GENERALIZED STOKES SYSTEM IN ORLICZ ...

Manuscript submitted to AIMS’ Journals Volume X, Number 0X, XX 200X

Website: http://AIMsciences.org pp. X–XX

GENERALIZED STOKES SYSTEM IN ORLICZ SPACES

Piotr Gwiazda Institute of Applied Mathematics and Mechanics University of Warsaw Banacha 2, 02-097 Warszawa, Poland

´ ´ blewska Agnieszka Swierczewska-Gwiazda and Aneta Wro Institute of Applied Mathematics and Mechanics University of Warsaw Banacha 2, 02-097 Warszawa, Poland

(Communicated by the associate editor name) Abstract. The paper concerns the generalized Stokes system with the nonlinear term having growth conditions prescribed by an N −function. Our main interest is directed to relaxing the assumptions on the N −function and in particular to capture the shear thinning fluids with rheology close to linear. The case of anisotropic functions is considered. The existence of weak solutions is the main result of the present paper. Additionally, for the purpose of the existence proof, a version of the Sobolev–Korn inequality in Orlicz spaces is proved.

1. Introduction. Our interest is directed to the generalized Stokes system ∂t u − div S (t, x, D u) + ∇p = f

in (0, T ) × Ω,

(1.1)

div u = 0

in (0, T ) × Ω,

(1.2)

u(0, x) = u0 u(t, x) = 0

in Ω, on (0, T ) × ∂Ω,

(1.3) (1.4)

n

where Ω ⊂ R is an open, bounded set with a sufficiently smooth boundary ∂Ω, (0, T ) is the time interval with T < ∞, Q = (0, T ) × Ω, u : Q → Rn is the velocity of a fluid and p : Q → R the pressure, S +IIp is the Cauchy stress tensor. We assume that S satisfies the following conditions (S1) S is a Carathéodory function (i.e., measurable w.r.t. t and x and continuous w.r.t. the last variable). (S2) There exists a function M : Rn×n sym → R+ and a constant c > 0 such that for all ξ ∈ Rn×n sym S(t, x, ξ ))) S (t, x, ξ ) · ξ ≥ c(M (ξξ ) + M ∗ (S

(1.5)

where M is an N − function, i.e. it is convex, has superlinear growth, M (ξξ ) = def

0 iff ξ = 0 and M (ξξ ) = M (−ξξ ), M ∗ (ηη ) = supξ ∈Rn×n (ηη · ξ − M (ξξ )) and Rn×n sym sym means the set of symmetric n × n−matrices. 2000 Mathematics Subject Classification. 35K55, 35K30, 76D07. Key words and phrases. Orlicz spaces, modular convergence, Sobolev-Korn inequality, generalized Stokes system, weak solutions, monotonicity, Prandtl-Eyring model.

1

´ ´ P. GWIAZDA, A. SWIERCZEWSKA-GWIAZDA AND A. WROBLEWSKA

2

(S3) For all ξ , η ∈ Rn×n sym and for a.a. t, x ∈ Q S(t, x, ξ ) − S (t, x, η )) · (ξξ − η ) ≥ 0. (S In the sequel we provide some examples of stress tensors satisfying the conditions collected above. First, we wish to point out that the case of stress tensors having convex potentials (additionally vanishing at 0 and symmetric w.r.t. the origin) provides an immediate verification of condition (S2). For finding N − functions M and M ∗ we notice that the following relation M (ξξ ) + M ∗ (∇M (ξξ )) = ξ · ∇M (ξξ )

(1.6)

Rn×n sym ,

holds for all ξ ∈ cf. [22]. The above identity corresponds to the case when the Fenchel-Young inequality for N − functions (Proposition 1) becomes the equality. Once we have a given function S , for simplicity consider it in the form Du) = 2µ(|D Du|2 )D Du S (D then choosing Z

|ξξ |2

M (ξξ ) =

α) dα α µ(α

(1.7)

0

provides that (S2) is satisfied with a constant c = 1. For such chosen M we only need to verify whether the N − function–conditions, i.e, behaviour in/near zero and near infinity, are satisfied. The monotonicity of S follows from the convexity of the potential. This observation makes the examples, which appear below, meaningful. By conditions (S1) − (S3) we can capture a wide class of models. Our particular interest is directed here to the rheology close to linear in at least one direction. We do not assume that the N − function satisfies ∆2 -condition1 in case of starshaped domains. For other domains we need to assume some conditions on the upper growth of M , however this does not contradict with a goal of describing the rheology close to linear. There is a wide range of fluid dynamics models obeying these conditions, we mention here two constitutive relations: Prandtl-Eyring model, cf. [6], where the stress tensor S is given by Du|) ar sinh(λ|D Du S = η0 (1.8) D λ|D u| and modified Powell-Eyring model cf. [19] Du|) ln(1 + λ|D Du Du|)m (λ|D where η∞ , η0 , λ, m are material constants. Our attention in the present paper is particularly directed to the case η∞ = 0 and m = 1. Both models are broadly used in geophysics, engineering and medical applications, e.g. for modelling of glacier ice, cf. [15], blood flow, cf. [20, 21] and many others, cf. [2, 17, 23, 29]. Our considerations concern the simplified system of equations of conservation of mass and momentum. Indeed, the convective term u · ∇u is not present in the equations. The motivation for considering such a simplified model is twofold. If the flow is assumed to be slow, then the inertial term u · ∇u can be assumed to be very small and therefore neglected, hence the whole system reduces to generalized Stokes S = η∞D u + (η0 − η∞ )

1

We say that an N -function M satisfies ∆2 –condition if for some constant C > 0 M (2ξξ ) ≤ CM (ξξ )

n×n for all ξ ∈ Rsym .


3

system (1.1)-(1.2). Another situation is the case of simple flows, e.g. Poisseuille type flow, between two fixed parallel plates, which is driven by constant pressure gradient (see [14]). With regards to blood flows the importance of considering simple flows arises since the geometry of vessels can be simplified to flow in a pipe. The analysis of both models in steady case (also without convective term) through variational approach was undertaken by Fuchs and Seregin in [9, 10]. The equations (1.1)-(1.2) with additional convective term u · ∇u in (1.1) have been extensively studied in [11, 13]. The appearance of the convective term enforced the restriction for the growth of an N -function, namely M (·) ≥ c| · |q for some ∗ exponent q ≥ 3n+2 n+2 . This automatically implied that M satisfied ∆2 -condition. Such a formulation allowed to capture shear thickening fluids, even very rapidly thickening (e.g. exponential growth). In the present paper we are able to skip the assumption on the lower growth of M (and consequently the bound for M ∗ ), which opens a possibility to include flows of different behaviour, in particular shear thinning fluids. Before stating the main result we introduce the notation of function spaces. The N -function M was defined in the condition (S2). Note, that contrary to a standard definition of Orlicz space, the N -function here has an anisotropic property and den×n pends on the whole tensor ξ ∈ Rn×n sym . By the anisotropic Orlicz class LM (Q; Rsym ) n×n we mean the set of all measurable functions ξ : Q → Rsym for which the modular Z ξ ρM (ξ ) = M (ξξ (t, x))dxdt Q

LM (Q; Rn×n sym )

we denote the generalized Orlicz space which is the set is finite. By ξ ) → 0 as α → 0. This is of all measurable functions ξ : Q → Rn×n sym for which ρM (αξ a Banach space with respect to the norm, cf. [18, Theorem 2.8], [24] Z n×n ∗ ∗ kξξ kM = sup η · ξ dxdt : η ∈ LM (Q; Rsym ), ρM (ηη ) ≤ 1 . Q n×n By EM (Q; Rn×n sym ) we mean the closure of bounded functions in LM (Q; Rsym ). The n×n n×n space LM ∗ (Q; Rsym ) is the dual space of EM (Q; Rsym ). We will say that a sequence ξ j converges modularly to ξ in LM (Q; Rn×n sym ) if there exists λ > 0 such that

