On the Domain Dependence of Solutions to the Compressible Navier-Stokes Equations of an Isothermal Fluid

Acta Appl Math (2013) 124:187–206DOI 10.1007/s10440-012-9775-2

On the Domain Dependence of Solutionsto the Compressible Navier-Stokes Equationsof an Isothermal Fluid

Nikola Hlavácová

Received: 4 October 2011 / Accepted: 16 July 2012 / Published online: 31 July 2012© Springer Science+Business Media B.V. 2012

Abstract The aim of this paper is to study the behaviour of the variational solutions to theNavier-Stokes equations describing viscous compressible isothermal fluids with nonlinearstress tensors in a sequence of domains {Ωn}∞

n=1. The sequence converges in sense of theSobolev-Orlicz capacity to domain Ω . We prove that the solutions of the equations in Ωn

converge to a solution of the respective equations in Ω . Moreover, The result can be appliedto generalization of the existence result.

Keywords Navier-Stokes equations · Viscous compressible isothermal fluids · Nonlinearstress tensor · Orlicz spaces · Sobolev-Orlicz capacity

1 Introduction

In continuum mechanics there are two basic models of compressible fluids: barotropic onewith linear stress tensor and isothermal one with nonlinear stress tensor. In case of thebarotropic fluids, the existence of a solution was proved in [3]. The result was further gen-eralized in [1] for Lipschitz domains. In case of the isothermal fluids the existence of asolution in a domain with smooth boundary was proved in [4, 5] but a similar relaxation ofthe assumptions on boundary as in the barotropic case remained an open problem.

The main object of the paper is to address the problem modifying the technique devel-oped in [1] for the isothermal fluids. In contrast to the barotropic fluids, the biggest problemin the limit passage in isothermal fluids is not caused by the pressure but the nonlinear stresstensor.

The considered model comprise the continuity equation

∂tρ + div(ρu) = 0 in Ω (1.1)

The work was partly supported by the grant of the Czech Science Foundation No. 201/09/0917 and bythe project LC06052 (Jindrich Necas Center for Mathematical Modeling) and by the grantSVV-2011-263316.

N. Hlavácová (�)Mathematical Institute, Charles University, Prague, Czech Republice-mail: [email protected]

mailto:[email protected]

188 N. Hlavácová

and the momentum equation

∂t (ρu) + div(ρu ⊗ u) + ∇ρ − div S(Du) = ρf in Ω, (1.2)

where ρ denotes the density, u velocity field, Du the symmetric part of the velocity gradientand S the Cauchy stress. Ω is a bounded domain in R

N . The system is completed by theboundary condition

u(t,x) = 0, t ∈ (0, T ), T > 0, x ∈ ∂Ω (1.3)

and the initial conditions

ρ(0,x) = ρ0(x) ≥ 0, x ∈ Ω, (1.4)

(ρu)(0,x) = q0, x ∈ Ω. (1.5)

2 Preliminaries

For the convenience of the reader we first give in this section a brief introduction to thetheory of Orlicz spaces. We also introduce here a basic notation used throughout the paper.A function u defined on an open set Ω ⊂ R

N belongs to Lp(Ω) if

∫Ω

∣∣u(x)∣∣p dx < ∞.

If we introduce the function Φ(t) = tp we can rewrite this condition in the form

ρ(u;Φ) :=∫

Ω

Φ(∣∣u(x)

∣∣)dx < ∞. (2.1)

If we replace the function Φ mentioned above by the so-called Young function (see [2] forits definition and properties), we can say that u belongs to Orlicz class LΦ(Ω). But thisset need not be a vector space and (2.1) does not induce a norm similarly as in the caseof Lebesgue spaces. Let us denote Ψ a complementary function to the Young function Φ .In case of Φ(t) = tp , Ψ (t) behaves asymptotically as tp

′, where p′ = p

p−1 . We refer thereader to [2] for more information about Ψ . The set of all Lebesgue-measurable functions u

defined almost everywhere in Ω such that

‖u‖Φ := sup

{∫Ω

∣∣u(x)v(x)∣∣dx;v ∈ LΨ (Ω),ρ(v;Ψ ) ≤ 1

}< ∞

is called the Orlicz space and is denoted by LΦ(Ω). The space EΦ(Ω) is defined to bethe closure of B(Ω), i.e. of the set of all bounded measurable functions defined in Ω , withrespect to the norm ‖ · ‖Φ . It generally holds

EΦ(Ω) ⊆ LΦ(Ω) ⊆ LΦ(Ω),

where the equality occurs if and only if Φ satisfies the Δ2-condition, i.e.

∃k > 0, z0 ≥ 0 : Φ(2z) ≤ kΦ(z), ∀z ≥ z0.

On the Domain Dependence of Solutions 189

For a pair of complementary Young functions Φ , Ψ and u,v ∈ [0,∞) the Young in-equality

uv ≤ Φ(u) + Ψ (v)

holds (see [2, 2.2.2]). It directly implies that for u ∈ LΦ(Ω) and v ∈ LΨ (Ω) we have uv ∈L1(Ω) and ∫

Ω

∣∣u(x)v(x)∣∣dx ≤

∫Ω

Φ(∣∣u(x)

∣∣)dx +∫

Ω

Ψ(∣∣v(x)

∣∣)dx. (2.2)

For u ∈ LΦ(Ω) and v ∈ LΨ (Ω) we also have the Hölder inequality (see [2, Theorem 3.7.5]),i.e. ∫

Ω

∣∣u(x)v(x)∣∣dx ≤ ‖u‖Φ‖v‖Ψ . (2.3)

We say that a sequence {un}∞n=1 ⊂ LΦ(Ω) converges EΨ -weakly to u ∈ LΦ(Ω), denoted

by un

Ψ⇀ u, for n → ∞ if ∫

Ω

unv dx →∫

Ω

uv dx

for any function v ∈ EΨ (Ω). Every sequence {un}∞n=1 bounded in LΦ(Ω) is EΨ -weakly

compact.We define Sobolev-Orlicz space WkLΦ(Ω) as the space of all functions u such that

‖u‖k,Φ :=( ∑

α,|α|≤k

∥∥Dαu∥∥2

Φ

) 12

< ∞,

the space WkEΦ(Ω) as the closure of C∞(Ω) with respect to the norm ‖ ·‖k,Φ and the spaceWk

0 LΦ(Ω) as the closure of C∞0 (Ω) with respect to the same norm. We denote

W−1LΦ(Ω) = [W 1

0 LΨ (Ω)]′,

where Ψ is the complementary function to Φ . We say that a sequence {un}∞n=1 ⊂ W 1LΦ(Ω)

converges EΨ -weakly to u ∈ W 1LΦ(Ω), un

Ψ⇀ u in W 1LΦ(Ω), if un

Ψ⇀ u and simultane-

ously ∇un

Ψ⇀ ∇u.

