). Then there exists some 0 < T0 < T such that the solution (as in Definition 2.1) of problem (12)–(14) exists and is unique in QT0 . The proof follows that of Theorem 2.1 with some additional estimates of terms involving F and G.

3

Solution in bounded domains, Dirichlet boundary conditions

Let Ω be a bounded domain in R3 and consider nematic liquid crystal flow in the cylinder Ω × R. From now on, the boundary ∂Ω is assumed to be Lipschitz and for almost all x ∈ ∂Ω, the boundary is the graph of a C 2 function in some neighborhood of x. 19

We study equations (12)–(14) in the domain (0, T ) × Ω with initial conditions (16) and additional boundary conditions u ∂Ω = 0, n − n1 ∂Ω = 0, ν|∂Ω = 0 for any t > 0, (45) where n1 is a given vector field on Ω. Condition u ∂Ω = 0 means that the domain has impenetrable boundary and that the fluid moves without slipping; n − n1 ∂Ω = 0 describes the director position at the boundary. The third condition comes from the original Ericksen-Leslie system and means that n˙ = 0 at the boundary.

3.1

Properties of the operator L in bounded domains

In this subsection we study some properties of the Vishik strong elliptic operator L in bounded domains, similar to §2.2 in the periodic case. Let ◦

C ∞(Ω) denote the smooth vector fields on a bounded domain Ω ⊂ R3 with ◦

◦

compact support and W2m (Ω) the closure of C ∞ (Ω) in the W2m (Ω)-norm, or, equivalently, the subspace of W2m (Ω) with zero boundary trace (see, e.g., [47], [39], [37, page 330] [2, Definition 3.2 and Theorem 5.37]). Recall that, by definition (e.g., [2, §§3.10–3.12]), ◦ ∗ 1 W2 (Ω) = W2−1 (Ω).

(46)

◦

Proposition 3.1. Let L :W21 (Ω) → W2−1 (Ω) be the strong Vishik elliptic ◦

e ∈C ∞(Ω) operator defined in (17). Then, for any v, v e ) = (Le (Lv, v v, v) = −

3 Z X i,j=1 Ω

αvxj j vexkk + vxkj vexkj dx.

(47)

Also, for any m ∈ N, m ≥ 1, Lm v = ((α + 1)m − 1)∆m−1 ∇(div v) + ∆m v

(48)

(−1)m (Lm v, v) ≥ CkDm vk22 ,

(49)

and where the constant C does not depend on v. Proof. This result is a consequence of [48, Theorem 1]. Define the bilinear form hu, vi := (u, v) +

3 Z X i,j=1 Ω

(47) αujxj vxkk + ukxj vxkj dx = (u, v) − (Lu, v).

20

◦

If Ω has C 1 -boundary, this bilinear form is a scalar product in W21 (Ω) inducing the same topology. Indeed, from Proposition 3.1, we deduce C −1 kvk2W 1 (Ω) ≤ hv, vi ≤ Ckvk2W 1 (Ω) 2

(50)

2

◦

◦

for any v ∈C ∞ (Ω). If Ω is bounded with C 1 boundary then C ∞ (Ω) is dense ◦

◦

in W21 (Ω), thus these inequalities hold for any v ∈W21 (Ω). Proposition 3.2. Let ϕ ∈ C ∞ (Ω) be a scalar function with compact support. Suppose that v ∈ C ∞ (Ω) satisfies Lm v|∂Ω = 0, m = 0, 1, 2. Then there is a constant C(ϕ) > 0 (depending only on ϕ) such that (L3 (ϕv), ϕv) ≤ C(ϕ) −(L(Lv), (Lv)) + kvk2L2 (Ω) =: C(ϕ)J (v). (51) If Ω0 is an open connected set satisfying Ω0 ⊂ Ω, then there is a constant C(Ω0 ) > 0 (depending only on Ω0 ) such that kvk2W 3 (Ω0 ) ≤ C(Ω0 )J (v). 2

◦

Proof. Since v ∈ C ∞ (Ω) and Lv|∂Ω = 0, we have Lv ∈W21 (Ω). Then (50) implies kLvk2W 1 (Ω) ≤ CJ (v), 2

and, consequently, kLvk2L2 (Ω) ≤ CJ (v). Also, any v ∈ W22 (Ω), satisfying v|∂Ω = 0, can be estimated in the W22 norm (G˚ arding’s Inequality, see, e.g., [26, §8], [17, inequality (5.58)]) in the following way kvkW22 (Ω) ≤ C kLvkL2 (Ω) + kvkL2 (Ω) . (52) Now we show that the W21 -norm of L(ϕv) can be estimated in terms of J (v). Indeed, i L(ϕv) = ϕ(Lv)i + α(ϕxi vxj j + ϕxj vxj i ) + 2ϕxj vxi j + (αv j φxi xj + v i ∆φ) and the W21 -norm of each term of the sum is bounded above by C(ϕ)J (v), for a constant C(ϕ) > 0 depending only of ϕ. Finally, (L3 (ϕv), ϕv) ≤ CkL(ϕv)k2W 1 (Ω) . 2

The first part of the proposition is proved. Now fix an open subset Ω0 ⊂ Ω with Ω0 ⊂ Ω and select ϕ ∈ C ∞ (Ω) with compact support such that ϕ(x) = 1 if x ∈ Ω0 . By (51) and (49) we have kvk2W 3 (Ω0 ) = kϕvk2W 3 (Ω0 ) ≤ kϕvk2W 3 (Ω) ≤ CJ (v) 2

2

2

for a constant C > 0 that depends only on Ω0 . 21

Proposition 3.3 below guarantees the existence of an eigenbasis for L. ◦

◦

Define Q :W21 (Ω) →W21 (Ω), by the formula ◦

hQu, vi = (u, v)

∀v ∈W21 (Ω).