ρM

ξj − ξ λ

→ 0. M

For the modular convergence we use the notation ξ j −→ ξ . The present paper consists of a new analytical approach to the existence problem. In the previous studies the main reason to assume that M ∗ satisfies a ∆2 -condition was providing that the solution is bounded in an appropriate Sobolev space W 1,q (Ω) which is compactly embedded in L2 (Ω). However, as a byproduct, we gained that n×n LM ∗ (Q; Rn×n sym ) = EM ∗ (Q; Rsym ) is a separable space. The naturally arising question is whether the existence of solutions can still be proved after omitting the convective term and relaxing the assumptions on M and M ∗ . The preliminary studies in this direction were done for an abstract parabolic equation, cf. [12]. In the present paper we present a non-trivial extension of these considerations for the system of equations. We study the problem in two different cases. In the first case the domain is star-shaped and the N -function is anisotropic with absolutely no restriction on the

4


growth. In the second case arbitrary domains with a sufficiently smooth boundary are considered. We define two functions m, m : R+ → R+ as follows m(r) := m(r) :=

min

M (ξξ ),

max

M (ξξ ).

ξ |=r ξ ∈Rn×n sym ,|ξ ξ |=r ξ ∈Rn×n sym ,|ξ

(1.9)

The existence result is formulated under the control of the spread between m and m. We use the following notation V := {u ∈ Cc∞ (Ω; Rn ); div u = 0}, L2div (Ω) := closure of V in L2 -norm, where Cc∞ (Ω; Rn ) is the set of compactly supported smooth functions. We also define the space of functions with symmetric gradient in LM (Ω; Rn×n sym ), namely BDM (Ω) := {u ∈ L1 (Ω)n | Du ∈ LM (Ω; Rn×n sym )}. The space BDM (Ω) is a Banach space with a norm DukM kukBDM (Ω) := kukL1 (Ω) + kD and it is a subspace of the space of bounded deformations BD(Ω) Du]i,j ∈ M(Ω), for i, j = 1, . . . , n}, BD(Ω) := {u ∈ L1 (Ω)n | [D ∂u

∂ui Du]i,j = 21 ( ∂x + ∂xji ). here M(Ω) denotes the space of bounded measures on Ω and [D j According to [26, Theorem 1.1.] there exists a unique continuous operator γ0 from BD(Ω) onto L1 (∂Ω)n such that the generalized Green formula Z Z Z ∂φ ∂φ Du]i,j dx = − 2 φ[D uj + ui dx + φ (γ0 (ui )νj + γ0 (uj )νi ) dHn−1 ∂xi ∂xj Ω Ω ∂Ω (1.10) holds for every φ ∈ C 1 (Ω), where ν is the unit outward normal vector on ∂Ω and γ0 (ui ) is the i-th component of γ0 (u) and Hn−1 is the (n − 1)−Hausdorff measure. Such a γ0 is a generalization of the trace operator in Sobolev spaces to the case of BD space. If additionally u ∈ C(Ω)n , then γ0 (u) = u|∂Ω . In case of u ∈ W01,1 (Ω)n it coincides with the classical trace operator in Sobolev spaces. Understanding the trace in this generalized sense we define the subspace and the subset of BDM (Ω) as follows

BDM,0 (Ω) := {u ∈ BDM (Ω) | γ0 (u) = 0}, BDM,0 (Ω) := {u ∈ BDM (Ω) | D u ∈ LM (Ω; Rn×n sym ) and γ0 (u) = 0}. Define also BDM (Q) := {u ∈ L1 (Q)n | D u ∈ LM (Q; Rn×n sym )} and corresponding subspace BDM,0 (Q) := {u ∈ BDM (Q) | γ0 (u) = 0} where γ0 has the following meaning Z Z ∂φ ∂φ Du]i,j dxdt = − dxdt 2 φ[D uj + ui ∂xi ∂xj Q Q Z + φ (γ0 (ui )νj + γ0 (uj )νi ) dHn−1 dt (0,T )×∂Ω

(1.11)


5

¯ If u ∈ BDM (Q), then for a.a. t ∈ (0, T ) we have u(t, ·) ∈ for all φ ∈ C 1 (Q). BDM (Ω). For such vector fields it is equivalent that u ∈ BDM,0 (Q) and that u(t, ·) ∈ BDM,0 (Ω) for a.a. t ∈ (0, T ). By [26, Proposition 1.1.] there exists an extension operator from BD(Ω) to BD(Rn ) and consequently we are able to extend the functions from BDM,0 (Q) by zero to the function in BDM ([0, T ] × Rn ). In what follows, the closure of Cc∞ (Ω; Rn ) with respect to two topologies will be considered, namely M 1. modular topology of LM (Q; Rn×n sym ), which we denote by Y0 , namely j ∞ ∞ Y0M ={u ∈ L∞ (0, T ; L2div (Ω)n ), D u ∈ LM (Q; Rn×n sym ) | ∃ {u }j=1 ⊂ Cc ((−∞, T ); V) : ∗

M

n×n uj * u in L∞ (0, T ; L2div (Ω)n ) and D uj −→ D u modularly in LM (Q; Rsym )} (1.12) M 2. weak-star topology of LM (Q; Rn×n sym ), which we denote by Z0 , namely j ∞ ∞ Z0M ={u ∈ L∞ (0, T ; L2div (Ω)n ), D u ∈ LM (Q; Rn×n sym ) | ∃ {u }j=1 ⊂ Cc ((−∞, T ); V) : ∗

∗

uj * u in L∞ (0, T ; L2div (Ω)n ) and D uj * D u weakly star in LM (Q; Rn×n sym )}. (1.13) The main result of the paper concerns the existence of weak solutions to the initial boundary value problem (1.1)–(1.4). Theorem 1.1. Let condition D1. or D2. be satisfied (D1) Ω is a bounded star-shaped domain, (D2) Ω is a bounded non-star-shaped domain and n

m(r) ≤ cm ((m(r)) n−1 + |r|2 + 1)

(1.14)

for all r ∈ R+ , and m satisfies ∆2 -condition. Let M be an N -function and S satisfy conditions (S1)-(S3). Then, for given u0 ∈ L2div (Ω)n and f ∈ Em∗ (Q; Rn ) there exists u ∈ Z0M such that Z Z Z −u · ∂t ϕ + S (t, x, D u) · D ϕdxdt = f · ϕdxdt − u0 ϕ(0)dx (1.15) Q

for all ϕ ∈

Q

Ω

Cc∞ (−∞, T ; V).

The paper is organized as follows: Section 2 is devoted to the Sobolev-Korn-type inequality in Orlicz spaces, which to the authors’ knowledge is a new result. In the third section we concentrate on showing that the spaces Y0M and Z0M defined above coincide and how this fact is used in the integration by parts formula. Also in the same section some facts about Orlicz spaces are collected. The last section contains the proof of Theorem 1.1, which essentially bases on the facts proved in previous sections. 2. Variant of the Sobolev-Korn inequality. The numerous classical results (e.g. Poincaré, Sobolev, Korn inequalities) are generalized from Lebesgue and Sobolev spaces to Orlicz spaces. Among others we find results of Cianchi on the Sobolev inequality, see [3, 4]. Other interesting results concern the embeddings of a very particular type of Orlicz-Sobolev space, namely BLD(Ω) := {u ∈ L1 (Ω)n : Du| ∈ Lm (Ω)} where Lm (Ω) is defined by the function m(ξ) = ξ ln(ξ + 1), ξ ∈ R+ , |D cf. Bildhauer and Fuchs [8].