We further define the Young functions we use throughout the paper. We put

Φ1(z) := z ln(1 + z), (2.4)

Φγ (z) := (1 + z) lnγ (1 + z) for γ > 1. (2.5)

and denote Ψ1, Ψγ their complementary functions. Next we define the Young function

M(z) := ez − z − 1 (2.6)

and M stands for its complementary function. Let us denote Φ 1α(z) the Young functions with

the asymptotic growth z ln1α (z) for z ≥ z0 > 0, α ∈ (1,∞), and Ψ 1

α(z) their complementary

functions. Let us define the spaces

X := {v : Ω → R

N ;v∣∣∂Ω

= 0,Dv ∈ LM(Ω)}, (2.7)

190 N. Hlavácová

Y := {v : (0, T ) × Ω → R

N ;v(t)∣∣∂Ω

= 0 for a.a. t ∈ (0, T ),Dv ∈ LM

((0, T ) × Ω

)}(2.8)

and their norms

‖v‖X := ‖Dv‖M,Ω, ‖v‖Y := ‖Dv‖M,(0,T )×Ω. (2.9)

There is no problem to verify that the growth of the functions Ψγ (z) is of the type ez1/γ

and that the function M is equivalent to the Young function Ψ1. It can be also shown that theYoung functions Φγ , γ ≥ 1, satisfy the global Δ2-condition. Relationship between Youngfunctions Φγ1 and Φγ2 is Φγ1 ≺≺ Φγ2 for 0 < γ1 < γ2, i.e.

limz→∞

Φγ1(z)

Φγ2(λz)= 0

for all λ > 0. For their complementary functions it then holds Ψγ2 ≺≺ Ψγ1 .Let us define the cut-off function as

Tk(z) = kT

(z

k

), k ≥ 1, (2.10)

with T ∈ C∞(R) such that T (−z) = T (z) for every z ∈ R, T being concave in (0,∞) and

T (z) ={

z for 0 ≤ z ≤ 1,

2 for z ≥ 3.

3 Main Result

In this section, we introduce the definition of a variational solution, the convergence of asequence of domains and properties of the stress tensor S. This enables us to formulate themain result of the paper.

Definition 3.1 Couple (ρ,u) is called the variational solution of system (1.1)–(1.3) if

• the density ρ is a nonnegative function,• continuity equation (1.1) is satisfied in the sense of distribution in R

N and in the sense ofrenormalized solutions, i.e.

∂tb(ρ) + div(b(ρ)u

) + (b′(ρ)ρ − b(ρ)

)divu = 0 in D′((0, T ) × R

N)

(3.1)

for any b ∈ C1([0,∞)) such that b and b′ are bounded provided ρ and u were extendedto be zero outside Ω ,

• u satisfies (1.3) in the sense of traces and equation (1.2) holds in space D′((0, T ) × Ω),• the energy inequality

E(τ) +∫ τ

0

∫Ω

S(u) : Dudx dt ≤ E(0) +∫ τ

0

∫Ω

ρf · udx dt (3.2)

holds for a.a. t ∈ [0, T ], where

E(t) = 1

2

∫Ω

(ρ(t)

∣∣u(t)∣∣2 + ρ(t) lnρ(t)

)dx,

E(0) = 1

2

∫Ω

( |q0|2ρ0

+ ρ0 lnρ0

)dx,


• the initial conditions (1.4)–(1.5) are satisfied in the sense

limt→0+

∫Ω

ρ(t)η dx =∫

Ω

ρ0η dx, ∀η ∈ D(Ω),

limt→0+

∫Ω

ρ(t)u(t) · η dx =∫

Ω

q0 · η dx, ∀η ∈ D(Ω),

Definition 3.2 Assume that Φ is a Young function. Let {Ωn}∞n=1 be a sequence of open sets

in RN . We say that Ωn converges to an open set Ω ⊂ R

N with respect to Φ , denoted by

Ωn

Φ−→ Ω , if

• for any compact set K ⊂ Ω there exists m ∈ N such that

K ⊂ Ωn ∀n ≥ m, (3.3)

• the sets Ωn \ Ω are bounded and

capΦ(Ωn \ Ω) → 0 pro n → ∞, (3.4)

where

capΦ(K) = inf

{∫RN

Φ(|∇v|)dx;v ∈ D

(R

N), v ≥ 1 in K

}

for each compact K ⊂ RN (see [7, page 134]).

We now specify assumptions on stress tensor S. We assume that

(1) S is coercive, i.e. ∫Ω

S(Dv) : Dv dx ≥∫

Ω

M(|Dv|)dx (3.5)

for any function v ∈ X and M defined by (2.6),(2) S is monotone, i.e.

∫Ω

(S(Dv) − S(Dw)

) : (Dv − Dw)ϕ dx ≥ 0 (3.6)

for any v,w ∈ X and ϕ ∈ C∞0 (Ω), ϕ ≥ 0,

(3) S is bounded in the following sense:

∫Ω

M(∣∣S(Dv)

∣∣)dx ≤ c

(1 +

∫Ω

M(|Dv|)dx

)(3.7)

for any functions v ∈ X

(4) S(Dv − εDw)M⇀ S(Dv) for ε → 0 and any function v ∈ Y and any w ∈ C∞

0 ((0, T ) ×Ω),

(5) if {Dun}∞n=1 ⊂ LM((0, T ) × Ω) is a sequence such that

Dun

M⇀ Du in LM

((0, T ) × Ω

)

192 N. Hlavácová

and ∫ T

0

∫Ω

S(Dun) : Dun dx dt ≤ c for all n ∈ N,

then

lim infn→∞

∫ t

0

∫Ω

S(Dun) : Dun dx dt ≥∫ t

0

∫Ω

S(Du) : Dudx dt (3.8)

for all t ∈ [0, T ] and any domain Ω .

Note that such a tensor really exist. As an example we can take

S(Du) ={

M(|Du|)Du

|Du|2 for Du �= 0,

0 for Du = 0.

It is easy to check that it satisfies the above mentioned conditions.The main result and its corollary state:

Theorem 3.3 Let {Ωn}∞n=1 be a sequence of nonempty open sets in R

N , N ≥ 2, such that

Ωn

Ψ2−→ Ω , where Ω is a nonempty open set. Assume that tensor S satisfies conditions(1)–(5). Let (ρn, un) be a variational solution of the problem (1.1)–(1.5) in (0, T )×Ωn withthe driving force f n ∈ EΨ β

r

((0, T ) × RN), β > 2, r ≥ 2, and initial data ρn

0 ∈ LΦβ(RN),

and |qn0|2 ∈ L1(RN) such that

|qn0 |2

ρn0

:= 0 if ρn0 = 0 and

|qn0 |2

ρn0

∈ L1(RN) for ρn0 > 0. Let

f n

∗⇀ f in LΨ β

r

(0, T ;LΨ β

r

(R

N))

, (3.9)

ρn0

∗⇀ ρ0 in LΦβ

(R

N), (3.10)

qn0

∗⇀ q0 in LΦ β

2

(R

N)

(3.11)

and

|qn0|2

ρn0

→ |q0|2ρ0

in L1(R

N). (3.12)

Then (passing to subsequences if necessary) we have

ρn → ρ in C([0, T ];Lweak∗

Φβ

(R

N))

,

un

∗⇀ un in LM

(0, T ;W 1

0 LΨ2

(R

N))

,

Dun

∗⇀ Du in LM

(0, T ;LM

(R

N))

and

ρnun → ρu in C([0, T ];Lweak∗

Φ β2

(R

N))

,

where (ρ, u) is a variational solution of the problem (1.1)–(1.5) in (0, T ) × Ω driven by theforce f with initial data ρ0 and q0.