(53) ◦ ∗ Since the functional fu (v) := (u, v) belongs to the dual space W21 (Ω) 1

1

(indeed, |fu (v)| ≤ kukL2 (Ω) kvk ≤ hu, ui 2 hv, vi 2 ), by the Riesz Representation Theorem it follows that the operator Qu is well defined for any ◦

u ∈W21 (Ω). ◦

◦

Proposition 3.3. The operator Q :W21 (Ω) →W21 (Ω), defined by (53), has the following properties. ◦

◦

(i) Q :W21 (Ω) →W21 (Ω) is a linear positive definite compact operator. (ii) Consider the spectral problem ◦

◦

hu, vi = λ(u, v), ∀v ∈W 12 (Ω) ⇐⇒ λ−1 hu, vi = hQu, vi, ∀v ∈W21 (Ω). ◦

Then there exists an orthogonal basis of eigenvectors uk ∈W21 (Ω) of the operator Q. (iii) If uk is an eigenfunction of Q, then uk ∈ W22 (Ω) and for some λ0k ∈ R we have L(uk ) = λ0k uk almost everywhere. Proof. Note that Q = (I − L)−1 AB, where A : L2 (Ω) ,→ ◦

◦

W21 (Ω)

∗

(46)

=

W2−1 (Ω) and B :W21 (Ω) ,→ L2 (Ω) are the standard embeddings given by ◦ ∗ (46) ◦1 inclusion. As in §2.2, the operator I −L :W 2 (Ω) → W21 (Ω) = W2−1 (Ω) is invertible. If Ω has a smooth boundary, the inclusions A : L2 (Ω) ,→ ◦

W2−1 (Ω) and B :W21 (Ω) ,→ L2 (Ω) are compact. Thus, Q is a linear compact positive definite (see (53)) operator and we can use the spectral theorem for compact operators to obtain part (i) and (ii) of the proposition. The first statement of part (iii) follows from (52). The identity Luk = λ0k uk for some λ0k ∈ R is a direct verification.

22

3.2

Existence and uniqueness theorem

In this section, we let QT := (0, T ) × Ω and introduce the function spaces ◦

Sol (Ω) := {v : Ω → R3 | v has compact support, div v = 0}, ◦ ◦ Sol (QT ) := v ∈ C ∞ (QT ) | v(t, ·) ∈ Sol (Ω), ∀t , ◦

◦

Sol2m (Ω) is the closure of Sol (Ω) in the norm W2m (Ω). The definition of a solution of the Ericksen-Leslie equations is quite similar to the one in Definition 2.1, with a few changes because of the boundary. Definition 3.1. The quadruple (u, ν, n, ∇p0 ) is a strong solution of problem (12)–(16), (45) in the domain QT if ◦

• u is a vector field in L2 ((0, T ); Sol21 (Ω)) ∩ L2 ((0, T ); W23 (Ω)), ut ∈ L2 (QT ); ◦

• ν is a vector field in L∞ ((0, T ); W21 (Ω)) ∩ L∞ ((0, T ); W22 (Ω)), νt ∈ L∞ ((0, T ); L2 (Ω)); ◦

• n − n1 is a vector field in L∞ ((0, T ); W21 (Ω)) ∩ L∞ ((0, T ); W23 (Ω)), where n1 is a given constant vector field, and nt ∈ L∞ ((0, T ); W21 (Ω)); • ∇p0 ∈ L2 (QT ); • u, n, ν satisfy the initial conditions (16), i.e., (u, n, ν) → (u0 , n0 , ν0 ) in L2 (Ω) as t → 0; • equations (12)–(14) hold almost everywhere. Theorem 3.1. Assume that Ω is a domain with C 2 -boundary. Let n1 = ◦ ◦ const, n0 ∈W23 (Ω), ν0 ∈ W22 (Ω), u0 ∈Sol21 (Ω) ∩ W22 (Ω), ∆u0 ∂Ω = 0, and assume that for some d > 0 we have n0 (x) = const,

ν0 (x) = 0

if

dist(x, ∂Ω) < d.

Then problem (12)–(16), (45) has a unique solution in QT for some T > 0. The time of existence T can be estimated from below T >

ku0 k2W 2 2

Const , + kν0 k2W 2 + kn0 k2W 3 + 1 2

2

where the constant depends only on the distance d, the domain Ω, and the positive constants J, K1 , K2 = K3 , µ. 23

The proof proceeds along the same lines as that of Theorem 2.1. We point out the necessary modifications. Instead of (12)–(14) we consider the system u˙ − µ∆u = −∇p0 + (Ln · ∇n)Ψ,

div u = 0,

(54)

J(νt + Ψui νxi ) = (Ln × n)Ψ,

(55)

nt + Ψui nxi = (ν × n)Ψ,

(56)

where Ψ(x) ∈ C ∞ (Ω) is a given smooth non-negative function with compact support. e , where n e is a new unknown function. Also, we replace n with n1 + n As in the proof of Theorem 2.1, select sequences of subspaces E k , F k , Gk . It is still useful to choose them to be the linear span of the first k ◦

