On compactness of solutions to the compressible isentropic Navier-Stokes equations when

On compactness of solutions to the compressible isentropic Navier-Stokes equations when

the density is not square integrable

Eduard Feireisl

Abstract. We show compactness of bounded sets of weak solutions to the isentropic compressible Navier-Stokes equations in three space dimensions under the hypothesis that the adiabatic constantγ >3/2.

Keywords: compressible flow, weak solutions, compactness Classification: 35Q30, 35B05

1. Introduction

In this paper, we generalize the result of Lions [9] concerning compactness of bounded sets of weak solutions to the Navier-Stokes equations of a compressible isentropic fluid flow:

∂_t̺+ div(̺u) = 0, (1.1)

∂_t(̺u) + div(̺u⊗u)−µ∆u−(λ+µ)∇ divu+a∇̺^γ= 0, (1.2)

where the density ̺ = ̺(t, x) and the velocityu =u(t, x) are functions of the time t ∈ (0, T) and the spatial coordinate x ∈ Ω. Here Ω ⊂ R³ is a bounded domain with regular boundary on whichusatisfies the standard no-slip boundary conditions:

(1.3) u|_∂Ω= 0.

Formally, multiplying the second equation byuand integrating by parts one obtains the energy inequality:

(1.4) d

dtE(t) + Z

Ω

µ|∇u|²+ (λ+µ)|divu|² dx≤0 where

E(t) =E[̺,u](t) = Z

Ω

1

2̺|u|²+ a

γ−1̺^γ dx.

Work supported by Grant 201/98/1450 GA ˇCR.

We assumea >0,γ >1,µ >0, andλ+ 2/3µ≥0 throughout the whole text.

In what follows, we shall deal withfinite energy weak solutions of the problem (1.1)–(1.3), specifically,̺,uwill comply with the following hypotheses:

• ̺,u= [u¹, u², u³] satisfy

̺≥0, ̺∈L^∞(0, T;L^γ(Ω)), uⁱ∈L²(0, T;W₀^1,2(Ω)), i= 1,2,3;

• the energyEis locally integrable on (0, T) and the energy inequality (1.4) holds inD^′(0, T);

• the equations (1.1), (1.2) are satisfied inD^′((0, T)×Ω) and, in addition, (1.1) holds in the sense of renormalized solutions, i.e.,

(1.5)

(∂_tb(̺) + div(b(̺)u) + (b^′(̺)̺−b(̺)) divu= 0 in D^′((0, T)×Ω) for any b∈C¹(R) such that |b^′(z)z|+|b(z)| ≤c for all z∈R.

Our main result reads as follows:

Theorem 1.1. Letγ > ³₂ andΩ⊂R³ be a bounded Lipschitz domain. Assume

̺_n,u_nis a sequence of finite energy weak solutions of the problem(1.1)–(1.3)on the set(0, T)×Ωsatisfying

(1.6) ess lim sup

t→0+ E[̺n,u_n](t)≤E0 uniformly in n= 1,2, . . . , and

(1.7) ̺n(0)→̺0 in L¹(Ω).

Then there exists a subsequence(not relabeled)such that

̺n→̺ in L¹((0, T)×Ω) and C([0, T];L^γ_weak(Ω)), u_n→u weakly in L²(0, T; [W₀^1,2(Ω)]³)

where̺,uis a finite energy weak solution of the problem(1.1)–(1.3).

Theorem 1.1 generalizes a similar result of Lions [9] where the adiabatic expo- nent satisfies the restriction

(1.8) γ≥ 9

5.

Using (1.8) together with estimates analogous to those presented in Section 3 below, one can prove that the sequence̺_n is bounded inL²((0, T)×Ω) and so is its weak limit̺. This in turn implies that the limit functions ̺, u represent a renormalized solution of (1.1) in the sense of DiPerna and Lions [3], i.e., they

satisfy (1.5). Such a result is far from being obvious under the sole hypothesis γ >3/2.

The rest of the paper is devoted to the proof of Theorem 1.1. In Section 2, we review some basic properties of renormalized solutions, in particular, we shall show that (1.5) holds, in fact, on the whole set (0, T)×R³ provided ̺, u are prolonged to be zero out of Ω.