Corollary 3.4 Assume that ∂Ω ∈ C0,1 and tensor S satisfies conditions (1)–(5). Let f ∈LΨ β

r

(0, T ;LΨ βr

(RN)), β > 2, r ≥ 2. For given initial data ρ0 ∈ LΦβ(Ω) and |q0|2 ∈ L1(Ω)

such that |q0|2ρ0

:= 0 if ρ0 = 0 and |q0|2ρ0

∈ L1(Ω) for ρ0 > 0 there exist functions

ρ ∈ L∞(0, T ;LΦβ

(Ω)), u ∈ Y,

such that couple (ρ,u) is the variational solution of (1.1)–(1.5). In addition

d

dt

(∫Ω

1

2ρ|u|2 + ρ lnρ dx

)+

∫Ω

S(Du) : Dudx =∫

Ω

ρf · udx

in D′((0, T ) × Ω).

4 Auxiliary Assertions

In this section we introduce several auxiliary results which will be useful in the proof of themain result.

Theorem 4.1 (Korn’s inequality) Let u ∈ W1,p

0 (Ω) for all p > 1. Then

‖u‖1,p ≤ cp2

p − 1‖Du‖p.

Proof See [6]. �

Lemma 4.2 Let u ∈ LΨ2(Ω), v ∈ LΨ1(Ω) and

‖u‖p ≤ cp‖v‖p, ∀p ≥ 2,

where the constant c does not depend on p. Then

‖u‖Ψ2 ≤ c‖v‖M. (4.1)

Proof See [9, Lemma 4.4]. �

Lemma 4.3 Let us denote B ⊂ RN an open ball such that Ω ∪ Ωn ⊂ B for all n ≥ m. Let

Ωn

Ψ2−→ Ω and

wn

Φ2⇀ w in W 1

0 LΨ2(B),

where wn ∈ W 10 LΨ2(Ωn). Then w ∈ W 1

0 LΨ2(Ω).

Proof Similarly as in [1] (3.4) implies the existence of functions ϕn ∈ D(B) such that

0 ≤ ϕn ≤ 1, ϕn ≡ 1 in Vn(Ωn \ Ω), ϕn → 0 in W 10 LΨ2(B),

where Vn(Ωn \ Ω) is an open neighbourhood of Ωn \ Ω . Put

vn = (1 − ϕn)Tk(wn) ∈ W 10 LΨ2(B), (4.2)

194 N. Hlavácová

where Tk are the cut-off functions. Functions Tk(wn) are obviously bounded in W 10 LΨ2(B)

thus (passing to subsequence if necessary)

Tk(wn)Φ2⇀ Tk(w) in W 1

0 LΨ2(B).

Furthermore, it is known from [2, Theorem 7.2.5] that W 10 LΨ2(B) ↪→↪→ C(B). Hence

wn → w in C(B)

and from the uniqueness of the limit we thus have

Tk(wn) → Tk(w) in C(B).

Now we show that vn

Φ2⇀ Tk(w) in W 1

0 LΨ2(B). For ψ ∈ EΦ2(B) one has

∣∣∣∣∫

B

(vn − Tk(w)

)ψ dx

∣∣∣∣(4.2)≤

∣∣∣∣∫

B

ϕnTk(wn)ψ dx

∣∣∣∣ +∣∣∣∣∫

B

(Tk(wn) − Tk(w)

)ψ dx

∣∣∣∣

≤ 2k‖ϕn‖Ψ2‖ψ‖Φ2 +∣∣∣∣∫

B

(Tk(wn) − Tk(w)

)ψ dx

∣∣∣∣ → 0 for n → ∞

by virtue of the EΦ2 -weak convergence of Tk(wn) and

∣∣∣∣∫

B

∇(vn − Tk(w)

)ψ dx

∣∣∣∣(4.2)≤

∣∣∣∣∫

B

∇ϕnTk(wn)ψ dx

∣∣∣∣ +∣∣∣∣∫

B

ϕn

(∇Tk(wn) − ∇Tk(w))ψ dx

∣∣∣∣

+∣∣∣∣∫

B

ϕnT′k (w)∇wψ dx

∣∣∣∣ +∣∣∣∣∫

B


∣∣∣∣

≤ 2k‖∇ϕn‖Ψ2‖ψ‖Φ2 +∣∣∣∣∫

B

ϕn


∣∣∣∣

+ ‖ϕn‖∞‖∇w‖Ψ2‖ψ‖Φ2 +∣∣∣∣∫

B


∣∣∣∣ → 0 for n → ∞

as a consequence of the EΦ2 -weak convergence of ∇Tk(wn), the strong convergence of ϕn tozero in L∞(B) which follows from imbedding theorems, and the EΦ2 -weak convergence of∇Tk(wn). Consequently Tk(w) ∈ W 1

0 LΨ2(Ω) because of vn ∈ W 10 LΨ2(Ω) for every n ∈ N.

But this also means that w ∈ W 10 LΨ2(Ω). �

Remark 4.4 If Ωn

Ψ2−→ Ω and

Dwn

M⇀ Dw in LM(B),


where Dwn ∈ LM(Ωn), then it follows from Lemma 4.2 that (passing to subsequence ifnecessary)

wn

Φ2⇀ w in W 1

0 LΨ2(B)

and according to the previous lemma w ∈ W 10 LΨ2(Ω). Thus Dw ∈ LM(Ω).

Lemma 4.5 Let u ∈ LΨγ ((0, T ) × Ω), γ ≥ 1. Then u ∈ LΨγ (0, T ;LΨγ (Ω)). Let furtherv ∈ LΦ1((0, T )×Ω). Then v ∈ LΦ 1

α

(0, T ;LΦ 1β

(Ω)) for α,β ∈ (1,∞) such that 1α

+ 1β

= 1.