˜ acting on the space Sol1 (Ω), ∆ eigenfunctions of the Stokes operator −∆ 2 ◦

◦

acting on the space W21(Ω), and the operator L acting on the space W21 (Ω), ˜ := −P∆, where P is the projector sending a respectively. Recall that −∆ vector field to its divergence free part in the Hodge decomposition . As a pseudo-differential operator, P = Id − grad ∆−1 div. The Stokes operator ˜ is an unbounded L2 -self-adjoint positive definite operator. −∆ e k ) ⊂ C 0 ((0, T ); E k × F k × Gk ) to be the solution of the Define (uk , νk , n finite dimensional system of ordinary differential equations (uk,t , ω) = −(ulk uk,xl , ω)+(µ∆uk , ω)−(ΨL(e nk +n1 )·∇(e nk +n1 ), ω), (57) nk + n1 ) × (e nk + n1 ), ζ), J(νk,t , ζ) = −(ΨJulk νk,xl k, ζ) + (ΨL(e

(58)

nk + n1 )xl , ψ) + (Ψνk × (e (e nk,t , ψ) = −(Ψulk (e nk + n1 ), ψ) ,

(59)

(uk , νk , nk )|t=0 = (u0k , ν0k , n0k ).

(60)

The identities (57)–(60) hold for all ω ∈ E k , ζ ∈ F k , ψ ∈ Gk . (For simplicity e k + n1 ). The solution of this finite-dimensional we use the notation nk = n problem, obviously, exists for some T > 0 and so the result of Lemma 2.1 still holds. Lemma 3.1. There exists T > 0 and C > 0 depending on Ψ, on the initial and on the boundary data, and on the positive constants J, Ki , µ, such that kuk kL2 ((0,T );W23 (Ω)) , k∇νk kL∞ ((0,T );L2 (Ω)) , knk kL∞ ((0,T );W22 (Ω)) ≤ C, kuk,t kL2 (QT ) , kνk,t kL2 (QT ) , k∇nk,t kL2 (QT ) ≤ C.

24

e ) instead of Proof. To simplify notation, in this proof we shall write (u, ν, n e k ). (uk , νk , n 2 2 3 e ); In equations (57)–(59) and substitute (ω, ζ, ψ) = (∆ q u, ∆ ν, K2 L n recall that the operator L was defined in (17) with α :=

3

K1 K2

− 1 . First, we

Fk

note that since is defined to be the linear span of the first k eigenfunctions {vm | m = 0, . . . , k} of the Laplacian, i.e., vm |∂Ω = 0,

∆vm = λvm , ◦

◦

we have vm ∈W21 (Ω). By regularity, we conclude that ∆vm ∈W21 (Ω) and ◦

hence ∆ν ∈W21 (Ω). In the same way, ˜ m u = 0, ∆m ν = 0, Lm n = 0, ∆ ∂Ω ∂Ω ∂Ω

m = 1, 2, 3.

and ˜ m ut = 0, ∆ ∂Ω

∆m νt ∂Ω = 0,

Lm nt ∂Ω = 0,

m = 0, 1, 2 . . .

˜ 2 u, ∆2 ν, K2 L3 n e ) is permitted. Thus, the substitution (ω, ζ, ψ) = (∆ 1 2 3 Let N = (N , N , N ) be an outward pointing unit normal vector to the boundary ∂Ω. We have Z

T

0

Z

Z

nt · L3 ndxdt

Ω T Z

nkt (α(L2 n)ixk + (L2 n)kxi )N i − nkxi t (α(L2 n)i N k + (L2 n)k N i ) dSdt 0 ∂Ω Z TZ + Lnt · L2 ndxdt 0 Ω Z TZ = αN k (Lnt )k div Ln + (Lnt ) · Lnxi N i dSdt 0 ∂Ω Z TZ − αdiv (Lnt )div (Ln) + (div (Lnt ))xk · (div (Ln))xk dx 0 Ω Z T 1 =− αdiv (Ln)div (Ln) + (div (Ln))xk · (div (Ln))xk dx 2 Ω 0 Z T 1 = L3 n · ndx , (61) 2 Ω 0 =

since the boundary integrals vanish because Lm n|∂Ω = 0 and Lm nt |∂Ω = 0. Similar identities hold for u and ν. Return to equation (59). Since we chose ψ = K2 L3 n ˜ (we assumed

25

Ln1 = 0, thus ψ = K2 L3 n as well), it can be rewritten as Z T K2 L3 n · ndx = − 2 Ω 0 Z − K2 Ψ (∆ν × n) · ∆(β∇div n + ∆n) + ∆uj nxj + 2ujxi nxj xi + uj ∆nxj QT + 2νxj × nxj + ν × ∆n x · (β∇div n + ∆n)xk dxdt k Z + (β∇div n + ∆n)xk · Ik dxdt, QT K1 were β = K − 1. The terms Ik depend only on the first, second, and third 2 derivatives of Ψ and on ν × n together with its first and second derivatives. All boundary integrals in this computation are equal to zero since Ψ vanishes in a neighborhood of the boundary. This identity is an analogue of (26). The analogues of (25) and (27) are be obtained in the same way and, consequently, we have an analogue of inequality (28): Z Z t 1 Jk∆νk2 + k∆uk2 − K2 L3 n · n dx + µk∇(∆u)k2 dxdt 2 Ω 0 Qt Z t ≤ C(Ψ) esssup k∇u(t)k kν(t)k2W 2 (Ω0 ) + ku(t)k2W 2 (Ω) + kn(t)k2W 3 (Ω0 ) 2 2 2 0 + ku(t)kW42 (Ω) + kν(t)kW41 (Ω0 ) kν(t)kW41 (Ω0 ) + ku(t)kW41 (Ω) + kn(t)kW42 (Ω0 ) kν(t)kW22 (Ω0 ) + ku(t)kW22 (Ω) + kn(t)kW23 (Ω0 )

+ ku(t)kW23 (Ω) kn(t)k2W 2 (Ω0 ) 4 + esssup k∇n(t)k kν(t)kW22 (Ω0 ) kn(t)kW23 (Ω0 ) Ω0 + esssup kν(t)kkn(t)k2W 3 (Ω0 ) dt.