In Section 3, we introduce a generalized inverse of the divergence operator and obtain furtherL^p-estimates of the density analogous to those presented in [6] and by Lions [10].

The limit passage forn→ ∞is carried out in Sections 4 and 5. Similarly as in [9] and [5], the main result (Proposition 5.1) asserts a sort of weak continuity of the quantitya̺^γ−(λ+ 2µ) divucalled the effective viscous flux. Here we give a simple proof based on Div-Curl Lemma of compensated compactness.

The main novelty of the present paper is the proof of strong convergence of the density inL¹((0, T)×Ω) under the sole hypothesisγ >3/2. This is done in Sections 6–8. The present approach is based on the cut-off operators introduced in [4] and [5]. More specifically, we consider a family of functions

T_k(z) =k T(z

k) for z∈R, k= 1,2, . . . whereT ∈C^∞(R) is chosen so that

T(z) =z for|z| ≤1, T(z) = 2 forz≥3, T concave on [0,∞), T(−z) =−T(z).

The main idea, formulated in Proposition 6.1, is to show that

(1.9) lim sup

n→∞ kT_k(̺_n)−T_k(̺)k_Lγ+1((0,T)×Ω)≤c(E₀)

where the constant c is independent of k. This is an estimate in the spirit of Jiang and Zhang [8] where the authors prove the existence of weak solutions to the problem (1.1)–(1.3) with radially symmetric initial data. They show, roughly speaking, that neither the sequence̺_n nor its weak limit ̺is square integrable but the amplitude of possible oscillations is.

The relation (1.9) is then used in Section 7 to prove that the limits ̺, u satisfy (1.5). The proof of Theorem 1.1 is completed in Section 8 by showing the strong convergence of the densities̺_n.

2. Basic properties of renormalized solutions

Lemma 2.1. Let̺,ube a finite energy weak solution of the problem(1.1)–(1.3) .

Then, prolonging̺,uto be zero outside Ωwe have

(2.1) ∂tb(̺) + div(b(̺)u) + (b^′(̺)̺−b(̺)) divu= 0 in D^′((0, T)×R³)

for anybas in(1.5).

Proof: We have to show Z _T

0

Z

R³

b(̺)ϕ_t+b(̺)u.∇ϕ+ (b(̺)−b^′(̺)̺) divuϕ dx dt= 0

for allϕ∈ D((0, T)×R³). To this end, consider a sequence of functionsφ_m ∈ D(Ω) such that

(2.2)







0≤φm≤1, φm(x) = 1 for all x such that dist[x, ∂Ω]≥ 1 m,

|∇φm(x)| ≤2m for all x∈Ω.

Now, we have Z T

0

Z

R³

b(̺)ϕ_tdx dt= Z T

0

Z

Ω

b(̺)(φ_mϕ)_t+b(̺)(1−φ_m)ϕ_t dx dt,

Z T 0

Z

R³

b(̺)u.∇ϕ dx dt= Z T

0

Z

Ω

b(̺)u.∇(φmϕ) +b(̺)(1−φm)u.∇ϕ−b(̺)u.∇φmϕ dx dt, and

Z _T

0

Z

R³

(b^′(̺)̺−b(̺)) divuϕ dx dt= Z T

0

Z

Ω

b^′(̺)̺−b(̺)

divu

φ_mϕ+ (1−φ_m)ϕ dx dt.

Since̺,usatisfy (1.5), one has Z _T

0

Z

Ω

b(̺)(φmϕ)t+b(̺)u.∇(φmϕ) + (b(̺)−b^′(̺)̺) divuφm ϕ dx dt= 0;

whence it is enough to show (2.3)

Z _T

0

Z

Ω

b(̺)u.∇φm ϕ dx dt→0 as m→ ∞.

The velocity componentsuⁱ, i= 1,2,3 belong toL²(0, T;W₀^1,2(Ω)) and, con- sequently,

|u| dist⁻¹[x, ∂Ω]∈L²(0, T;L²(Ω)).