Proof For functions ϕ ∈ LΦγ (Ω),∫

ΩΦγ (|ϕ|)dx ≤ 1, and ψ ∈ LΦγ (0, T ),

∫ T

0 Φγ (|ψ |)dt ≤1 we have

supψ

∣∣∣∣∫ T

0ψ sup

ϕ

∫Ω

uϕ dx dt

∣∣∣∣

≤∫ T

0

∫Ω

Ψγ

(|u|)dx dt + supψ

∫ T

0sup

ϕ

∫Ω

|ϕψ | lnγ(1 + |ϕψ |)dx dt + c

≤ c(u)

(supψ

∫ T

0|ψ |dt sup

ϕ

∫Ω

|ϕ| lnγ(1 + |ϕ|)dx

+ supψ

∫ T

0|ψ | lnγ

(1 + |ψ |)dt sup

ϕ

∫Ω

|ϕ|dx + 1

)< ∞,

where we have used the Young inequality and Jensen inequality. To prove the second part ofthe lemma we proceed in a similar way. �

Lemma 4.6 If Du ∈ LM((0, T ) × Ω) and u = 0 at ∂Ω , then u ∈ Lq(0, T ;L∞(Ω)) forq ∈ [1,∞).

Proof First we show that Du ∈ Lq(0, T ;Lp(Ω)) for arbitrary p ∈ [1,∞). Indeed,

∫Ω

|Du|p dx ≤ c

(∫Ω

M1q(|Du|)dx + 1

)≤ c

((∫Ω

M1q(|Du|)dx

)p

+ 1

),

because |z|p ≤ e|z|q + c and |a| ≤ |a|p + 1, and thus we can write

∫ T

0

(∫Ω

|Du|p dx

) qp

dt ≤ c

(∫ T

0

(∫Ω

M1q(|Du|)dx

)q

dt + 1

)

≤ c

(|Ω|q−1

∫ T

0

∫Ω

M(|Du|)dx dt + 1

)

where we have used the Jensen inequality. From the fact that Du ∈ Lq(0, T ;Lp(Ω)) itfollows from the Korn inequality (4.1) that u ∈ Lq(0, T ;W 1,p

0 (Ω)) for arbitrary p ∈ [1,∞).Since W 1,p(Ω) ↪→ L∞(Ω) for p > N , we get u ∈ Lq(0, T ;L∞(Ω)). �

196 N. Hlavácová

Lemma 4.7 Let Du ∈ LM((0, T ) × Ω) and ρ ∈ L∞(0, T ;LΦβ(Ω)), β > 2, be a solution

of (1.1) in the sense of distributions. Then

∫Ω

ρ dx =∫

Ω

ρ0 dx for a.e. t ∈ (0, T ).

Proof Let us recall the fact that if we extend the functions ρ and u to be zero outside Ω ,then the equation (1.1) is satisfied in the space D′((0, T ) × R

N) (see [9, Lemma 6.1]). In asimilar way like in [1, p. 5] we take a sequence

ϕj ∈ D(R

N), ϕj ≥ 0, sup

x∈RN

∣∣∇ϕj (x)∣∣ < 1/j, ϕj ↑ 1 for j → ∞

and the test functions of the form ϕj (x)ψ(t) with ψ ∈ D(0, T ) to deduce

∫RN

ρ(τ)ϕj dx =∫

RN

ρ(0)ϕj dx +∫ τ

0

∫RN

ρu · ∇ϕj dx dt

for a.a. τ ∈ (0, T ). Since ρ ∈ L∞(0, T ;LΦβ(RN)) and u ∈ L2(0, T ;L∞(RN)), then ρu ∈

L2(0, T ;L1(RN)). Recall that ρ(t) ∈ L1(RN) because it is extended to be zero outside Ω ,which is a bounded domain. Thus for j → ∞ we infer

∫Ω

ρ(τ)dx =∫

RN

ρ(τ)dx = limj→∞

(∫RN

ρ(0)ϕj dx +∫ τ

0

∫RN

ρu · ∇ϕj dx dt

)

= limj→∞

∫RN

ρ(0)ϕj dx =∫

RN

ρ(0)dx =∫

RN

ρ0 dx. �

5 Proof of the Main Result

In this section we prove the main result. First we derive apriori estimates. Such estimateswill allow us to pass to a limit which represents the variational solution of system (1.1)–(1.5)in Ω .

5.1 Apriori Estimates

Let us denote

θk(z) := Tk

(Φβ(z)

),

where Tk(z) are the cut-off functions defined by (2.10). Consider equation (1.1) and putb(ρn) = θ ′

k(ρn) in the renormalized continuity equation (3.1) to infer

d

dt

∫Ωn

θk

(ρn(t)

)dx −

∫Ωn

(θk

(ρn(t)

) − θ ′k

(ρn(t)

))divun(t)dx = 0

(see [9, Lemma 6.2]). Now letting k → ∞ we obtain

d

dt

∫Ωn

Φβ

(ρn(t)

)dx −

∫Ωn

(Φβ

(ρn(t)

) − ρn(t)Φ′β

(ρn(t)

))divun(t)dx = 0.


In the next step we use the fact that εM( zc) and εM(2cz) are for arbitrary ε and c the

complementary Young functions, M is equivalent with Φ1 and Φ1 satisfies the Δ2-condition.Employing the inequality

Φ1

(zΦ ′

γ (z) − Φγ (z)) ≤ Φγ (z) + c, ∀z ≥ 0, γ > 1,

we have

d

dt

∫Ωn

Φβ

(ρn(t)

)dx

=∫

Ωn

(Φβ

(ρn(t)

) − ρn(t)Φ′β

(ρn(t)

))divun(t)dx

≤ c(ε1)

∫Ωn

Φ1(Φβ

(ρn(t)

) − ρn(t)Φ′β

(ρn(t)

))dx + ε1

∫Ωn

M

( |divun(t)|c

)dx

≤ c(ε1)

(∫Ωn

Φβ

(ρn(t)

)dx + 1

)+ ε1

∫Ωn

M(∣∣Dun(t)

∣∣)dx,

where the constant c comes from the inequality

|divun| ≤ c|Dun|.We add the obtained inequality to (3.2) and estimate the remaining term on the right-hand

side as follows∣∣∣∣∫ τ

0

∫Ωn

ρnf n · un dx dt

∣∣∣∣

=∣∣∣∣∫ τ

0

∫Ωn

r√

ρnf n · r′√ρnun dx dt

∣∣∣∣

≤ c(ε2, r)

∫ τ

0

∫Ωn

Φβ(ρn)dx dt + c(ε2, r)

∫ τ

0

∫Ωn

Ψβ

(|f n|r)

dx dt

+ 1

r ′ ε2

∫Ωn

ρn0 dx

∫ τ

0‖un‖r ′

L∞(Ωn) dt

≤ c(ε2, r)

∫ τ

0

∫Ωn

Φβ(ρn)dx dt + ε2c(r,N,ρn

0

)

×∫ τ

0

∫Ωn

M(|Dun|

)dx dt + c(ε2, r,N,f ), (5.1)

where N appears from the Sobolev embedding theorem for p ≥ N . Using (3.5) we arrive at

1

2

∫Ωn

(ρn(τ )

∣∣un(τ )∣∣2 + ρn(τ ) lnρn(τ )

)dx +

∫Ωn

Φβ

(ρn(τ )

)dx

+ (1 − ε1 − ε2c(r,N)

)∫ τ

0

∫Ωn

M(|Dun|

)dx dt

≤ c(ε1, ε2)

∫ τ

0

∫Ωn

Φβ(ρn)dx dt + c(ε1, ε2, r,N,ρn

0 ,f).