6

The Korn inequality is a standard tool used in problems arising from fluid mechanics to provide the estimate of the gradient by symmetric gradient in appropriate norms. The generalization of the Korn inequality, namely Z Z Du|)dx m(|∇u|)dx ≤ c m(|D Ω

Ω ∗

is valid for the case of m and m satisfying ∆2 condition, see e.g. [7]. Since this is not the case of our considerations we will concentrate on generalizing the result of Strauss, cf. [25], namely n DukL1 (Ω) ≤ kD kukL n−1 (Ω)

to the case of integrability of appropriate N −functions.2 Indeed the following fact holds: ¯ ⊂ [− 1 , 1 ]n , Lemma 2.1. Let m be an N −function and Ω be a bounded domain, Ω 4 4 and u ∈ BDM,0 (Ω). Then n Du|)kL1 (Ω) . ≤ Cn km(|D km(|u|)kL n−1 (Ω)

(2.16)

The proof is presented in two R parts. First, we show the validity of (2.16) for Dϕ|)dx < ∞} and then the result is extended u ∈ X(Ω) = {ϕ ∈ Cc1 (Ω; Rn ); Ω m(|D for u ∈ BDM,0 (Ω). Proof. Step 1. Assume that u ∈ X(Ω) and supp u ⊂ [− 41 , 41 ]n . Let us denote δn = (1, 1, ..., 1). Then by the mean value theorem in the integral form (see e.g. [1]) it follows Z 0 X Z 21 X n n ∂j ui (x + sδn )ds (2.17) ui (x) = ∂j ui (x + sδn )ds = − − 12 j=1

0

j=1

and n X

Z ui (x) =

0

n X

Z ∂j ui (x + sδn )ds = −

− 12 i,j=1

i=1

0

1 2

n X

∂j ui (x + sδn )ds.

(2.18)

i,j=1

Hence 2

n X

Z

n X

0

ui (x) =

i=1

1 2

Z =− 0

and consequently we obtain Z n X 4| ui (x)| ≤ i=1

(∂j ui (x + sδn ) + ∂i uj (x + sδn ))ds

− 12 i,j=1

1 2

n X

(2.19) (∂j ui (x + sδn ) + ∂i uj (x + sδn ))ds

i,j=1

n X

|∂j ui (x + sδn ) + ∂i uj (x + sδn )|ds.

(2.20)

− 21 i,j=1

2 In the current section the N −function has the same properties as before with only one difference - it is defined on R+ . To distinguish, we will denote it with a small letter m, contrary to M defined on Rn×n sym .


7

Applying N -function m : R+ → R+ to the above inequality, using convexity of m and the Jensen inequality (here we use the fact that the support of u is in [− 14 , 14 ]n ) we observe that

m |

n X

1 !! n−1

ui (x)|

i=1

 Z ≤

(2.21)

1   n−1 n X 1 m . |∂j ui (x + sδn ) + ∂i uj (x + sδn )| ds 4 i,j=1



1 2

− 12

Let ek = (0, ..., 0, 1, 0, ..., 0) be a unit vector along the xk -axis and fk = δn − ek = (1, ..., 1, 0, 1, ..., 1) for k ∈ {1, ..., n − 1}. Obviously

n X

Z

n X

0

ui (x) =

Z

Z

n X

1 2

=− 0

∂k uk (x + sek )ds − 21

− 12 i,j=1,i6=k,j6=k

i=1

0

∂j ui (x + sfk )ds + Z ∂j ui (x + sfk )ds −

(2.22)

1 2

∂k uk (x + sek )ds. 0

i,j=1,i6=k,j6=k

Consequently

n X

m(|

1 ! n−1

≤

ui (x)|)

i=1

hZ



1 2

1 m 4

− 12

1 n−1 1 ≤ 2

 1 i n−1 1 |∂j ui (x + sfk ) + ∂i uj (x + sfk )| + |∂k uk (x + sek )| ds 2 i,j=1,i6=k,j6=k   n h Z 12 X 1 m |∂j ui (x + sfk ) + ∂i uj (x + sfk )| 2 − 12 n X

i,j=1,i6=k,j6=k

1 i n−1 + m (|∂k uk (x + sek )|) ds   1 n−1 h Z 12 1 1 ≤ C  m 2 2 − 12

1  n−1

n X

|∂j ui (x + sfk ) + ∂i uj (x + sfk )|

i,j=1,i6=k,j6=k

Z

1 2

+

1 ! n−1

m (|∂k uk (x + sek )|) ds

i .

− 21

(2.23)


8

1 Pn Next, we multiply expression (m(| i=1 ui (x)|)) n−1 by itself n times and conclude that n !! n−1 Z n X dx1 ...dxn m | ui (x)|

Rn

i=1

 Z ≤C



Rn n−1 Yh

 



− 12

k=1

Z

− 21

1 2

Z

1   n−1 n X 1 |∂j ui (x + sδn ) + ∂i uj (x + sδn )| ds m 4 i,j=1



1 2

Z

m

1 2

1  n−1

n X

|∂j ui (x + sfk ) + ∂i uj (x + sfk )|

i,j=1,i6=k,j6=k 1 ! n−1

1 2

+

i


(2.24)

dx1 ...dxn

− 12

 =C

XZ 

Rn

σ

 Z 

n−1 Y k=1,k∈σ n−1 Y

Z

− 21



1 2

− 12

1   n−1 n X 1 |∂j ui (x + sδn ) + ∂i uj (x + sδn )| ds m 4 i,j=1



1 2

Z

m

1 2

1  n−1

n X

|∂j ui (x + sfk ) + ∂i uj (x + sfk )|

i,j=1,i6=k,j6=k 1 ! n−1

1 2


dx1 ...dxn

− 12

k=1,k∈σ /

where σ runs over possible subsets of {1, 2, ..., n − 1}. Since supp u ⊂ [− 41 , 14 ]n , then by the Fubini theorem it is easy to notice that n !! n−1 Z n X ui | m | dx1 ...dxn Rn

i=1 1   n−1 n X 1 |∂j ui (x) + ∂i uj (x)| dx m 4 i,j=1 Rn

 X Z  ≤C σ n−1 Y



 Z



Rn

k=1,k∈σ n−1 Y k=1,k∈σ /



Z

m

1 2

n X



1  n−1

(2.25)

|∂j ui (x) + ∂i uj (x)| dx

i,j=1,i6=k,j6=k

1 n−1 m (|∂k uk (x)|) dx .