(62)

2

Ω0

Here Ω0 is a subdomain of Ω, such that supp(Ψ) ⊂ Ω0 ⊂ Ω. Due to Proposition 3.2, Z 2 knkW 3 (Ω0 ) ≤ C L3 n · n dx + knk2L2 (Ω) . 2

Ω

Next, we need [27, Lemma 8.1, Chapter III] to estimate kf kW22 (Ω) in terms of k∆f kL2 (Ω) and kf kL2 (Ω) , namely, kf kW22 (Ω) ≤ C(k∆f kL2 (Ω) + kf kL2 (Ω) ). We also need inequality [27, (11.8), Chapter III] to estimate kf kWp2 (Ω) (and, consequently, max k∇f k ) in terms of k∆f kW21 (Ω) and lower derivatives, namely, kf kWp2 (Ω) ≤ C(k∆f kLp (Ω) + kf kL2 (Ω) ), (63) 26

where f |∂Ω = 0, f ∈ Wp2 (Ω), Ω is a domain, with C 2 boundary; the constant C depends on p > 1 and Ω. Finally, we obtain an inequality similar to (32) and, consequently, (35). This proves Lemma 3.1. Proof of Theorem 3.1. Due toLemma 3.1, the sequences uk , νk ,◦nk areweakly◦ 2 precompact in the spaces L2 (0, T ); Sol2 (Ω) , L∞ (0, T ); W21 (Ω) , and ◦ 2 L∞ (0, T ); W2 (Ω) . From the embedding theorems, uk * u

◦ 2 weakly in L2 (0, T ); Sol2 (Ω ,

uk,t * ut

weakly in Sol2 (QT ),

∗

◦ 1 *-weakly in L∞ (0, T ); W2 (Ω) ,

∗

◦ 2 *-weakly in L∞ (0, T ); W2 (Ω) ,

νk * ν nk * n νk,t * νt ,

∇nk,t * ∇nt

uk → u strongly in Sol2 (QT ),

weakly in L2 (QT ).

νk → ν

strongly in L2 (QT ),

and ∇nk → ∇n

strongly in L2 (QT )

for some subsequence of (uk , νk , nk ). This proves existence of solutions of the modified problem (54), (55), (56) for some fixed Ψ. To end the proof of Theorem 3.1 we need a final statement. ◦

Theorem 3.2. Fix u ∈ L2 ((0, T ); Sol21 (Ω)) ∩ L2 ((0, T ); W22 (Ω)) and Ψ ∈ C ∞ (Ω) with compact support, 0 ≤ Ψ ≤ 1. Consider the equations (55), (56) for this given vector field u. Suppose, in addition, that for some 1 < α ≤ ∞ and for all i, j, there are constants m > 0, M > 0 such that the vector field u satisfies k esssup |uixj (x, t)| kLα (0,T ) ≤ M

and

ku(x, t)k ≤ m,

x

∀(x, t) ∈ QT .

Assume also that the initial conditions n0 and ν0 of (55), (56), with this given vector field u, are such that ∇n0 and ν0 vanish for kx − x0 k < r. Then there exist constants m0 , t0 > 0 such that ∇n and ν are equal to zero for all (x, t) satisfying kx − x0 k < r − m0 t,

t < t0 .

The time t0 can be estimated from below with C(M α +m−1 ), where C depends on Ψ, α, K1 , K2 = K3 , and J. 27

Proof. Let x0 = 0 and assume that K1 ≥ K2 . Taking ζ = νϕ and ψ = Lnϕ in (58) and (59) we get Z t Z J J 2 j j 2 kνk ϕdx = (ϕt + ψu ϕxj + u Ψxj ϕ)kνk + Ψ(Ln × n) · νϕ dxdt, 2 2 0 Z t 1 (K1 − K2 )(div n)2 + K2 k∇nk2 ϕdx = 2 0 Z K (ϕt + uj ϕxj + uj Ψxj ϕ) (K1 − K2 )(div n)2 + K2 k∇nk2 2 + ϕΨ(ν × n) · Ln − (K1 − K2 )ϕ∇ Ψuj · nxj div n − K2 ϕ(uj Ψ)xk nxj · nxk − (nt + ui Ψnxi ) · (K1 − K2 )div n∇ϕ + K2 nxj ϕxj dxdt. Add these two identities. Taking into account (56) and knk = 1, we can estimate the result as Z t 1 Z t J 2 kνk ϕdx + (K1 − K2 )(div n)2 + K2 k∇nk2 ϕdx ≤ 2 2 0 0 Z " 1 ϕ˙ Jkνk2 + (K1 − K2 )(div n)2 + K2 k∇nk2 2 + mC(Ψ) + C max |uixj | k∇nk2 ϕ i,j,x # √ (64) + (K2 + 3(K1 − K2 ))k∇ϕk kνk k∇nk dxdt, where C(Ψ) is a constant, depending on K1 , K2 and sup k∇Ψk. Ω