On the other hand, by virtue of (2.2),

dist[x, ∂Ω]|∇φ_m| →0 in L^p(Ω) for any 1≤p <∞,

yielding (2.3).

With the conclusion of Lemma 2.1 at hand, we can regularize the equation (2.1) in the spirit of DiPerna and Lions [3]. Introducing a regularizing sequence ϑε(x), one obtains

(2.4) ∂_tS_ε[b(̺)] + div(S_ε[b(̺)]u) +S_εh

(b^′(̺)̺−b(̺)) divui

=r_ε whereS_ε[v] =ϑ_ε∗v. By virtue of [9, Lemma 2.3], we have

(2.5) rε→0 in L²(0, T;L²(R³)) as ε→0+

sincebis uniformly bounded.

3. More about integrability of the density

For any finite energy weak solution of (1.1), (1.2), the pressure term p(̺) =a̺^γ belongs a priori only to the space L^∞(0, T;L¹(Ω)). We shall show that one can control possible concentration effects up to the boundary. To this end, we introduce the operatorB= [B₁,B₂,B₃] enjoying the properties:

•

B:n

g∈L^p(Ω)| Z

Ωg= 0o

7→[W₀^1,p(Ω)]³ is a bounded linear operator, i.e.,

kB[g]k_W1,p

0 (Ω)≤c(p)kgk_L^p_(Ω) for any 1< p <∞;

• the functionv=B[g] solves the problem divv=g in Ω, v|_∂Ω= 0;

• if, moreover, g can be written in the form g = divh for a certain h ∈ [L^r(Ω)]³,h.n|_∂Ω= 0, then

kB[g]k_L^r_(Ω)≤c(r)khk_L^r_(Ω)

for arbitrary 1< r <∞.

The operatorBwas introduced by Bogovskii [1]. A complete proof of the above mentioned properties may be found in Galdi [7, Theorem 3.3] or Borchers and Sohr [2, Proof of Theorem 2.4].

At this stage we can use the operatorBto construct multipliers of the form ϕ_i(t, x) =ψ(t)B_ih

S_ε[b(̺)]− I

Ω

S_ε[b(̺)]dxi

, i= 1,2,3, ψ∈ D(0, T) whereSεare the smoothing operators introduced in (2.4) andH

Ωvdx=_|Ω|¹ R

Ωvdx.

The functions ϕ_i are smooth with respect to the x-variable while ∂tϕ_i are bounded inL²(0, T;W₀^1,2(Ω)) in view of (2.4), (2.5). Consequently, the quantities ϕ_i, i= 1,2,3 may be used as test functions for the equations (1.2) and, after a bit lengthy but straightforward computation where (2.4) is taken into account, one arrives at the following formula:

a Z T

0

Z

Ωψ̺^γSε[b(̺)] dxdt= (3.1)

Z _T

0

ψZ

Ω

a̺^γ dxI

Ω

Sε[b(̺)] dx

dt+ (λ+µ) Z _T

0

Z

Ω

ψ Sε[b(̺)] divudxdt−

Z T 0

Z

Ω

ψ_t̺uⁱ B_in

S_ε[b(̺)]− I

Ω

S_ε[b(̺)]dxo

dxdt+

Z T 0

Z

Ω

ψ

µ∂xjuⁱ−̺uⁱu^j

∂xjB_in

Sε[b(̺)]− I

Ω

Sε[b(̺)] dxo

dxdt+

Z T 0

Z

Ω

ψ̺uⁱB_in S_εh

(b(̺)−b^′(̺)̺) divui

− I

Ω

S_εh

(b(̺)−b^′(̺)̺) divui dxo

dxdt+

Z T 0

Z

Ωψ̺uⁱ B_in rε−

I

Ωrε dxo dx−

Z T 0

Z

Ωψ̺uⁱ B_in div

Sε[b(̺)]uo dxdt (the summation convention has been used).