Rn

In a similar way, by integration over lines (1, −1, 1, ...., −1) etc., instead of these Pn n/(n−1) we can obtain the same bound for any km( i=1 vi (x)ui )kLn/(n−1) (Rn ) where vi ∈ {±1, 0}. Now, let vi vary by setting vi (x) = sgn ui (x), and then n n !! n−1 !! n−1 Z Z n n X X m |ui (x)| dx1 ...dxn ≤ m |vi (x)ui (x)| dx1 ...dxn Rn

i=1

Rn

i=1


9

has the same bound (up to a constant 2n ). Indeed, let Υ = {γ = (γ1 , γ2 , γ3 ) : γi ∈ {−1, 0, 1}, i = 1, 2, 3}, Aγ = {x ∈ Rn : sgn ui (x) = vi (x) = γi , i = 1, 2, 3}. Estimates (2.23), (2.25) are also valid if we integrate over any measurable subset of Rn instead of the whole Rn . One easily observes that {Aγ }γ is a devision of Rn on measurable subsets. Obviously |

n X

vi (x)ui (x)| =

i=1

n X

vi (x)ui (x) ≥ 0.

i=1

Thus Z | Rn

n X

vi (x)ui (x)|dx =

i=1

XZ γ∈Υ

=

|

Aγ

vi (x)ui (x)|dx

i=1 n X

XZ γ∈Υ

n X

(2.26)

|vi (x)ui (x)|dx = I1

Aγ i=1

where vi (x) is constant on any subset of devision Aγ . Hence all expressions in summation over γ are positive and independent of vi (x). Therefore we obtain I1 =

n X

XZ γ∈Υ

Z |ui (x)|dx =

n X

|ui (x)|dx

Rn i=1

Aγ i=1

and on the other hand n

Z

I1 ≤ 2

m(| Rn

n X

vi (x)ui (x)|)dx.

i=1

Finally we deduce that Z n X m( |ui (x)|)dx ≤ 2n

Z Rn

Rn

i=1

m(|

n X

vi ui (x)|)dx.

i=1

In the end, since the geometric mean of nonnegative numbers is no greater than the arithmetic mean, we estimate the right hand side of (2.25) Z m(| Rn

n X

n ! n−1

ui |)

dx1 ...dxn

i=1

  n X 1hZ X 1 ≤C m |∂j ui (x) + ∂i uj (x)| dx n 4 i,j=1 n R σ   n−1 n X Z X 1 + m |∂j ui (x) + ∂i uj (x)| dx 2 n R k=1,k∈σ

+

n−1 X k=1,k∈σ /

i,j=1,i6=k,j6=k

Z m Rn

i n 1 n−1 |∂k uk (x)| dx = I2 . 2

(2.27)

10


Since m is convex and m(0) = 0, then   n X 1 h1 Z X 1 I2 ≤ C m |∂j ui (x) + ∂i uj (x)| dx n 2 2 i,j=1 n R σ   Z n−1 n X X 1 |∂j ui (x) + ∂i uj (x)| dx + m 2 Rn i,j=1,i6=k,j6=k

k=1,k∈σ n−1 X

+

Z m

k=1,k∈σ /

Rn

(2.28)

i n 1 n−1 |∂k uk (x)| dx 2



 n n i n−1 X 1 . ≤ K(n) m |∂j ui (x) + ∂i uj (x)| dx 2 i,j=1 Rn Z

h

Step 2. ˜ Rn ) be the set of smooth functions in Rn with ˜ ⊃ Ω ¯ and Cc∞ (Ω; Let [− 14 , 14 ] ⊃ Ω ˜ Step 1 provides that u ∈ C ∞ (Ω; ˜ Rn ) with supp u ∈ [− 1 , 1 ]n satisfies support in Ω. c 4 4 n Du|)kL1 (Rn ) . ≤ Cn km(|D km(|u|)kL n−1 (Rn )

(2.29)

To deduce the validity of (2.29) for all u ∈ BDM,0 (Ω), we extend u by zero outside ˜ Now u can be regularized as follows of the set Ω. Obviously u ∈ BDM,0 (Ω). uε (x) := %ε ∗ u(x) ˜ Ω) and %ε is a standard mollifier, the convolution is done where ε < 21 dist (∂ Ω, ε ˜ inequality (2.29) is w.r.t. x. Since u (x) is smooth with compact support in Ω, ε provided for u . Passing to the limit with ε → 0 yields that uε → u, Duε → Du a.e. in Rn and the continuity of an N -function m provides that m(|uε |) → m(|u|), Duε |) → m(|D Du|) a.e. in Rn . m(|D To conclude the strong convergence in L1 (Ω) of the sequence {m(|uε |)} we start with an abstract fact concerning the uniform integrability. Observe that the following two conditions are equivalent for any measurable sequence {z j } Z |z j (x)|dx ≤ ε, (2.30) ∀ ε > 0 ∃ δ > 0 : sup sup ˜ j∈N A⊂Ω,|A|≤δ

∀ε > 0

∃δ > 0 :

A

Z j 1 sup |z (x)| − √δ dx ≤ ε, ˜ j∈N Ω +

(2.31)

where |ξ|+ = max{0, ξ}. The implication (2.30)⇒(2.31) is obvious. To show that also (2.31)⇒(2.30) holds let us estimate Z Z j 1 1 j sup sup |z |dx ≤ sup |A| · √ + sup |z | − √ dx δ j∈N Ω˜ δ + j∈N |A|≤δ A |A|≤δ Z √ j |z | − √1 dx. ≤ δ + sup ˜ δ + j∈N Ω Since m is a convex function, then the following inequality holds for all δ > 0 Z Z m(|u|) − √1 dx ≥ m(|%ε ∗ u|) − √1 dx. (2.32) ˜ ˜ δ + δ + Ω Ω


11

R R Finally, since Ω˜ m(|u|)dx < ∞, then also Ω˜ |m(|u|) − √1δ |+ dx is finite and hence taking supremum over ε > 0 in (2.32) we prove the uniform integrability of {m(|uε |)}. Duε |)}. Finally, by the Vitali lemma we The same considerations are valid for {m(|D conclude that m(|uε |) → m(|u|) strongly in L1 (Rn ), Duε |) → m(|D Du|) m(|D

strongly in L1 (Rn ).

Consequently, the limit u satisfies inequality (2.29). Remark: If Ω is bounded, we can rescale the space variables. Then we have n km(|u|)kL n−1 ≤ Cn km(|CrD u|)kL1 (Ω) . (Ω)

(2.33)

where Cr is a constant dependent on the jacobian of rescaling. 3. Domains and closures. In the present section we concentrate on the issue of closures of smooth functions w.r.t. various topologies. In the introduction we defined the spaces Y0M and Z0M . Our interest is directed to the equivalence between these two spaces. The simplest proof is provided in the case of star-shaped domains. For extending the result for arbitrary domains with regular boundary, the set Ω is considered as a sum of star-shaped domains. In this case the Sobolev-Korn inequality (2.16) provides an essential estimate. Another requirement appearing for non-star-shaped domains is the constrain on the spread between m and m and also on the growth of m, i.e. (D2). In the present section the integration by parts is also considered as the main issue, where the equivalence between the spaces Y0M and Z0M is crucial. Lemma 3.1 (star-shaped domains). Let M : Rn×n sym → R+ be an N -function, Ω be M M a bounded star-shaped domain and Y0 , Z0 be the function spaces defined by (1.12) and (1.13). Then Y0M = Z0M . n×n ), f ∈ Lm∗ (Q; Rn ) and Moreover, if u ∈ Y0M , χ ∈ LM ∗ (Q; Rsym ∂t u − div χ = f

in D0 (Q),

then Z sZ Z sZ 1 1 ku(s)k2L2 (Ω) − ku(s0 )k2L2 (Ω) + χ · D udxdt = f · udxdt 2 2 s0 Ω s0 Ω

(3.34)

(3.35)

for a.a. s0 , s : 0 < s0 < s < T . Proof. Since the modular topology is stronger than weak-star, obviously we have Y0M ⊂ Z0M . Therefore we focus on proving the opposite inclusion, namely Z0M ⊂ Y0M .