198 N. Hlavácová

From the integral Gronwall inequality we have

∫Ωn

Φβ

(ρn(τ )

)dx ≤ c

(ε1, ε2, r,N,T ,ρn

0 ,f).

Altogether

1

2

∫Ωn

(ρn(τ )

∣∣un(τ )∣∣2 + ρn(τ ) lnρn(τ )

)dx +

∫Ωn

Φβ

(ρn(τ )

)dx

+ (1 − ε1 − ε2c(r,N)

)∫ τ

0

∫Ωn

M(|Dun|

)dx dt

≤ c(ε1, ε2, r,N,T ,ρn

0 ,f). (5.2)

5.2 Limit Passages

The apriori estimate derived in the previous section implies convergences of the respectivesolutions passing to subsequences if necessary. If we show that the limit functions satisfy thecontinuity equation in the sense of distribution and in the sense of renormalized solution, themomentum equation in the sense of distribution and the energy inequality, the main resultwill be proved.

5.2.1 Continuity Equation

Consider sequence {(ρn,un)}∞n=1 of variational solutions of the problem (1.1)–(1.5) on cor-

responding sets Ωn. We extend this functions to be zero in RN \ Ωn and take arbitrary open

ball B ⊂ RN such that Ω ∪ Ωn ⊂ B for all n ≥ m for some m ∈ N. We can see from (5.2)

that (passing to a subsequence if necessary)

ρn

∗⇀ ρ in L∞(

0, T ;LΦβ(B)

). (5.3)

In the same way we get from (5.2), Lemma 4.5 and Lemma 4.2 that (passing to a subse-quence if necessary)

Dun

∗⇀ Du in LM

(0, T ;LM(B)

), un

∗⇀ u in LM

(0, T ;W 1

0 LΨ2(B)), (5.4)

where u ∈ LM(0, T ;W 10 LΨ2(Ω)) from Remark 4.4.

From continuity equation (1.1) we obtain for ϕ ∈ W 10 LΨβ

(B) and ψ ∈ L2(0, T ) the esti-mate

∣∣∣∣∫ T

0ψ(t)〈∂tρn,ϕ〉dt

∣∣∣∣ =∣∣∣∣∫ T

0ψ(t)

∫B

ρnun · ∇ϕ(x)dx dt

∣∣∣∣≤ ‖ψ‖2‖ρn‖L∞(0,T ;LΦβ

(B))‖un‖L2(0,T ;L∞(B))‖∇ϕ‖Ψβ< ∞

independently of n, where we have used the conclusion of Lemma 4.6. Hence ∂tρn areuniformly bounded in L2(0, T ;W−1LΦβ

(B)). Since ρn are uniformly bounded in the spaceL∞(0, T ;LΦβ

(B)) and

W 10 LΨβ

(B) ↪→↪→ EΨβ(B),

[EΨβ

(B)]′ = LΦβ

(B) ↪→↪→ W−1LΦβ(B),


one has from [8, p. 85] that (passing to a subsequence if necessary)

ρn → ρ in C([0, T ];W−1LΦβ

(B))

and thus

ρn → ρ in C([0, T ];Lweak∗

Φβ(B)

), (5.5)

i.e. ∥∥∥∥∫

B

(ρn − ρ)ψ dx

∥∥∥∥C([0,T ])

→ 0 for n → ∞

for every ψ ∈ EΨβ(B). Since ρn are extended to be zero outside Ωn the same holds in

C([0, T ];Lweak∗Φβ

(RN)).Now we are going to show the weak-∗ convergence of functions ρnun to ρu. At first we

deduce for ϕ ∈ LΨ β2

(B) the estimate

∣∣∣∣∫

B

ρn(t)un(t) · ϕ dx

∣∣∣∣ ≤∫

B

ρn(t)∣∣un(t)

∣∣2dx +

∫B

ρn(t)|ϕ|2 dx

≤ c +∫

B

Φβ

(ρn(t)

)dx +

∫B

Ψβ

(|ϕ|2)dx (5.6)

for a.a. t ∈ [0, T ]. In view of estimate (5.2), {ρnun}∞n=1 is bounded in L∞(0, T ;LΦ β

2

(B))

and thus (passing to a subsequence if necessary)

ρnun

∗⇀ ρu in L∞(

0, T ;LΦ β2

(B)).

It remains to show that ρu = ρu. Take arbitrary open ball B1 ⊂ B1 ⊂ Ω (this and Def-inition 3.2 imply B1 ⊂ Ωn for every n ≥ m for some m ∈ N). For ϕ ∈ W 1

0 LΨβ(B1) and

ψ ∈ LΦ1(0, T ) it holds

∣∣∣∣∫ T

0ψ(t)

∫B1

(ρnun − ρu) · ϕ(x)dx dt

∣∣∣∣

≤∣∣∣∣∫ T

0ψ(t)

∫B1

(ρn − ρ)un · ϕ(x)dx dt

∣∣∣∣︸︷︷︸I1

+∣∣∣∣∫ T

0ψ(t)

∫B1

ρ(un − u) · ϕ(x)dx dt

∣∣∣∣︸︷︷︸I2

.

For the first integral we have

I1 ≤∫ T

0

∣∣ψ(t)∣∣∥∥ρn(t) − ρ(t)

∥∥W−1LΦβ

(B1)

∥∥un(t) · ϕ∥∥W1

0 LΨβ(B1)

dt,

where

∥∥un(t) · ϕ∥∥W1

0 LΨβ(B1)

= ∥∥un(t) · ϕ∥∥Ψβ ,B1

+ ∥∥∇(un(t) · ϕ)∥∥

Ψβ,B1

200 N. Hlavácová

≤ ∥∥Dun(t)∥∥

M,Ωn‖ϕ‖Ψβ,B1 + ∥∥∇un(t)

∥∥Ψβ ,Ωn

‖ϕ‖∞,B1 + ∥∥un(t)∥∥

∞,Ωn‖∇ϕ‖Ψβ,B1

≤ c∥∥Dun(t)

∥∥M,Ωn

‖ϕ‖W10 LΨβ

(B1)

using the assumption that β > 2 and Lemma 4.2, thus

I1 ≤ c∥∥ψ(t)

∥∥Φ1

‖ρn − ρ‖C([0,T ];W−1LΦβ(B1))‖Dun‖LM(0,T ;LM(Ωn))‖ϕ‖W1

0 LΨβ(B1).

In case of the second integral I2 we use weak-∗ convergences (5.4). It only remains to checkthat ρ(t)ϕ ∈ EΦ2(B1) = LΦ2(B1) (Φ2 satisfies the Δ2-condition). But this is easy becausefor σ ∈ LΨ2(B1) it holds