Now, making use of (2.5), we can pass to the limit forε→0 in (3.1). Moreover, approximating the functionz7→z^θby a sequence of functionsb_nsatisfying (1.5), we deduce the following result (see [6] for details):

Proposition 3.1. Letγ > ³₂ and let̺,ube a finite energy weak solution of the problem(1.1)–(1.3)such that

ess lim sup

t→0+

E(t)≤E₀.

Then there existθ >0, depending only onγ, andc=c(T, E₀), such that Z T

0

Z

Ω

̺^γ+θ dx dt≤c(T, E0).

Remark. It can be shown (cf. Lions [9]) that the optimal value ofθisθ= ²₃γ−1.

Thus forγ≥9/5, one getsγ+θ≥2.

4. The limit passage

The uniform energy estimates induced by (1.4) and the hypothesis (1.6) together with (1.1), (1.2) yield

̺n→̺ in C([0, T];L^γ_weak(Ω)), u_n→u weakly in L²(0, T; [W₀^1,2(Ω)]³), (4.1)

̺nu_n→̺u in C([0, T];L

2γ γ+1

weak(Ω)), (4.2)

and, by virtue of Proposition 3.1,

̺^γ→̺^γ weakly in L

γ+θ

γ ((0, T)×Ω)

passing to subsequences as the case may be. Moreover (4.1) together with (4.2) imply

̺uⁱ_nu^j_n→̺uⁱu^j, i, j= 1,2, . . . in, say, D^′((0, T)×Ω) and, consequently,̺,usatisfy

∂_t̺+ div(̺u) = 0, (4.3)

∂t(̺u) + div(̺u⊗u)−µ∆u−(λ+µ)∇ divu+a∇̺^γ= 0 (4.4)

in D^′((0, T)×Ω). Thus the only thing to prove is the strong convergence of ̺n

inL¹ or, equivalently,̺^γ=̺^γ. By virtue of Lemma 2.1, we have

(4.5) ∂_tT_k(̺_n) + div(T_k(̺_n)u_n) + (T_k^′(̺_n)̺_n−T_k(̺_n)) divu_n= 0

in D^′((0, T)×R³) whereT_kare the cut-off functions introduced in Section 1.

Passing to the limit forn→ ∞we obtain

(4.6) ∂_tT_k(̺) + div(T_k(̺)u) + (T_k^′(̺)̺−T_k(̺)) divu= 0 in D^′((0, T)×R³) where

(T_k^′(̺n)̺n−T_k(̺n)) divu_n→(T_k^′(̺)̺−T_k(̺)) divu weakly in L²((0, T)×Ω) and

(4.7) T_k(̺n)→T_k(̺) in C([0, T];L^p_weak(Ω)) for all 1≤p <∞.

5. The effective viscous flux

We shall investigate the properties of the quantity a̺^γ −(λ+ 2µ) divu called usually the effective viscous flux. It turns out that it is “more regular” than its components, in particular, it exhibits certain weak continuity. This is the crucial property used in the proof of existence of weak solutions as presented in Lions [9].

Proposition 5.1. Under the hypotheses of Theorem1.1, we have

n→∞lim Z T

0

Z

Ω

ψφ

a̺^γ_n−(λ+ 2µ) divu_n

T_k(̺_n)dx dt= Z T

0

Z

Ω

ψφ

a̺^γ−(λ+ 2µ) divu

T_k(̺)dx dt

for anyψ∈ D(0, T),φ∈ D(Ω).

Remark. Similar assertion withT_k(̺) replaced by̺^θmay be found in Lions [9].

Here we give a different proof based on Div-Curl Lemma.

Proof: Consider the operators

A_j[v] = ∆⁻¹∂_xj(v), j= 1,2,3, specifically,

A_j[v] =F⁻¹n−iξ_j

|ξ|² F {v}(ξ)o

, j= 1,2,3,

whereF denotes the Fourier transform.

By means of the Mikhlin multiplier theorem, we have

k∂_xiA_j[v]k_Lp(Ω)≤c(p)kvk_Lp(R³) for any 1< p <∞ and

kA_i[v]k_L^q_(Ω)≤c(q, r)kvk_Lr(R³)

wherer≤q≤_3−r^3r if 1< r <3, qarbitrary finite ifr= 3,q=∞forr >3.