(3.36)

Towards this goal we want to extend u by zero outside of Ω to the whole Rn and then mollify it. To extend u we observe that Z0M ⊂ BDM,0 (Q). By definition it is obvious that each u ∈ Z0M is an element of BD(Q), hence let us concentrate on showing that it vanishes on the boundary. Recall that for u formula (1.11) is satisfied. Take the sequence {uk } of compactly supported smooth functions with the properties prescribed in the definition of the space Z0M . After inserting this sequence into (1.11) we obtain Z Z k k ∂φ k ∂φ Du ]i,j dxdt = − 2 φ[D uj + ui dxdt (3.37) ∂xi ∂xj Q Q


12

Now we can easily pass to the weak-star limit in (3.37) because of the linearity of all terms. As a consequence we conclude that the boundary term vanishes. Next we introduce uλ , where the index λ for any function v we understand as follows v λ (t, x) := v(t, λ(x − x0 ) + x0 ) where x0 is a vantage point of Ω and λ ∈ (0, 1). Let ελ = λΩ := {y = λ(x − x0 ) + x0 | x ∈ Ω}. Define then

(3.38) 1 2 dist (∂Ω, λΩ)

uδ,λ,ε (t, x) := σδ ∗ ((%ε ∗ uλ (t, x)) 1l(s0 ,s) )

where (3.39)

where %ε = ε1n %( xε ) is a standard regularizing kernel on Rn (i.e. % ∈ C ∞ (Rn ), R % has a compact support in B(0, 1) and Rn %(x)dx = 1, %(x) = %(−x)) and the convolution is done w.r.t. space variable x, ε < ε2λ and R σδ is a regularizing kernel on R (i.e. σ ∈ C ∞ (R), σ has a compact support and R σ(τ )dτ = 1, σ(t) = σ(−t)) and convolution is done w.r.t. time variable t with δ < min{s0 , T − s}. The approximated function uδ,λ,ε also has zero trace. ε→0 First we pass to the limit with ε → 0 and hence D uδ,λ,ε −−−→ D uδ,λ in L1 (Q)n×n . ε→0 For a.a. t ∈ [0, T ] the function Duδ,λ,ε (t, ·) ∈ L1 (Ω)n×n and %ε ∗ Duδ,λ (t, ·) −−−→ ε→0 D uδ,λ (t, ·) in L1 (Ω)n×n and hence %ε ∗ D uδ,λ −−−→ D uδ,λ in measure on the set [0, T ] × Ω. Duδ,λ,ε )}ε>0 we use the analogous arTo show the uniform integrability of {M (D gumentation as in the proof of Lemma 2.1, i.e. the equivalence of the following two conditions for any measurable sequence {z j } R (a) ∀ > 0 ∃ θ > 0 : sup sup |z j (t, x)|dxdt ≤ , A j∈N A⊂Q, |A|≤θ R (b) ∀ > 0 ∃ θ > 0 : sup Q |z j (t, x)| − √1δ dxdt ≤ . +

j∈N

Notice that since M is a convex function, then the following inequality holds for all δ>0 Z Z 1 1 δ,λ δ,λ M (D √ √ (3.40) Du ) − θ dxdt ≥ M (%ε ∗ D u ) − θ dxdt. Q Q + + R Duδ,λ ) − Duδ,λ ∈ LM (Q; Rn×n Finally, since βD sym ) for some β > 0, then also Q |M (βD √1 |+ dxdt is finite and hence taking supremum over ε ∈ (0, ελ ) in ?? we prove 2 θ Duδ,λ,ε )}ε>0 is uniformly integrable. Finally, Lemma 3.3 that the sequence {M (βD ε→0 provides that D uδ,λ,ε −−−→ D uδ,λ modularly in LM (Q; Rn×n sym ). Next, we pass to λ→1

λ→1

the limit with λ → 1 and obtain that D uδ,λ −−−→ D uδ in L1 (Q)n×n and D uδ,λ −−−→ + D uδ modularly in LM (Q; Rn×n we employ similar sym ). To converge with δ → 0 + arguments as to converge with ε → 0 . Finally we observe that Y0M = Z0M . The forthcoming part of the proof is devoted to the integration by parts formula. Let us define now uδ,λ,ε (t, x) := σδ ∗ ((σδ ∗ %ε ∗ uλ (t, x)) 1l(s0 ,s) ) ελ 2

(3.41)

1 2

where ε < and σ < min{s0 , T − s}. We test each equation in (3.34) with uδ,λ,ε (which is a sufficiently regular test function) Z sZ Z TZ Z TZ λ,ε δ,λ,ε (u ∗ σδ ) · ∂t (u ∗ σδ )dxdt = χ · Du dxdt − f · uδ,λ,ε dxdt. s0

Ω

0

Ω

0

Ω

(3.42)


13

Rs R The left-hand side of (3.42) is equivalent to s0 Ω (u ∗ σδ ) · (uλ,ε ∗ ∂t σδ )dxdt, hence to pass to the limit with ε → 0 and λ → 1 we use the fact that uλ,ε → u in L∞ (0, T ; L2div (Ω)n ). To handle the right-hand side of (3.42) we use the results shown in the first part of the proof. For proving the convergence of the term RT R f · uδ,λ,ε dxdt we apply Lemma 2.1 to m and observe that 0 Ω Z

δ,λ,ε

(m(|u

(t, x)|))

n n−1

n−1 Z n Duδ,λ,ε (t, x)|)dx) ≤ Cn dx m(|D Ω

Ω

for a.a. t ∈ [0, T ]. Consequently the Hölder inequality implies that Z TZ Z TZ δ,λ,ε Duδ,λ,ε (t, x)|)dxdt. (m(|u (t, x)|))dxdt ≤ CΩ,n (m(|D 0

Ω

0

Using definition of m we obtain Z TZ Z δ,λ,ε (m(|u (t, x)|))dxdt ≤ CΩ,n 0

Ω

0

T

Ω

Z

Duδ,λ,ε (t, x))dxdt M (D

(3.43)

ε→0

λ→1

Ω

Hence (3.43) provides that modular convergences D uδ,λ,ε −−−→ D uδ,λ , D uδ,λ −−−→ λ→1 δ,λ,ε ε→0 D uδ in LM (Q; Rn×n −−−→ uδ,λ , uδ,λ −−−→ uδ modularly in sym ) imply that u Lm (Q; Rn ). Using Proposition 2 for N -functions m∗ and m we obtain Z Z f · uδ,λ,ε dxdt = f · uδ dxdt. lim ε→0,λ→1

Q

Q

In a similar way Proposition 2 for N -functions M and M ∗ provides the convergence Z Z lim χ · D uδ,λ,ε dxdt = χ · D uδ dxdt. ε→0,λ→1

Q

Q

Note that for all 0 < s0 < s < T it follows Z sZ Z s 1 d (σδ ∗ u) · ∂t (σδ ∗ u)dxdt = kσδ ∗ uk2L2 (Ω) dt s0 Ω s0 2 dt 1 1 = kσδ ∗ u(s)k2L2 (Ω) − kσδ ∗ u(s0 )k2L2 (Ω) . 2 2 We pass to the limit with δ → 0 and obtain for almost all s0 , s, namely for all Lebesgue points of the function u(t) that the following identity Z sZ 1 1 lim (u ∗ σδ ) · ∂t (u ∗ σδ ) = ku(s)k2L2 (Ω) − ku(s0 )k2L2 (Ω) (3.44) δ→0 s 2 2 Ω 0 holds. Observe now the term Z TZ Z χ · (σδ ∗ ((σδ ∗ D u) 1l(s0 ,s) ))dxdt = 0

Ω

s

s0

Z (σδ ∗ χ ) · (σδ ∗ D u)dxdt. Ω

Both of the sequences {σδ ∗ χ } and {σδ ∗ D u} converge in measure in Q. Moreover, the assumptions u ∈ Y0M and χ ∈ LM ∗ (Q; Rn×n sym ) provide that the integrals Z TZ Z TZ D χ)dxdt M (D u)dxdt and M ∗ (χ 0

Ω

0

Ω

are finite. Hence using the same method as before we conclude that the sequences {M ∗ (σδ ∗ χ )} and {M (σδ ∗ D u)} are uniformly integrable and by Lemma 3.3 we


14

have M

σδ ∗ D u−→ D u modularly in LM (Q; Rn×n sym ), M∗

σδ ∗ χ −→ χ

modularly in LM ∗ (Q; Rn×n sym ).