∣∣∣∣∫

B1

ρ(t)ϕσ dx

∣∣∣∣ ≤ c‖ρ‖L∞(0,T ;LΦβ(B1))‖ϕ‖W1

0 LΨβ(B1)‖σ‖Ψ2

as a consequence of embedding W 10 LΨβ

(B1) ↪→ L∞(B1). Altogether we have

ρnun

∗⇀ ρu in LM

(0, T ;W−1LΦβ

(B1))

for arbitrary B1 ⊂ B1 ⊂ Ω and consequently from the uniqueness of the limit, the definitionof open sets and the prolongation of ρn, un, ρ and u we have

ρnun

∗⇀ ρu in L∞(

0, T ;LΦ β2

(B)). (5.7)

Since the ball B was arbitrary, we have deduced that couple (ρ,u) satisfies the equation

∂tρ + div(ρu) = 0 in D′((0, T ) × RN)

(5.8)

and also in the sense of renormalized solution (see [9, Lemma 6.2]).

5.2.2 Momentum Equation

In the foregoing part we assumed that the open ball B ⊂ RN satisfies Ω ∪Ωn ⊂ B for n ≥ m

for some m ∈ N. But in this subsection we will consider the case B ⊂ B ⊂ Ω .Now we are going to show the uniform boundedness of ρnun ⊗ un in the space

Lq(0, T ;LΦβ(B)), q ∈ [1,∞). Let ϕ ∈ LΨβ

(B) and ψ ∈ Lq ′(0, T ), 1

q+ 1

q ′ = 1. Then ac-cording to Lemma 4.6 and (5.2) for u = un and Ω = Ωn the boundedness of ρnun ⊗ un inLq(0, T ;LΦβ

(B)) follows from the estimate

∣∣∣∣∫ T

0ψ(t)

∫B

ρn|un|2ϕ(x)dx dt

∣∣∣∣ ≤ ‖ψ‖q ′ ‖ρn‖L∞(0,T ;LΦβ(B))‖un‖2

L2q (0,T ;L∞(Ωn))‖ϕ‖Ψβ

and thus we have (passing to a subsequence if necessary)

ρnun ⊗ un ⇀ ρu ⊗ u in Lq(0, T ;LΦβ

(B)), q ∈ [1,∞). (5.9)

In the next step we obtain from the momentum equation for ϕ ∈ W 10 LΨ 1

2

(B) and ψ ∈LΨ 1

2

(0, T ) the estimate


∣∣∣∣∫ T

0ψ(t)

⟨∂t (ρnun),ϕ(x)

⟩dt

∣∣∣∣

≤∣∣∣∣∫ T

0ψ(t)

∫B

(ρnun ⊗ un) : Dϕ(x)dx dt

∣∣∣∣︸︷︷︸I1

+∣∣∣∣∫ T

0ψ(t)

∫B

ρn divϕ(x)dx dt

∣∣∣∣︸︷︷︸I2

+∣∣∣∣∫ T

0ψ(t)

∫B

S(Dun) : Dϕ(x)dx dt

∣∣∣∣︸︷︷︸I3

+∣∣∣∣∫ T

0ψ(t)

∫B

ρnf n · ϕ(x)dx dt

∣∣∣∣︸︷︷︸I4

.

We now estimate integrals Ii , i = 1,2,3,4, one by one:

I1 =∣∣∣∣∫ T

0ψ(t)

∫B

(ρnun ⊗ un) : Dϕ(x)dx dt

∣∣∣∣≤ ‖ψ‖q ′ ‖ρnun ⊗ un‖Lq(0,T ;LΦβ

(B))‖Dϕ‖Ψβ

because we have already proved that ρnun ⊗ un are uniformly bounded in the spaceLq(0, T ;LΦβ

(B)), see (5.9),

I2 =∣∣∣∣∫ T

0ψ(t)

∫B

ρn divϕ(x)dx dt

∣∣∣∣ ≤ ‖ψ‖1‖ρn‖L∞(0,T ;LΦβ(B))‖divϕ‖Ψβ

,

where we have used the fact that β > 2, and therefore LΨ 12

(B) ⊂ LΨβ(B),

I3 =∣∣∣∣∫ T

0ψ(t)

∫B

S(Dun) : Dϕ(x)dx dt

∣∣∣∣≤ ‖ψ‖Ψ 1

2

∥∥S(Dun)∥∥

LΦ 12

(0,T ;LΦ 12

(B))‖Dϕ‖Ψ 1

2

≤ ‖ψ‖Ψ 12

‖Dϕ‖Ψ 12

c2

(∫ T

0

∫Ωn

M(|Dun|

)dx dt + 1

),

which follows from Lemma 4.5 and the third assumption on the stress tensor S,

I4 =∣∣∣∣∫ T

0ψ(t)

∫B

ρnf n · ϕ(x)dx dt

∣∣∣∣ ≤∫ T

0

∣∣ψ(t)∣∣∥∥ρn(t)

∥∥Φβ

∥∥f n(t)∥∥

Ψβ‖ϕ‖∞ dt

≤ c‖ψ‖Φ βr

‖ρn‖L∞(0,T ;LΦβ(B)‖f n‖LΨ β

r

(0,T ;LΨ βr

(B))‖ϕ‖∞

in view of Lemma 4.5 and the fact that LΨ βr

(B) ↪→ LΨβ(B) which gives the result that

sequence {∂t (ρnun)}∞n=1 is uniformly bounded in LΦ 1

2

(0, T ;W−1LΦ 12

(B)). Since ρnun are

moreover uniformly bounded in L∞(0, T ;LΦ β2

(B)), see (5.6), and

LΦ 12

(B) ↪→↪→ W−1LΦβ(B) ↪→ W−1LΦ 1

2

(B)),

we arrive at (see [8, p. 85])

ρnun → ρu in C([0, T ];W−1LΦβ

(B))

(5.10)

202 N. Hlavácová

and

ρnun → ρu in C([0, T ];Lweak∗

Φ β2

(B)). (5.11)

Now we are going to show the convergence of ρnun ⊗ un. For ϕ ∈ W 10 LΨβ

(B) and ψ ∈LΦ1(0, T ) we have

∣∣∣∣∫ T

0ψ(t)

∫B

(ρnun ⊗ un − ρu ⊗ u) : ϕ(x)dx dt

∣∣∣∣

≤∣∣∣∣∫ T

0ψ(t)

∫B

((ρnun − ρu) ⊗ un

) : ϕ(x)dx dt

∣∣∣∣︸︷︷︸I1

+∣∣∣∣∫ T

0ψ(t)

∫B

(ρu ⊗ (un − u)

) : ϕ(x)dx dt

∣∣∣∣︸︷︷︸I2

.