Define ϕ(x, t) := φ(kxk √ + m0 t), where φ(x) ∈ C 1 (R), φ = 0 for kxk > r, φ0 ≤ 0. Let m0 := m +

K2 + 3(K1 −K2 ) √ JK2

and estimate

Z " ϕ˙ 2 2 2 Jkνk + (K1 − K2 )(div n) + K2 k∇nk 2 # √ + (K2 + 3(K1 − K2 ))k∇ϕk kνk k∇nk) dxdt Z " ui xi J K1 − K2 K2 0 0 0 2 2 2 ≤ φ (kxk + m t) m + kνk + (div n) + k∇nk kxk 2 2 2 # √ |φ0 (kxk + m0 t)| 2 2 √ + (K2 + 3(K1 − K2 )) Jkνk + K2 k∇nk dxdt ≤ 0. 2 JK2

28

Consequently, 1 J K2 1 kν(t)ϕ 2 k22 + kϕ 2 ∇n(t)k22 ≤ 2 2

Z

mC(Ψ) + C

≤ Ct

α−1 α

max |uixj | i,j,x

k∇nk2 ϕdxdt

1

esssup kϕ 2 ∇nk22 , t

which proves the statement. Now assume that K1 < K2 . Then the term (K1 − K2 )nixj njxi ϕ appears in the left-hand side of (64) instead of (div n)2 , i.e., Z t 1 Z t J kνk2 ϕdx + (K1 − K2 )nixj njxi + K2 k∇nk2 ϕdx = 2 2 0 0 Z 1 (ϕt + uj ϕxj + uj Ψxj ϕ) Jkνk2 + (K1 − K2 )nkxi nixk + K2 k∇nk2 2 − (K1 − K2 )ϕ∇ Ψuj · nkxj nxk − K2 ϕ(uj Ψ)xk nxj · nxk − (nt + ui Ψnxi ) · (K1 − K2 )∇nk ϕxk + K2 nxj ϕxj dxdt ≤ Z " 1 ϕ˙ Jkνk2 + (K1 − K2 )nixj njxi + K2 k∇nk2 2 i + mC(Ψ) + C max |uxj | k∇nk2 ϕ i,j,x # + (2K2 − K1 )k∇ϕk kνk k∇nk dxdt. As in the first case, define ϕ(x, t) := φ(kxk + m0 t), where φ(x) ∈ C 1 (R), √2 −K1 . Since φ = 0 for kxk > r, φ0 ≤ 0; m0 := m + 2K JK 2

K1 nixj nixj

≤

K2 nixj nixj

+ (K1 − K2 )nixj njxi ,

we have 1 2

Z " ϕ˙ Jkνk2 + (K1 − K2 )nixj njxi + K2 k∇nk2 # + (2K2 − K1 )k∇ϕk kνk k∇nk dxdt ≤ 0

and, consequently, 1 J K1 1 kν(t)ϕ 2 k22 + kϕ 2 ∇n(t)k22 2 2 Z t 1 Z t J ≤ kνk2 ϕdx + (K1 − K2 )nixj njxi + K2 k∇nk2 ϕdx 2 2 0 0 Z ≤ mC(Ψ) + C max |uixj | k∇nk2 ϕdxdt i,j,x

≤ Ct

α−1 α

1

esssup kϕ 2 ∇nk22 . t

29

Thus ϕν and ϕ∇n are equal to zero for any t < t0 , where t0 is a sufficiently small positive constant, t0 ∼ M −α + 1/m. That means that nt is also equal to zero if t < t0 , kx − x0 k < r − m0 t. Due to the initial data and the equation (56) we have proved the statement of the Theorem. Remark 3.1. A similar result, with identical proof, holds in a periodic domain. In this case, we assume u ∈ L2 ((0, T ); Sol21 (T)) ∩ L2 ((0, T ); W22 (T)) and take Ψ ≡ 1. Corollary 3.1. Consider the solution (u, ν, n, p0 ) of problem (12)–(14), (16) in a periodic domain as given in Definition 2.1. Assume also that the initial conditions n0 and ν0 are such that ∇n0 and ν0 vanish for kx − x0 k < r. Then there exist constants m0 , t0 > 0 such that ∇n and ν are equal to zero for all (x, t) satisfying kx − x0 k < r − m0 t,

t < t0 .

If (u, ν, n, p0 ) is the solution of problem (12)–(16), (45) in a bounded domain, we need to assume, in addition, that ν and ∇n vanish in some neighborhood of the boundary ∂Ω. Return now to the proof of the Theorem 3.1. Let Ψ be a smooth nonnegative function with compact support such that Ψ(x) ≡ 1 in a δ-neighborhood Vδ of supp (ν0 , ∇n0 ). At this point, the existence of the solution (u, ν, n, ∇p0 ) of (54), (55), (56) is proved. Therefore, this solution satisfies equations (12), (13), (14) in the domain (0, T ) × Vδ almost everywhere. Moreover, since u ∈ W23 (QT ), we have ∇u ∈ L2 (0, T ; L∞ ) and 1

max kuk + k∇ukL2 (0,T ;L∞ ) ≤ Const(ku0 k2W 2 + kν0 k2W 2 + kn0 k2W 3 + 1) 2 , Qt