Applying Proposition 2 allows to conclude Z sZ Z lim (σδ ∗ χ ) · (σδ ∗ D u)dxdt = δ→0

s0

Ω

s

Z χ · D udxdt.

s0

(3.45)

Ω

In the same manner we treat the source term, just instead of the N -function M we consider m. Hence we have Z sZ Z TZ (σδ ∗ f ) · (σδ ∗ u)dxdt. f · (σδ ∗ ((σδ ∗ u) 1l(s0 ,s) ))dxdt = 0

s0

Ω

Ω

Then we observe that m

σ δ ∗ u−→ u modularly in Lm (Q : Rn ), m∗

σ δ ∗ f −→ f

modularly in Lm∗ (Q; Rn ).

and we conclude that Z sZ Z lim (σ δ ∗ f ) · (σ δ ∗ u)dxdt = δ→0

s0

Ω

s

Z

s0

f · udxdt.

(3.46)

Ω

Combining (3.44), (3.45) and (3.46) we obtain after passing to the limit with ε, λ and δ in (3.42) that Z sZ Z sZ 1 1 ku(s)k2L2 (Ω) − ku(s0 )k2L2 (Ω) + χ · D udxdt = f · udxdt (3.47) 2 2 s0 Ω s0 Ω for almost all 0 < s0 < s < T . Lemma 3.2 (Non-star-shaped domain with the control of anisotropy). Let M be an n N -function such that m(r) ≤ cm ((m(r)) n−1 + |r|2 + 1) for r ∈ R+ and let m satisfy ∆2 -condition. Let Ω be a bounded domain with a sufficiently smooth boundary, Y0M , Z0M be the function spaces defined by (1.12) and (1.13). Then Y0M = Z0M . n Moreover, let u ∈ Y0M , χ ∈ LM ∗ (Q; Rn×n sym ), f ∈ Lm∗ (Q; R ) and ∂t u − div χ = f

in D0 (Q).

Then Z sZ Z sZ 1 1 ku(s)k2L2 (Ω) − ku(s0 )k2L2 (Ω) + χ · D udxdt = f · udxdt 2 2 s0 Ω s0 Ω

(3.48)

(3.49)

holds for a.a. s0 , s: 0 ≤ s0 < s ≤ T . Remark: An obvious example for Lemma 3.2 is M (ξξ ) := M (|ξξ |) which simplifies the proof. Proof. Already for Lipschitz domains there exists a finite family of star-shaped domains {Ωi }i∈J such that [ Ω= Ωi i∈J


15

see e.g. [16]. We introduce the partition of unity θi with 0 ≤ θi ≤ 1, θi ∈ P Cc∞ (Ωi ), supp θi = Ωi , i∈J θi (x) = 1 for x ∈ Ω. Applying Lemma 2.1. to m, we obtain n Z n−1 Z n Duδ,λ,ε (t, x)|)dx (m(|uδ,λ,ε (t, x)|)) n−1 dx ≤ Cn m(|D Ω

Ω δ,λ,ε

for a.a. t ∈ [0, T ] (here u is defined as in (3.39)). Consequently n n−1 Z TZ Z T Z n δ,λ,ε δ,λ,ε n−1 Du (m(|u dt. (t, x)|)) dxdt ≤ Cn (m(|D (t, x)|)dx 0

Ω

0

Ω

Using definition of m and assumption that T < ∞ we obtain n n−1 Z TZ Z T Z n Duδ,λ,ε (t, x))dx (m(|uδ,λ,ε (t, x)|)) n−1 dxdt ≤ Cn M (D dt 0

Ω

0

Ω

Z ≤ CT,n sup t∈[0,T ]

δ,λ,ε

Du M (D

n n−1 (t, x))dx .

Ω

(3.50) To show boundedness of the right-hand side of (3.50) for fixed δ we use the Jensen inequality, the Fubini theorem and nonnegativity of M in the following way Z Z Z δ,λ,ε Du Duλ,ε (t − τ, x))σδ (τ )dτ dx M (D (t, x))dx ≤ M (D Ω Ω Bδ Z Z Duλ,ε (t − τ, x))σδ (τ )dxdτ ≤ M (D (3.51) Bδ

Ω

Duλ,ε )kL1 (Bδ ×Ω) ≤ kσδ kL∞ (Bδ ) kM (D Duλ,ε )kL1 (Q) . ≤ kσδ kL∞ (Bδ ) kM (D n

Since m(r) ≤ cm ((m(r)) n−1 + |r|2 + 1) and ∇θ ∈ L∞ (Ω)n we obtain 1 1 D (uδ θi )λ,ε + (uδ ∇T θi )λ,ε + (∇θi (uδ )T )λ,ε = D (uδ θi )λ,ε ∈ LM ((0, T )×Ωi ; Rn×n sym ), 2 2 where Ωi = supp θi . λ P We observe now the function uδ,λ,ε (t, x) = i∈J %ε ∗ σδ ∗ u 1l(s0 ,s) θi , where {·}λ is defined by (3.38). Since uδ,λ,ε is in general not divergence-free, we introduce n (Ω; Rn ) which for a.a. t ∈ (0, T ) for a.a. t ∈ (0, T ) the function ϕλ,ε (t, ·) ∈ Lm n−1 is a solution to the problem X λ div ϕλ,ε = %ε ∗ σδ ∗ u 1l(s0 ,s) · ∇θi in Ω i∈J (3.52) λ,ε ϕ = 0 on ∂Ω The existence of such ϕλ,ε is provided by Theorem 3.5 applied to the N -function n m n−1 which satisfies ∆2 -condition. The quasiconvexity condition is obviously satisfied with γ = n−1 n . Then we follow the case of star-shaped domains to complete the proof, but instead of the sequence defined by (3.39), we consider X λ ψ δ,λ,ε (t, x) := %ε ∗ σδ ∗ u 1l(s0 ,s) θi − ϕλ,ε (x) i∈J


16

It remains to show that ϕλ,ε vanishes in the limit as λ → 1 and ε → 0. Indeed, Theorem 3.5 implies the estimate Z Z n n Dϕλ,ε |)dx ≤ m n−1 (|D m n−1 (|∇ϕλ,ε |)dx Ω Ω Z (3.53) X λ n %ε ∗ σδ ∗ u 1l(s0 ,s) · ∇θi |) ≤c m n−1 (| Ω

i∈J

for a.a. t ∈ (0, T ). Let us integrate (3.53) over time interval (0, T ). Since for every i ∈ J the sequence %ε ∗

n λ m n−1 n (Q) σδ ∗ u 1l(s0 ,s) · ∇θi −→ σδ ∗ u 1l(s0 ,s) · ∇θi modularly in Lm n−1