For the first integral

I1 ≤∫ T

0

∣∣ψ(t)∣∣∥∥ρn(t)un(t) − ρ(t)u(t)

∥∥W−1LΦβ

(B)

∥∥un(t)ϕ∥∥

W10 LΨβ

(B)dt

≤ c‖ψ‖Φ1‖ρnun − ρu‖C([0,T ];W−1LΦβ(B))‖Dun‖LM(0,T ;LM(Ωn))‖ϕ‖W1

0 LΨβ(B),

which converges to zero according to (5.10). For the convergence of the second integral wecan use weak-∗ convergence (5.4) because of the estimate

∣∣∣∣∫ T

0

(∫B

ρnun · ϕ(x)dx

)2

dt

∣∣∣∣ ≤ ∥∥ρn(t)∥∥2

L∞(0,T ;LΦβ(B))

∥∥un(t)∥∥2

L2(0,T ;L∞(B))‖ϕ‖2

Ψβ,

which implies that sequence ρnun∞n=1 is bounded in L2(0, T ;LΦβ

(B)). Altogether we have

ρnun ⊗ un

∗⇀ ρu ⊗ u in LM

(0, T ;W−1LΦβ

(B)),

which together with (5.9) and the uniqueness of the limit function give

ρnun ⊗ un

∗⇀ ρu ⊗ u in Lq

(0, T ;LΦβ

(B))

for q ∈ [1,∞).In the case of ρnf n we have for ψ ∈ EΦβ

(0, T )

∣∣∣∣∫ T

0ψ(t)

∫B

(ρnf n − ρf )dx

∣∣∣∣

≤∣∣∣∣∫ T

0ψ(t)

∫B

(ρn − ρ)f n dx

∣∣∣∣ +∣∣∣∣∫ T

0ψ(t)

∫B

ρ(f n − f )dx

∣∣∣∣ (5.12)

and we can thus apply weak-∗ convergence (5.3) because it can be easily checked thatψ(t)f n ∈ L1(0, T ;EΨβ

(B)) and weak-∗ convergence (3.9) because ψ(t)ρ ∈ LΦβ(0, T ;

LΦβ(B)) ↪→ EΦ β

r

(0, T ;EΦ βr

(B)).


Next obviously

S(Dun)M⇀ S(Du) in LM

((0, T ) × B

)from (5.2) and the properties of S.

From definition of the open set it follows that ρ, u satisfy the equation

∂t (ρu) + div(ρu ⊗ u) + ∇ρ − div S(Du) = ρf in D′((0, T ) × Ω). (5.13)

Finally, we must check that S(Du) = S(Du). Similarly as in [6] we use Steklovov’sfunctions to prove that couples (ρ,u) and (ρn,un) satisfy in the sense of distributions theidentities

d

dt

∫Ω

1

2ρ|u|2 dx +

∫Ω

S(Du) : Dudx −∫

Ω

ρ divudx =∫

Ω

ρu · f dx, (5.14)

d

dt

∫Ωn

1

2ρn|un|2 dx +

∫Ωn

S(Dun) : Dun dx −∫

Ωn

ρn divun dx =∫

Ωn

ρnun · f n dx, (5.15)

respectively. After subtraction of the identities we obtain

∫ τ

0ϕh(t)

(∫Ωn

S(Dun) : Dun dx −∫

Ω

S(Du) : Dudx

)dt

=∫ τ

0ϕ′

h(t)

(∫Ωn

1

2ρn|un|2 dx −

∫Ω

1

2ρ|u|2 dx

)dt

︸︷︷︸I1

+∫ τ

0ϕh(t)

(∫Ωn

ρn divun dx −∫

Ω

ρ divudx

)dt

︸︷︷︸I2

+∫ τ

0ϕh(t)

(∫Ωn

ρnf n · un dx −∫

Ω

ρf · udx

)dt

︸︷︷︸I3

, (5.16)

where ϕh ∈ C∞0 (0, T ), 0 ≤ ϕh ≤ 1, ϕh ↑ 1 for h → 0 a.e. in [0, T ]. Let Q, Q ⊂ Ω , be

a suitable set generated by a finite unification of balls B ⊂ B ⊂ Ω . We treat the aboveintegrals one by one

I1 =∫ τ

0ϕ′

h(t)

(∫Ωn

1

2ρn|un|2 dx −

∫Ω

1

2ρ|u|2 dx

)dt

=∫ τ

0ϕ′

h(t)

(∫Ωn\Q

1

2ρn|un|2 dx −

∫Ω\Q

1

2ρ|u|2 dx

)dt

+∫ τ

0ϕ′

h(t)

∫Q

1

2

(ρn|un|2 − ρ|u|2)dx dt,

where ∫ τ

0ϕ′

h(t)

∫Q

1

2

(ρn|un|2 − ρ|u|2)dx dt → 0 for n → ∞

204 N. Hlavácová

as a consequence of the weak-∗ convergence of ρn|un|2 in Lq(0, T ;LΦβ(Q)), see (5.9), and

∫ τ

0ϕ′

h(t)

(∫Ωn\Q

1

2ρn|un|2 dx −

∫Ω\Q

1

2ρ|u|2 dx

)dt

≤ c(h)(‖χΩn\Q‖Ψβ

+ ‖χΩ\Q‖Ψβ

)< ε,

which follows from the boundedness ρn|un|2 in Lq(0, T ;LΦβ(Q)) and the suitable choice

of Q depending on h. We argue similarly for I3. The convergence of ρnf n ·un can be shownin the same way as convergence (5.12) employing (3.9) and (5.7). For the remaining term I2

we use the same method as we introduced at the beginning of Section 5.1 to derive from thecontinuity equations the identities

∫Ωn

(ρn(τ ) ln

(ρn(t) + δ

) − ρn0 ln

(ρn

0 + δ))

dx = −∫ τ

0

∫Ωn

ρ2n

δ + ρn

divun dx dt, (5.17)

and∫

Ω

(ρ(τ) ln

(ρ(t) + δ

) − ρ0 ln(ρ0 + δ))

dx = −∫ τ

0

∫Ω

ρ2

δ + ρdivudx dt, (5.18)

where δ ∈ (0,1). We can now take a ball B , Ω ∪ Ωn ⊂ B . From zero prolongation of ρ andρn and the convexity of functional

∫B

ρ ln(ρ + δ)dx we have

∫Ω

ρ(τ) ln(ρ(τ) + δ

)dx −

∫Ωn

ρn(τ ) ln(ρn(τ ) + δ

)dx

=∫

B

(ρ(τ) ln

(ρ(τ) + δ

) − ρn(τ ) ln(ρn(τ ) + δ

))dx

≤∫

B

(ln(ρ + δ) + ρ

ρ + δ

)(ρ − ρn)dx → 0 for n → ∞ (5.19)

as a consequence of (5.5) and similarly

∫Ωn

ρn0 ln

(ρn

0 + δ)

dx −∫

Ω

ρ0 ln(ρ0 + δ)dx → 0 for n → ∞ (5.20)

as a consequence of (3.10). It follows from (5.3), (5.4) and (5.17)–(5.20)