Now, we use the quantities

ϕ_i(t, x) =ψ(t)φ(x)A_i[T_k(̺n)], ψ∈ D(0, T), φ∈ D(Ω), i= 1,2,3 as test functions for (1.2) (as always,̺n is prolonged by zero outside Ω):

Z T 0

Z

Ω

ψφh

a̺^γ_n−(λ+ 2µ) divu_ni

T_k(̺_n)dx dt= (5.1)

Z T 0

Z

Ωψh

(λ+µ) divu_n−a̺^γ_ni

∂xiφA_i[T_k(̺n)]dx dt+

µ Z _T

0

Z

Ω

ψn

∇φ.∇uⁱ_nA_i[T_k(̺n)]−uⁱ_n∂xjφ ∂xjA_i[T_k(̺n)]o

dx dt+

µ Z T

0

Z

Ω

ψu_n.∇φ T_k(̺_n)dx dt−

Z T 0

Z

Ω

φ̺nuⁱ_nn

∂tψA_i[T_k(̺n)] +ψA_i[(T_k(̺n)−T_k^′(̺n)̺n) divu_n]o

dx dt−

Z _T

0

Z

Ω

ψ ̺_nuⁱ_nu^j_n ∂_xjφA_i[T_k(̺_n)]dx dt+

Z T 0

Z

Ωψuⁱ_nn

T_k(̺n)R_i,j[φ̺nu^j_n]−φ̺nu^j_nR_i,j[T_k(̺n)]o dx dt where the operatorsR_i,j are defined as

R_i,j[v] =F⁻¹nξ_iξ_j

|ξ|²F {v}(ξ)o . Here, we have used the summation convention and (4.5).

Analogously, we can repeat the above arguments considering the equations (4.4), (4.6) and the test functions

ϕ_i(t, x) =ψφA_i[T_k(̺)], i= 1,2,3 to deduce

Z T 0

Z

Ω

ψφh

a̺^γ−(λ+ 2µ) divui

T_k(̺)dx dt= (5.2)

Z T 0

Z

Ωψh

(λ+µ) divu−a̺^γi

∂xiφA_i[T_k(̺)]dx dt+

µ Z _T

0

Z

Ω

ψn

∇φ∇uⁱ A_i[T_k(̺)]−uⁱ ∂xjφ ∂xjA_i[T_k(̺)] +u.∇φ T_k(̺)o

dx dt−

Z T 0

Z

Ω

φ̺ uⁱn

∂_tψA_i[T_k(̺)] +ψA_i[(T_k(̺)−T_k^′(̺)̺) divu]o

dx dt−

Z T 0

Z

Ω

ψ ̺ uⁱu^j ∂xjφA_i[T_k(̺)]dx dt+

Z _T

0

Z

Ω

ψuⁱn

T_k(̺)R_i,j[φ̺ u^j]−φ̺ u^jR_i,j[T_k(̺)]o dx dt.

It can be proved that all the terms on the right-hand side of (5.1) converge to their counterparts in (5.2) which yields the desired conclusion. Of course, the hardest term is the last integral in (5.1), (5.2) respectively, i.e., one has to show:

(5.3)

Z _T

0

Z

Ω

ψuⁱ_nn

T_k(̺_n)R_i,j[φ̺_nu^j_n]−φ̺_nu^j_nR_i,j[T_k(̺_n)]o

dx dt→ Z T

0

Z

Ωψuⁱn

T_k(̺)R_i,j[φ̺u^j]−φ̺u^jR_i,j[T_k(̺)]o dx dt.

In view of (4.1), (4.2), and (4.7), the relation (5.3) is a consequence of the following assertion:

Lemma 5.1. Suppose

v_n→v weakly in L^p(R³), w_n→w weakly in L^q(R³) where1/p+ 1/q= 1/r <1.

Then

vnRi,j[wn]−wnRi,j[vn]→vRi,j[w]−wRi,j[v] weakly in L^r(R³), i, j= 1,2,3.