(3.54) P as ε → 0 and λ → 1 and i∈J σδ ∗ u 1l(s0 ,s) · ∇θi = 0, we immediately conclude that n X λ m n−1 n (Q) (3.55) %ε ∗ σδ ∗ u 1l(s0 ,s) · ∇θi −→ 0 modularly in Lm n−1 i∈J

as ε → 0 and λ → 1. Consequently n

m n−1

n×n n (Q; R ). D ϕλ,ε −→ 0 modularly in Lm n−1

(3.56)

Employing the same argumentation, instead of the function defined by (3.41), we test (3.48) with X λ ζ δ,λ,ε (t, x) := %ε ∗ σδ ∗ σδ ∗ u 1l(s0 ,s) θi − σδ ∗ σδ ∗ ϕλ,ε (t, x) 1l(s0 ,s) . i∈J

(3.57) Passing to the limit with λ → 1 and ε → 0 in (3.42) it again remains to show that the second term on the right-hand side of (3.57) converges to zero, i.e., the following three limits vanish Z sZ lim (u ∗ σδ ) · ∂t σδ ∗ ϕλ,ε (t, x) 1l(s0 ,s) dxdt = 0, ε→0,λ→1

s0 Ω T Z

Z lim

ε→0,λ→1

0

Z 0

(3.58)

Ω T

Z

f · σδ ∗ σδ ∗ ϕλ,ε (t, x) 1l(s0 ,s) dxdt = 0.

lim

ε→0,λ→1

χ · σδ ∗ σδ ∗ D ϕλ,ε (t, x) 1l(s0 ,s) dxdt = 0,

Ω

To show (3.58)1 we apply Theorem 3.5 with N –function m = | · |2 and the Poincaré inequality, which allow to conclude that X λ kϕλ,ε kL2 (Ω) ≤ c1 k∇ϕλ,ε kL2 (Ω) ≤ c2 k %ε ∗ σδ ∗ u 1l(s0 ,s) · ∇θi kL2 (Ω) (3.59) i∈J

for term on the left-hand side of (3.58)1 is equivalent to R s Ra.a. t ∈ (0, T ).λ,εSince the (u ∗ σ ) · ϕ ∗ ∂ σ dxdt, we pass to the limit using the fact that δ t δ s0 Ω X λ ∗ %ε ∗ σδ ∗ u 1l(s0 ,s) · ∇θi * 0 weakly-star in L∞ (0, T ; L2 (Ω)), (3.60) i∈J

thus

∗

ϕλ,ε * 0 weakly-star in L∞ (0, T ; L2 (Ω)n ) as ε → 0 and λ → 1.

(3.61)


17

n (Q; Rn×n ) (see (3.56)), m(r) ≤ Since D ϕλ,ε converges modularly to zero in Lm n−1 sym n

Dϕλ,ε ) is uniformly intecm ((m(r)) n−1 + |r|2 + 1) and (3.59) holds, then M (αD grable with some α > 0. Moreover, by Lemma 3.3 the modular convergence in n (Q; Rn×n ) to zero implies the convergence in measure to zero. Hence using Lm n−1 sym again Lemma 3.3 with a function M we conclude that Dϕλ,ε → 0 modularly in LM (Q; Rn×n sym ) as ε → 0 and λ → 1. Therefore (3.58)2 is satisfied. Finally, the convergence passage in (3.58)3 is a consequence of (3.56). It implies that ∇ϕλ,ε → 0 modularly in Lm (Q; Rn×n ) and since ϕ = 0 on ∂Ω we obtain ϕλ,ε → 0 modularly in Lm (Q; Rn ) as ε → 0 and λ → 1. Now we follow the case of star-shaped domains to complete the proof. 3.1. A few facts about Orlicz spaces. In this short subsection we collect some facts about N –functions and Orlicz spaces, which are used in the proof of the main theorem. The proofs to the collected lemmas and propositions or further references can be found e.g in [11]. Proposition 1 (Fenchel-Young Inequality). Let M be an N –function and M ∗ a complementary to M . Then the following inequality is satisfied |ξξ · η | ≤ M (ξξ ) + M ∗ (ηη ) for all ξ , η ∈ Rn×n sym . M

Lemma 3.3. Let z j : Q → Rn×n be a measurable sequence. Then z j −→ z in LM (Q; Rn×n ) modularly if and only if z j → z in measure and there exists some λ > 0 such that the sequence {M (λzzj )} is uniformly integrable, i.e., ! Z j lim sup M (λzz )dxdt = 0. R→∞

j∈N

{(t,x):|M (λzzj )|≥R}

Lemma 3.4. Let M be an N –function and for all j ∈ N let Then the sequence {zzj } is uniformly integrable.

R Q

M (zzj )dxdt ≤ c.

Proposition 2. Let M be an N –function and M ∗ its complementary function. Suppose that the sequences ψ j : Q → Rn and φj : Q → Rn are uniformly M bounded in LM (Q; Rn ) and LM ∗ (Q; Rn ) respectively. Moreover ψ j −→ ψ moduM∗

larly in LM (Q; Rn ) and φj −→ φ modularly in LM ∗ (Q; Rn ). Then ψ j · φj → ψ · φ strongly in L1 (Q). Theorem 3.5. Let Ω be a bounded domain with a Lipschitz boundary. Let m be an N –function satisfying ∆2 -condition and such that mγ is quasiconvex for some γ ∈ (0, 1). Then, for any f ∈ Lm (Ω) such that Z f dx = 0, Ω

the problem of finding a vector field v : Ω → Rn such that div v = f in Ω v=0

on ∂Ω

(3.62)

has at least one solution v ∈ Lm (Ω; Rn ) and ∇v ∈ Lm (Ω; ; Rn×n ). Moreover, for some positive constant c Z Z m(|∇v|)dx ≤ c m(|f |)dx. Ω

Ω


18

For the proof see e.g. [27, 5]. 4. Existence result. The first part of the proof of Theorem 1.1 is standard. However we recall it for completeness of the paper. Proof. We construct Galerkin approximations to (1.1) - (1.4) using basis {ω i }∞ i=1 k P constructed with eigenvectors of the Stokes operator. We define uk = αik (t)ω i , i=1

where αik (t) solve the system Z Z Z d k u · ωi + S (t, x, D uk ) · D ω i dx = f · ω i dx, Ω dt Ω Ω

(4.63)

uk (0) = P k u0 . where i = 1, . . . , k and by P k we denote the orthogonal projection of L2div (Ω)n on conv{ω 1 , . . . , ω k }. Multiplying each equation of (4.63) by αik (t), summing over i = 1, . . . , k we obtain Z Z 1 d k 2 k k ku kL2 (Ω) + S (t, x, D u ) · D u dx = f · uk dx. (4.64) 2 dt Ω Ω The Fenchel-Young inequality, the Hölder inequality, Lemma 2.1 and convexity of the N -function provide that Z Z 2˜ c f · c uk dx f · uk dx ≤ c 2˜ c Ω Ω Z Z c 2˜ c ∗ ≤ m |f | dx + m |uk | dx c 2˜ c Ω Ω Z Z n−1 c n 1 2˜ c ∗ k (4.65) n ≤ m |f | dx + |Ω| m |u | dx c 2˜ c Ω Ω Z Z c 1 2˜ c ∗ n Duk | dx ≤ m |f | dx + |Ω| Cn m |D c 2˜ c Ω Ω Z Z 2˜ c c ≤ m∗ |f | dx + M D uk dx. c 2 Ω Ω 1

In the above considerations we choose a constant such that max(|Ω| n Cn , 2c ) < c˜ < ∞, where Cn is coming from Lemma 2.1. The last inequality follows since M is a convex function, M (0) = 0 and 0 < c < 1, which is an obvious consequence of combining (1.5) with the Fenchel-Young inequality. Integrating (4.64) over the time interval (0, t) with t ≤ T , using estimate (4.65) and the coercivity condition (S2) on S we obtain Z Z Z tZ 1 k c t Duk )dxdt + c S(t, x, D uk ))dxdt ku (t)k2L2 (Ω) + M (D M ∗ (S 2 2 0 Ω 0 Ω (4.66) Z tZ 1 c ∗ 2˜ 2 ≤ m ( |f |)dxdt + ku0 kL2 (Ω) , c 2 0 Ω for all t ∈ (0, T ]. Hence there exists a subsequence such that ∗

D uk * D u

weakly-star in LM (Q; Rn×n sym )

and ∗

S (·, ·, D uk ) * χ

weakly-star in LM ∗ (Q; Rn×n sym ).