∫ τ

0ϕh(t)

(∫Ωn

ρn divun dx −∫

Ω

ρ divudx

)dt

=∫ τ

0

(ϕh(t) − 1

)∫Ωn

ρn divun dx dt −∫ τ

0

(ϕh(t) − 1

)∫Ω

ρ divudx dt

+∫ τ

0

∫Ωn

δρn

δ + ρn

divun dx dt −∫ τ

0

∫Ω

δρ

δ + ρdivudx dt

+∫ τ

0

∫Ωn

ρ2n

δ + ρn

divun dx dt −∫ τ

0

∫Ω

ρ2

δ + ρdivudx dt

≤ ‖ϕh − 1‖M‖ρn‖L∞(0,T );LΦβ(Ωn))‖Dun‖LM(0,T ;LM(Ωn))


+ ‖ϕh − 1‖M‖ρ‖L∞(0,T );LΦβ(Ω))‖Du‖LM(0,T ;LM(Ω))

+ δ‖Dun‖LM(0,T ;LM(Ωn)) + δ‖Du‖LM(0,T ;LM(Ω)) + c(n)

≤ K(c1(h) + c2(n) + δ

),

where c1(h) → 0 for h → 0, c2(n) → 0 for n → ∞ and δ can be arbitrarily small. Altogether

lim infn→∞

∫ τ

0ϕh(t)

∫Ωn

S(Dun) : Dun dx dt −∫ τ

0ϕh(t)

∫Ω

S(Du) : Dudx dt

≤ c1(h) + c2(n) (5.21)

Now we take a sequence of suitable sets Qk generated by a finite unification of ballsB ⊂ B ⊂ Ω such that |Ω \ Qk| → 0, and functions

ψk ∈ C∞0 (Qk), 0 ≤ ψk ≤ 1, ψk ↑ 1 a.e. in Ω.

From the monotonicity of the stress tensor S we infer

∫ τ

0ϕh(t)

∫Ωn

(S(Dun) − S(Dv)

) : (Dun − Dv)ψk dx dt ≥ 0

and using the properties of S we conclude

∫ τ

0ϕh(t)

∫Ωn

S(Dun) : Dun dx dt ≥∫ τ

0ϕh(t)

∫Ωn

S(Dun) : Dunψk dx dt

≥∫ τ

0ϕh(t)

∫Ωn

(S(Dun) : Dv + S(Dv) : Dun − S(Dv) : Dv

)ψk dx dt.

By virtue of (5.21), we get after the limit passage n → ∞∫ τ

0ϕh(t)

∫Ω

S(Du) : Dudx dt

≥∫ τ

0ϕh(t)

∫Ω

(S(Du) : Dv + S(Dv) : Du − S(Dv) : Dv

)ψk dx dt.

In view of the Lebesgue theorem for k → ∞ and all foregoing estimates

∫ τ

0ϕh(t)

∫Ω

(S(Du) − S(Dv)

) : (Du − Dv)dx dt ≥ −c(δ,h),

where c(δ,h) is arbitrarily small. Thus we get using the Lebesgue theorem

∫ τ

0

∫Ω

(S(Du) − S(Dv)

) : (Du − Dv)dx dt ≥ 0.

Then put v = u − λψ , where ψ ∈ C∞0 ((0, T ) × Ω) and λ > 0. As a consequence to the

monotonicity and the fourth property of the stress tensor S we infer

divS(Du) = div S(Du) in LΦ 1α

(0, T ;W−1LΦ 1

2

(Ω)), α > 2.

206 N. Hlavácová

Moreover, S(Du) ∈ LM((0, T ) × Ω) in view of (3.5), (3.7) and (3.8).In the same way as (5.14) we deduce

d

dt

(∫Ω

1

2ρ|u|2 + ρ lnρ dx

)+

∫Ω

S(Du) : Dudx =∫

Ω

ρf · udx.

5.2.3 Energy Inequality

We know, that the energy inequality (3.2) holds for ρn, un, i.e.

1

2

∫Ωn

(ρn(τ )

∣∣un(τ )∣∣2 + ρn(τ ) lnρn(τ )

)dx +

∫ τ

0

∫Ωn

S(un) : Dun dx dt

≤ 1

2

∫Ωn

( |qn0|2

ρn0

+ ρn0 lnρn

0

)dx +

∫ τ

0

∫Ωn

ρnf n · un dx dt.

If we multiply this inequality by ψ ∈ D(0, T ), ψ ≥ 0, pass to the limit for n → ∞ anduse convergences proved in forgoing sections and the fifth condition on stress tensor S, wededuce

1

2

∫ T

0ψ(τ)

∫Ω

(ρ|u|2 + ρ lnρ

)dx dτ +

∫ T

0ψ(τ)

∫ τ

0

∫Ω

S(u) : Dudx dt dτ

≤ 1

2

∫Ω

( |q0|2ρ0

+ ρ0 lnρ0

)dx

∫ T

0ψ(τ)dτ +

∫ T

0ψ(τ)

∫ τ

0

∫Ω

ρf · udx dt dτ.

Since ψ was arbitrary, we deduce the energy inequality (3.2) for a.a. τ ∈ (0, T ) and Theo-rem 3.3 is proved.

References

1. Feireisl, E., Novotný, A., Petzeltová, H.: On the domain dependence of solutions to the compressibleNavier-Stokes equations of a barotropic fluid. Math. Methods Appl. Sci. 25(12), 1045–1073 (2002)

2. Kufner, A., John, O., Fuèík, S.: Function Spaces. Czech. Acad. Sci., Prague (1977). XV, 454 pp. Dfl.120.00

3. Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. Oxford LectureSeries in Mathematics and its Applications., vol. 10. Clarendon Press, Oxford (1998)

4. Mamontov, A.E.: Global solvability of the multidimensional Navier-Stokes equations of a compressiblefluid with nonlinear viscosity I. Sib. Math. J. 40(2), 351–362 (1999)

5. Mamontov, A.E.: Global solvability of the multidimensional Navier-Stokes equations of a compressiblefluid with nonlinear viscosity II. Sib. Math. J. 40(3), 541–555 (1999)

6. Mamontov, A.E.: Orlicz spaces in the existence problem of global solutions of viscous compressiblenonlinear fluid equations. Sb. Math. 190(8), 1131–1150 (1999)

7. Rudd, M.: A direct approach to Orlicz–Sobolev capacity. Nonlinear Anal., Theory Methods Appl. 60(1),129–147 (2005)

8. Simon, J.: Compact sets in the space Lp(0, T ;B). Ann. Mat. Pura Appl. 146, 65–96 (1987)9. Vodák, R.: Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids

with a nonlinear stress tensor. J. Evol. Equ. 4(2), 213–247 (2004)

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On the Domain Dependence of Solutions to the Compressible Navier-Stokes Equations of an Isothermal Fluid