2

2

2

where the constant depends on the domain Ω, the function Ψ, and J, K1 , K2 , µ. Theorem 3.2 guarantees that for sufficiently small T1 ≤ T , the vector field ν and the differential ∇n vanish on (0, T1 )×(Ω\Vδ ). Therefore, (u, ν, n, ∇p0 ) also satisfies (12), (13), (14) in (0, T1 ) × (Ω \ Vδ ) almost everywhere. This shows that (u, ν, n, ∇p0 ) is a strong solution of the original problem in the domain (0, T1 ) × Ω. Remark 3.2. As can be seen from the proof, the only hypothesis on n1 is that it is piecewise constant in a neighborhood of the boundary ∂Ω. The same result holds if external forces are present. Theorem 3.3. Suppose Ω, n0 , ν0 , u0 , n1 satisfy the conditions of Theorem 3.1. Assume also that F ∈ L2 ((0, T ); W21 (Ω)), G ∈ L1 ((0, T ); W22 (Ω)), and G is equal to zero in a neighborhood of ∂Ω. Then the solution of (12)– (16), (45) exists and is unique for some T > 0. 30

4

Solution in bounded domains, Neumann boundary conditions

The problem considered in Section 3 has an important restriction: the director vector field is assumed to be constant near the boundary (see the hypotheses in Theorem 3.1). We now show that this condition can be eliminated by changing to Neumann boundary conditions. Suppose K1 = K2 = K3 and consider a bounded domain Ω with boundary ∂Ω. Suppose Ω to be a polyhedron or Ω = T × Ω1 , where Ω1 is a polygon, or Ω = T2 × (a, b). Let N be the outward unit normal defined almost everywhere on ∂Ω having the same differentiability class as ∂Ω. The system under investigation is given by equations (12)–(14) with initial conditions (16), but instead of the boundary conditions (45), we impose = 0, ∀τ ⊥ N = 0. (65) = 0 and (uixj + ujxi )τ i N j u · N (0,T )×∂Ω

(0,T )×∂Ω

The first boundary condition says that the surface is impenetrable. The second says that the fluid moves near the boundary without adhesion. Due to the definition of Ω, we have u · N = 0 on the boundary and N is piecewise constant, so ∂(u·N) = 0, which then implies (u · N)xj τ j = uixj τ j N i ∂τ on the boundary. Thus the second condition in (65) is equivalent to uixj τ i N j = 0, (0,T )×∂Ω

which is the form of the boundary condition with which we shall work below. In addition, we require that the director n and the variable ν satisfy the following boundary conditions nxi N i = 0 and νxi N i (0,T )×∂Ω = 0. (66) (0,T )×∂Ω

4.1

An identity

Denote by E k the linear span of the first k eigenfunctions of the spectral problem ∆v = λv, div v = 0, viN i = 0 and τ i vxi j N j = 0, ∀τ ⊥ N , (67) (0,T )×∂Ω

and by

Fk

(0,T )×∂Ω

the linear span of the first k eigenfunctions of the spectral problem ∆w = λw, wxi N i = 0. (68) (0,T )×∂Ω

In the case of arbitrary Ω one should replace the Laplacian in (67) with the Stokes operator and use the boundary conditions (65). 31

Lemma 4.1. Consider a domain Ω satisfying the following conditions: 1. Ω is a polyhedron or Ω = T × Ω1 , where Ω1 is a polygon, or Ω = T2 × (a, b). 2. Any solution to the spectral problems (67) or (68) belongs to W24 (Ω). Then ∀k > 0, ∀v ∈ E k ∪ F k Z Z ∆vxi · ∆vxi dx. vxi xj xk · vxi xj xk dx = Ω

Ω

Proof. We have Z Z ∆vxi · ∆vxi dx =

j

vxi xj · ∆vxi xj dx.

vxi xj · ∆vxi N dS − Ω

∂Ω

Ω

Z

If v ∈ F k , then ∆vxi N i = 0, so the boundary integral vanishes. If v ∈ E k , we fix the point x0 ∈ ∂Ω and choose coordinates in which ∂Ω = {x3 = 0} in the neighborhood of x0 (this is possible for almost every the x0 ). We want to prove vxl i x3 ∆vxl i = 0 if x3 = 0, and consequently boundary integral vanishes. Indeed, if l 6= 3 and i 6= 3, then vxl i x3 x =0 = 3 = 0. If l 6= 3 and i = 3, then ∆vxl 3 x3 =0 = 0. If l = 3 vxl 3 x3 =0 xi and i 6= 3, then ∆vx33 x3 =0 = 0. If l = 3 and i = 3, then vx33 x3 x3 =0 = P = 0. Here we used the fact that v is a linear ∆v 3 − j6=3 vx3j xj x3 =0

combination of eigenfunctions corresponding to the spectral problem (67). Next, Z Z Z k vxi xj · ∆vxi xj dx = vxi xj · vxi xj xk N dS − vxi xj xk · vxi xj xk dx. Ω

∂Ω

Ω

The boundary integral also vanishes: in the case v ∈ F k we have (in appropriate coordinates) vxi xj x3 = 0 if i, j 6= 3, vxi xj = 0 if either i = 3 or j = 3, and vx3 x3 x3 = 0. If v ∈ E k , for the first and second coordinates we apply the same equations as above; for the third coordinate we have vx3i xj = 0 if i, j 6= 0, vx33 x3 = 0 if i = j = 3, and vx3i x3 x3 = 0 if i 6= 3, j = 3. Thus Z Z vxi xj xk · vxi xj xk dx = ∆vxi · ∆vxi dx, Ω

Ω

as stated. The second condition in Lemma 4.1 holds, for example, if Ω = (a1 , b1 ) × (a2 , b2 ) × (a3 , b3 ) or Ω1 is a rectangle. (In these cases, any component of the eigenfunction is a trigonometric function.)