(4.67)


19

Moreover from (4.66) we conclude the uniform boundedness of the sequence {uk }∞ k=1 in the space L∞ (0, T ; L2div (Ω)n ) and as an immediate conclusion, we have at least for a subsequence ∗

uk * u

weakly-star in L∞ (0, T ; L2div (Ω)n ).

After passing to the limit we obtain the following limit identity Z Z Z Z − u · ∂t ϕdxdt + χ · D ϕdxdt = f · ϕdxdt − u0 · ϕ(0, x)dx Q

Q

Q

(4.68)

Ω

for all ϕ ∈ Cc∞ ((−∞, T ); V). In the remaining steps we will concentrate on characterizing the limit χ . Since the weak-star and modular limits coincide, Lemma 3.1 for star-shaped domains or Lemma 3.2 for non-star-shaped domains and the equality (4.68) provide Z sZ Z sZ 1 1 ku(s)k2L2 (Ω) − ku(s0 )k2L2 (Ω) + χ · D udxdt = f · udxdt (4.69) 2 2 s0 Ω s0 Ω for a.a. 0 < s0 < s < T . To pass to the limit with s0 → 0 we need to establish the weak continuity of u in L2 (Ω)n w.r.t. time. For this purpose we consider again k ∞ the sequence { du dt } and provide the uniform estimates. Let ϕ ∈ L (0, T ; Vr ), kϕkL∞ (0,T ;Vr ) ≤ 1, where Vr := closure of V w.r.t. the W r,2 (Ω)−norm where r > n2 + 1 and observe that k k Z Z du du n k k ,ϕ = , P ϕ = − S (t, x, D u ) · D (P ϕ)dx + f · (P k ϕ)dx. dt dt Ω Ω Since kP k ϕkVr ≤ kϕkVr and W r−1,2 (Ω) ⊂ L∞ (Ω) we estimate as follows Z T Z Z T k k S(t, ·, D uk )kL1 (Ω) kD D(P k ϕ)kL∞ (Ω) dt S (t, x, D u ) · D (P ϕ)dxdt kS ≤ 0 Ω 0 Z T Z T k k S(t, ·, D u )kL1 (Ω) kP ϕkVr dt ≤ c S(t, ·, D uk )kL1 (Ω) kϕkVr dt ≤c kS kS 0

0

S(·, ·, D uk )kL1 (Q) kϕkL∞ (0,T ;Vr ) ≤ ckS (4.70) and Z

0

T

Z T f · P ϕdxdt ≤ kf kL1 (Ω) kP k ϕkL∞ (Ω) dt Ω 0 Z T Z T k ≤c kf kL1 (Ω) kP ϕkVr dt ≤ c kf kL1 (Ω) kϕkVr dt

Z

k

0

(4.71)

0

≤ ckf kL1 (Q) kϕkL∞ (0,T ;Vr ) . The assumptions on f and uniform estimates for S (·, ·, D uk ) in a proper Orlicz class k provide integrability of the above functions. Hence we conclude that du dt is bounded in L1 (0, T ; Vr∗ ). By (4.66) and assumptions on f there exist a constant C > 0 such that Z S(t, x, D uk ) + m∗ (|f |) dxdt ≤ C, sup M (S k∈N

Q

20


consequently using the Jensen inequality we obtain T

Z

S(t, ·, D uk kL1 (Ω) ) + m∗ (kf kL1 (Ω) ) dt < C m(kS

sup |Ω| k∈N

0

and hence we conclude by Lemma 3.4, there exists a monotone, continuous function L : R+ → R+ , with L(0) = 0 which is independent of k and Z

s2

S(t, ·, D uk )kL1 (Ω) + kf kL1 (Ω) dt ≤ L(|s1 − s2 |) kS

s1

for any s1 , s2 ∈ [0, T ]. Consequently, estimates (4.70)-(4.71) provide that Z

s2

s1

duk , ϕ dt ≤ L(|s1 − s2 |) dt

for all ϕ with supp ϕ ⊂ (s1 , s2 ) ⊂ [0, T ] and kϕkL∞ (0,T ;Vr ) ≤ 1. The following estimates kuk (s1 ) − uk (s2 )kVr∗

= sup uk (s1 ) − uk (s2 ), ψ = (Z ≤ sup 0

sup kψkVr ≤1

ψ∈Vr

kψkVr ≤1 T

Z

s2

s1

duk (t) , ψ dt

) k du (τ ) , ϕ dt : kϕkL∞ (0,T ;Vr ) ≤ 1, supp ϕ ⊂ (s1 , s2 ) dτ (4.72)

imply that sup kuk (s1 ) − uk (s2 )kVr∗ ≤ L(|s1 − s2 |).

(4.73)

k∈N

Since u ∈ L∞ (0, T ; L2div (Ω)n ), we can choose a sequence {si0 }i , si0 → 0+ as i → ∞. Thus {u(si0 )}i is weakly convergent in L2div (Ω)n . The estimate (4.73) provides that the family of functions uk : [0, T ] → Vr∗ is equicontinuous. Using the uniform bound in L∞ (0, T ; L2div (Ω)n ) and the compact embedding L2div (Ω)n ⊂⊂ Vr∗ we conclude by means of the Arzel` a-Ascoli theorem that the sequence {uk } is relatively compact ∗ in C([0, T ]; Vr ) and u ∈ C([0, T ]; Vr∗ ). Consequently we obtain that i→∞

u(si0 ) −→ u(0)

in Vr∗ .

(4.74)

2 n The limit coincides with the weak limit of {u(si0 )}∞ i=1 in Ldiv (Ω) and hence we conclude

lim inf ku(s0 )kL2 (Ω) ≥ ku0 kL2 (Ω) . i→∞

(4.75)


21

Let s be any Lebesgue point of u. Integrating (4.64) over the time interval (0, s) gives Z sZ lim sup S (t, x, D uk ) · D uk k→∞ 0 Ω Z sZ 1 1 f · udxdt + ku0 k2L2 (Ω) − lim inf kuk (s)k2L2 (Ω) = k→∞ 2 2 Z0 s ZΩ 1 1 f · udxdt + ku0 k2L2 (Ω) − ku(s)k2L2 (Ω) ≤ 2 2 0 Ω ! Z Z s (4.75) 1 1 i 2 2 ≤ lim inf f · udxdt + ku(s0 )kL2 (Ω) − ku(s)kL2 (Ω) i→∞ 2 2 si0 Ω Z sZ Z sZ (4.69) χ · D udxdt = χ · D udxdt. = lim i→∞

si0

Ω

0

Ω

(4.76) The monotonicity of S yields Z sZ S(t, x, ¯v ) − S (t, x, D uk )) · (¯v¯ − D uk )dxdt ≥ 0 (S 0

(4.77)

Ω

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