Proof of Lemma 5.1: It is easy to see that the conclusion of Lemma 5.1 is a particular case of a more general statement:

(5.4)

3

X

i,j=1

v_nⁱR_i,j[w^j_n]−w_n^jR_i,j[v_nⁱ]→

3

X

i,j=1

vⁱR_i,j[w^j]−w^jR_i,j[vⁱ] in D^′(R³).

providedv_n = [v¹_n, v_n², v_n³], w_n = [w¹_n, w_n², w_n³] are sequences of vector functions satisfying

v_n→v weakly in [L^p(R³)]³, w_n→w weakly in [L^q(R³)]³.

Indeed, Lemma 5.1 follows from (5.4) taking v_n =v_ne_i, w_n =w_ne_j where e_i, i= 1,2,3 is the orthogonal basis of R³.

To show (5.4), one can use the symmetryR_i,j =R_j,i to deduce

3

X

i,j=1

vⁱ_nR_i,j[w^j_n]−w^j_nR_i,j[vⁱ_n] =

3

X

i=1

hvⁱ_n−(

3

X

k=1

R_i,k[v^k_n]) (

3

X

j=1

R_i,j[w_n^j])i

−

3

X

j=1

hw_n^j −(

3

X

k=1

R_k,j[w^k_n]) (

3

X

i=1

R_i,j[vⁱ_n])i

= Un.Vn−Xn.Yn

where

divU_n=

3

X

i=1

∂_xi v_nⁱ −(

3

X

k=1

R_i,k[v_n^k])

=

div Xn=

3

X

j=1

∂xj

w_n^j −(

3

X

k=1

R_j,k[w_n^k])

= 0

and

V_n=∇(∆⁻¹

3

X

j=1

∂_xjw^j_n), Y_n=∇(∆⁻¹

3

X

i=1

∂_xivⁱ_n), i.e., curl(V_n) = curl(Y_n) = 0.

Consequently, it is possible to use theL^p−L^qversion of Div-Curl Lemma (see e.g. Yi [11]) to conclude

Un.Vn→U.V, Xn.Yn→X.Y in D^′(R³) where

Uⁱ= vⁱ−(

3

X

k=1

R_i,k[v^k]) , Vⁱ =

3

X

j=1

R_i,j[w^j],

X^j= w^j−(

3

X

k=1

R_j,k[w^k]) , Y^j=

3

X

i=1

R_j,i[vⁱ], i, j= 1,2,3.

We have proved Proposition 5.1.

6. The amplitude of oscillations

The main result of this section is inspired by the paper of Jiang and Zhang [8].

Proposition 6.1. Under the hypotheses of Theorem1.1, let̺be a weak limit of the sequence̺n.

Then

lim sup

n→∞ kT_k(̺n)−T_k(̺)k_L^γ+1_((0,T)×Ω)≤c(E0) where the constantc(E₀)is independent of k.

Proof: One has

n→∞lim Z T

0

Z

Ω

̺^γ_nT_k(̺n)−̺^γ T_k(̺)dx dt= (6.1)

n→∞lim Z _T

0

Z

Ω

(̺^γ_n−̺^γ)(T_k(̺n)−T_k(̺))dx dt+

Z T 0

Z

Ω

(̺^γ−̺^γ)(T_k(̺)−T_k(̺))dx dt≥

n→∞lim Z T

0

Z

Ω

(̺^γ_n−̺^γ)(T_k(̺n)−T_k(̺))dx dt≥ lim sup

n→∞

Z _T

0

Z

Ω

|T_k(̺_n)−T_k(̺)|^γ+1 dx dt asz7→z^γ is convex,T_kconcave on [0,∞), and

(z^γ−y^γ)(T_k(z)−T_k(y))≥ |T_k(z)−T_k(y)|^γ+1 for all z, y≥0.

On the other hand,

n→∞lim Z _T

0

Z

Ω

divu_n T_k(̺n)−divuT_k(̺)dx dt= (6.2)

n→∞lim Z T

0

Z

Ω

T_k(̺_n)−T_k(̺) +T_k(̺)−T_k(̺)

divu_n dx dt≤ 2 sup

n kdivu_nk_L²_((0,T)×Ω)lim sup

n→∞ kT_k(̺_n)−T_k(̺)k_L²_((0,T)×Ω).