32

4.2

Existence and uniqueness theorem

Definition 4.1. The quadruple (u, ν, n, ∇p0 ) is a strong solution of problem (12)–(16), (65), (66) in the domain QT if • u is a vector field in L2 ((0, T ); Sol21 (Ω)) ∩ L2 ((0, T ); W23 (Ω)), ut ∈ L2 (QT ); • ν is a vector field in L∞ ((0, T ); W22 (Ω)), νt ∈ L∞ ((0, T ); L2 (Ω)); • n is a vector field in L∞ ((0, T ); W23 (Ω)), where n1 is a given constant vector field, and nt ∈ L∞ ((0, T ); W21 (Ω)); • ∇p0 ∈ L2 (QT ); • u, n, ν satisfy the initial conditions (16), i.e., (u, n, ν) → (u0 , n0 , ν0 ) in L2 (Ω) as t → 0; • equations (12)–(14) and boundary conditions (65),(66) hold almost everywhere. Theorem 4.1. the conditions of Lemma 4.1. Let Consider Ω satisfying ∂n 3 2 n0 ∈ W2(Ω), ∂N ∂Ω = 0, ν0 ∈ W2 (Ω), u0 ∈ Sol21(Ω) ∩ W22 (Ω), uj0 N j |∂Ω = 0, (uixj + ujxi )τ i N j |∂Ω = 0. Then problem (12)–(16), (65), (66), has a unique solution in QT for some T > 0. The time of existence T can be estimated from below T >

ku0 k2W 2 2

Const , + kν0 k2W 2 + kn0 k2W 3 + 1 2

2

where the constant depends only on the domain Ω and the positive constants J, K1 = K2 = K3 , and µ. Proof. Apply the same method we used in Section 3. Let E k and F k are the spaces defined in Section 4.1. Suppose a triple (uk , νk , nk ) to be Galerkin approximation of the solution, i.e. it satisfies equations (21)–(23) and initial data (24) for all ω ∈ E k , ζ, ψ ∈ F k . The main step of the proof is the following lemma. Lemma 4.2. There exist T > 0 and C > 0, depending only on the initial data and the constants J, Ki , µ > 0 given in the system (12)–(14), such that kuk kL2 ((0,T );W 3 ) , kνk kL∞ ((0,T );W 2 ) , knk kL∞ ((0,T );W 3 ) ≤ C, 2

2

2

kuk,t kL2 (QT ) , kνk,t kL2 (QT ) , k∇nk,t kL2 (QT ) ≤ C.

33

To prove the Lemma we repeat the procedure from Lemma 2.1. Set (ω, ψ, ζ) = (∆2 uk , ∆3 nk , ∆2 ν k ) and show that the inequalities (25)–(27) still hold. It is enough to check that boundary integrals occurring in the computations of Lemma 2.1 vanish. The boundary integral appearing in the equation (23) is Z

T

Z

0

∂Ω

(nit ∆2 nixj N j − nitxj ∆2 nN j + ∆nit ∆nixj N j + uk nixk ∆2 nixj N j − (ukxj nixk + uk nixk xj )∆2 ni N j + ∆nixj N j ∆(uk nixk )) dS dt.

(69)

(Here we omitted the lower index in uk , nk and νk to simplify notation.) We show now that integral term vanishes. Since n ∈ F k , and, consequently, ∆m nxj N j = 0, ∆m ntxj N j = 0, m = 0, 1, 2, all we need to prove is (ukxj nixk + uk nixk xj )∆2 ni N j = 0. To see this, we fix the point ∂Ω x0 ∈ ∂Ω, x0 does not lie on an edge, and choose Cartesian coordinates in which ∂Ω = {x3 = 0}. Thus, for any i = 1, . . . , n, we have (ukxj nixk + uk nixk xj )N j = u3x3 nix3 + u3 nix3 x3 +

2 X

(ukx3 nixk + uk nix3 xk ) = 0.

k=1

Here we used the director boundary conditions nxi N i |∂Ω = 0 and smooth i which guarantees that the tangential derivatives ni ness of n = x x x 3 2 3 ∂Ω i nx1 x3 ∂Ω = 0 at the point x0 . Consequently, boundary integral (69) vanishes, so we have inequality (26). (We remark that if the domain is not a polygon, this boundary integral does not necessarily vanish.) The next inequality we want to prove is (25). Considering equation (22) with ζ = ∆2 ν, we obtain the following boundary integral Z TZ i νti ∆νxi j − νtx ∆ν i + uk νxi k ∆νxi j − (uk νxi k )xj ∆ν i j 0

∂Ω

+ (∆n × n) · ∆νxj − (∆n × n)xj · ∆ν − (∆2 n × n) · νxj +(∆2 n × n)xj · ν N j dS dt (70) Since we choose ν ∈ F k , it follows that ∆m νxj N j = 0, ∆m νtxj N j = 0 on (0, T ) × ∂Ω for m = 0, 1. In addition, n ∈ F k , so the sixth and eighth terms in the integrand vanish. Indeed, recall that, in the formulas above, n and ν stand for the Galerkin approximations nk and νk , so, by definition, we have nk (t), νk (t), ∆nk (t), ∆νk (t) ∈ F k , for all t. Thus, we can use the boundary conditions. For example, on ∂Ω we get, (∆n × n)xj N j = ((∆n)xj N j ) × n + ∆n × (nxj N j ) = 0 × n + ∆n × 0 = 0. The fourth term in (70) can be represented in appropriate coordinates around the point x0 as follows: 2 X