The relations (6.1), (6.2) combined with Proposition 5.1 yield the desired con- clusion.

7. The renormalized solutions

Proposition 7.1. Under the hypotheses of Theorem1.1, the limit functions̺, usolve(4.3) in the sense of renormalized solutions, i.e.,

(7.1) ∂tb(̺) + div(b(̺)u) + (b^′(̺)̺−b(̺)) divu= 0

holds inD^′((0, T)×R³)for anyb∈C¹(R),|b^′(z)z|+|b(z)| ≤cprovided̺,uare set zero outsideΩ.

Proof: It is enough to prove (7.1) for anybsatisfying, in addition to the above hypotheses,

b^′(z) = 0 for all z large enough, say,z≥M

whereM is a certain constant. The rest follows by a simple density argument.

Regularizing (4.6) one gets

(7.2) ∂_tS_ε[T_k(̺)] + div(S_ε[T_k(̺)]u) +S_ε[(T_k^′(̺)̺−T_k(̺)) divu] =r_ε whererε→0 inL²(0, T;L²(R³)) for any fixedk.

Multiplying (7.2) byb^′(Sε[T_k(̺)]) and lettingε→0 we deduce (7.3) ∂tb(T_k(̺)) + div(b(T_k(̺))u) +

b^′(T_k(̺))T_k(̺)−b(T_k(̺))

divu= b^′(T_k(̺))[(T_k(̺)−T_k^′(̺)̺) divu]

inD^′((0, T)×R³).

At this stage, the idea is to pass to the limit in (7.3) fork→ ∞. We have T_k(̺)→̺ as k→ ∞ in L^p((0, T)×Ω) for any 1≤p < γ since

kT_k(̺)−̺k_Lp((0,T)×Ω)≤lim inf

n→∞ kT_k(̺_n)−̺_nk_Lp((0,T)×Ω)

and

(7.4) kT_k(̺n)−̺nk^p_Lp((0,T)×Ω)≤2^pk^p−γk̺nk^γ_Lγ((0,T)×Ω). Thus (7.3) will imply (7.1) provided we show

(7.5) b^′(T_k(̺))[(T_k^′(̺)̺−T_k(̺) divu]→0 in L¹((0, T)×Ω) as k→ ∞.

Denoting

Q_k,M={(t, x)∈(0, T)×Ω|T_k(̺)≤M}, we can estimate

Z _T

0

Z

Ω

b^′(T_k(̺))[(T_k^′(̺)̺−T_k(̺)) divu]

dx dt≤ sup

0≤z≤M

|b^′(z)|

Z Z

Qk,M

(T_k^′(̺)̺−T_k(̺)) divu

dx dt≤ sup

0≤z≤M

|b^′(z)| sup

n ku_nk_L²_(0,T;W1,2(Ω)) lim inf

n→∞ kT_k^′(̺_n)̺_n−T_k(̺_n)k_L²_(Qk,M). Now, by interpolation,

kT_k^′(̺n)̺n−T_k(̺n)k²_L2(Qk,M)≤ (7.6)

kT_k^′(̺n)̺n−T_k(̺n)k^α_L1((0,T)×Ω)kT_k^′(̺n)̺n−T_k(̺n)k^{(1−α)(γ+1)}_Lγ+1(Qk,M), α=γ−1 γ

where, similarly as in (7.4),

(7.7) kT_k^′(̺n)̺n−T_k(̺n)k_L¹_((0,T)×Ω)≤2^γk^1−γsup

n k̺nk^γ_Lγ((0,T)×Ω), and

kT_k^′(̺n)̺n−T_k(̺n)k_L^γ+1_(Qk,M)≤ (7.8)

2

kT_k(̺n)−T_k(̺)k_L^γ+1_((0,T)×Ω)+kT_k(̺)k_L^γ+1_(Qk,M)

≤ 2

kT_k(̺n)−T_k(̺)k_L^γ+1_((0,T)×Ω)+kT_k(̺)−T_k(̺)k_L^γ+1_((0,T)×Ω)+ kT_k(̺)k_L^γ+1_(Qk,M)

≤

2kT_k(̺n)−T_k(̺)k_L^γ+1_((0,T)×Ω)+ 2kT_k(̺)−T_k(̺)k_L^γ+1_((0,T)×Ω)+ 2M|Ω|.