(ukx3 νxi k + uk νxk x3 )∆ν i + (u3x3 νxi 3 + u3 νx3 x3 )∆ν i .

k=1

34

On ∂Ω this term vanishes. (For details, see how we have proved earlier that (ukxj nixk + uk nixk xj )∆2 ni N j = 0). ∂Ω

Therefore, the boundary integral (70) vanishes. The last step in the proof is to obtain the identity (27) and, consequently, the inequality (28). Setting ω = ∆2 u in equation (21), we have the boundary integral Z TZ uit ∆uixj − uitxj ∆ui + uk uixk ∆uixj − ∆ui (uk uixk )xj 0 ∂Ω −(µ∆ui + K2 ∆n · nxi )∆uixj N j dS dt = 0, by the boundary conditions (65) and (66). Thus all the boundary integrals vanish and we have inequality (28). Now, we need inequality (32). Due to Lemma 4.1, we can replace the left-hand side of (28) with X X X kuxi xj k2L2 (Ω) + J kν(t)xi xj k2L2 (Ω) + K knxi xj xk (t)k2L2 (Ω) i,j

i,j

+µ

X

i,j,k

knxi xj xk k2L2 (Qt ) .

i,j,k

To estimate the norms kukW42 , knkW42 , and kνkW41 in terms of kukW23 , knkW23 , and kνkW22 , we apply the Sobolev embedding theorems. So, we can repeat the procedure in Lemma 2.1 to obtain inequality (32). Now use Lemma 4.2, instead of Lemma 2.1, and repeat the structure of the proof of Theorem 3.1. Convergence of the approximations and uniqueness is proved with standard methods. This concludes the proof of Theorem 4.1. Theorem 4.2. Suppose Ω, n0 , ν0 , u0 , n1 satisfy the conditions of Theorem 4.1. Assume also that F ∈ L2 ((0, T ); W21 (Ω)), G ∈ L1 ((0, T ); W22 (Ω)), and Gi N i |∂Ω = 0. Then the solution of (12)–(16), (65), (66) exists and is unique for some T > 0.

5

Discussion

While the Ericksen-Leslie director model for liquid crystal nematodynamics is quite popular, it has serious limitations. Because the Oseen-Zöcher-Frank free energy (2) depends on the director n and its first derivatives, this induces problems in the choice of the boundary conditions, depending on the specific physical boundary properties of the domain. Thus, when dealing with the Ericksen-Leslie model, an important question arising from the underlying physics, is the choice of boundary conditions. 35

Experiments indicate that the boundary affects the orientation of the director. Prior research shows that the director in nematic liquid crystals is often oriented either perpendicularly to the boundary surface or in a direction parallel to it (see, e.g., [40], [41], [4]). This justifies Dirichlet-type boundary conditions. On the other hand, in certain situations, boundary conditions at the substrate are qualitatively closer to a Neumann, rather than to a Dirichlet, condition (see [51], [52]). For instance, paper [1] supposes that Dirichlet conditions correspond to the case of strong anchoring, while the weak anchoring yields Neumann boundary conditions, and mixed boundary conditions apply between these two extremes. In this paper, we study both cases. In Section 3 we considered Dirichlet boundary conditions. We showed the solvability of the problem under artificial conditions, namely the director field should be constant in a neighborhood of the boundary. Our method of proof of the local-in-time solutions does not allow to obtain better results and these are the best ones nowadays, to our knowledge. At the moment when perturbations of the director reach the boundary, we can not affirm the existence of the solution. We do not know if this is caused by the model itself, or if this is an indication, the previously cited papers notwithstanding, that Dirichlet boundary conditions are not natural for the Ericksen-Leslie director model. The Neumann boundary conditions considered in Section 4 seem to be more natural, since the initial director field is not necessarily constant near the boundary. We proved local-in-time existence of solutions only for a special type of domains (see Lemma 4.1), but we hope that it is possible to adapt our proof to the case of general smooth boundaries. Perhaps, the same proof can be applied to Robin boundary conditions. One way to solve the question of the physically relevant boundary conditions is to use alternative models of liquid crystals that are more elaborate. For example, well-posedness of the equations of motion with various boundary conditions for the Eringen (micromorphic, microstrech, or micropolar) [14] and the Lhuillier-Rey [32] models has not been done, to our knowledge. We do not know how serious the boundary conditions question is for these models, nor are we aware of an in-depth analysis of this problem in the literature. We plan to study the well posedness of these equations of motion with various boundary conditions. A considerably more drastic way to solve the issue of the boundary conditions was taken in [42] and [43] and it is based on an approach for modeling a class of isotropic and anisotropic viscoelastic fluids based on the notion of an “evolving natural configuration”. The proposed model appears not to be a director theory; no balance laws for directors are posited, nor is there a notion of a director body force, director stress, or director kinetic energy. In particular, this approach does not require specifying any additional 36

boundary conditions. It would be very interesting to study the equations of motion of this model mathematically and compare its predictions with the Ericksen-Leslie director theory and also with experimental data. However, this is the object of future research.

Acknowledgments GAC was partially supported by RFBR grant 15-01-07920. TSR was partially supported by the grant NCCR SwissMAP of the Swiss National Science Foundation. All authors were partially supported by the Government grant of the Russian Federation under the Resolution No. 220 “On measures designed to attract leading scientists to Russian institutions of higher education” according to the Agreement No. 11.G34.31.0054, signed by the Ministry of Education and Science of the Russian Federation, the leading scientist, and Lomonosov Moscow State University (on the basis of which the present research is organized).

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