By virtue of Proposition 6.1 and (7.8), one gets lim sup

n→∞ kT_k^′(̺n)̺n−T_k(̺n)k_L^γ+1_(Qk,M)≤2c+ 2M|Ω|

which, together with (7.6), (7.7), completes the proof of (7.5).

8. Strong convergence of the density

We introduce a family of functionsL_k: L_k(z) =

(zlog(z) for 0≤z < k, zlog(k) +zRz

kT_k(s)/s² ds for z≥k.

Seeing thatL_k can be written as

(8.1) L_k(z) =β_kz+b_k(z)

where |b_k(z)| ≤ c(k) and b^′_k(z)z−b_k(z) =T_k(z) for all z > 0, we can combine (1.1), (1.5) to deduce

(8.2) ∂_tL_k(̺_n) + div(L_k(̺_n)u_n) +T_k(̺_n) divu_n= 0 and, by virtue of (4.3) and Proposition 7.1,

(8.3) ∂_tL_k(̺) + div(L_k(̺)u) +T_k(̺) divu= 0 inD^′((0, T)×Ω).

Consequently, we can assume

(8.4) L_k(̺n)→L_k(̺) in C([0, T];L^γ_weak(Ω)) and, approximatingzlog(z)≈L_k(z),

̺nlog(̺n)→̺log(̺) in C([0, T];L^α_weak(Ω)) for any 1≤α < γ.

Taking the difference of (8.2) and (8.3) and integrating with respect to t we get

Z

Ω

(L_k(̺_n)−L_k(̺))(t)φ dx= Z

Ω

(L_k(̺_n)(0)−L_k(̺₀))φ dx+

Z _t

0

Z

Ω

(L_k(̺n)un−L_k(̺)u).∇φ+ (T_k(̺) divu−T_k(̺n) divu_n)φ dx dt for any φ ∈ D(Ω). Passing to the limit for n → ∞ and making use of the hypothesis (1.7) together with (8.4), one obtains

Z

Ω

(L_k(̺)−L_k(̺))(t)φ dx= Z t

0

Z

Ω

(L_k(̺)−L_k(̺))u).∇φ dx dt+

n→∞lim Z t

0

Z

Ω

(T_k(̺) divu−T_k(̺_n) divu_n)φ dx dt

Takingφ=φmthe sequence approximating the characteristic function of Ω as in (2.2) and making use of the boundary conditions (1.3), one derives

(8.5)

Z

Ω

(L_k(̺)−L_k(̺))(t)dx= Z _t

0

Z

Ω

T_k(̺) divudx dt− lim

n→∞

Z _t

0

Z

Ω

T_k(̺n) divu_n dx dt.

Observe that the termL_k(̺)−L_k(̺) is bounded in view of (8.1).

Finally, making use of Proposition 5.1 and the monotonicity of the pressure, we cane estimate the right-hand side of (8.5):

(8.6)

Z _t

0

Z

Ω

T_k(̺) divudx dt− lim

n→∞

Z _t

0

Z

Ω

T_k(̺n) divu_ndx dt≤ Z t

0

Z

Ω

(T_k(̺)−T_k(̺)) divudx dt.

By virtue of Proposition 6.1, the right-hand side of (8.6) tends to zero as k→ ∞. Accordingly, one can pass to the limit fork→ ∞in (8.5) to conclude

̺log(̺)(t) =̺log(̺)(t) for all t∈[0, T]

which implies strong convergence of the sequence̺_n inL¹((0, T)×Ω).

Theorem 1.1 has been proved.

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Institute of Mathematics AV ˇCR, ˇZitn´a 25, 115 67 Praha 1, Czech Republic (Received August 7, 2000)