Existence of weak solutions to the three-dimensional steady compressible Navier–Stokes equations

www.elsevier.com/locate/anihpc

Existence of weak solutions to the three-dimensional steady compressible Navier–Stokes equations

Song Jiang, Chunhui Zhou

LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China Received 23 September 2010; received in revised form 16 December 2010; accepted 28 February 2011

Available online 16 March 2011

Dedicated to Professor Rolf Leis on the occasion of his 80th birthday

Abstract

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier–Stokes equations in R³for any specific heat ratioγ >1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.

©2011 Elsevier Masson SAS. All rights reserved.

Keywords:Steady compressible Navier–Stokes equations; Existence forγ >1; Potential estimate; Effective viscous flux

1. Introduction

In this paper we prove the existence of a spatially periodic weak solution(ρ,u)to the following steady isentropic compressible Navier–Stokes equations inR³for any specific heat ratioγ >1:

div(ρu)=0, (1.1)

−μu− ˜μ∇divu+div(ρu⊗u)+ ∇P =ρf. (1.2)

Hereu=(u1, u2, u3)is the velocity andρ is the density, the viscosity constantsμandμ˜ satisfyμ >0,μ˜ =μ+λ withλ+2μ/30, the pressureP for isentropic flows is given byP (ρ)=aρ^γ witha being a positive constant and γ >1 being the specific heat ratio.f=(f1, f2, f3)is the external force, and for simplicity, we assume that

f∈L^∞ R³

.

Besides, we consider that(ρ,u)andfare periodic in eachx_i with period 2π for all 1i3.

For simplicity, throughout this paper, we denote byΩthe periodic cell(−π, π )³.

In general, there could be no solution for arbitrary f, since for a (smooth) solution, which is periodic inx with period 2π,fhas to satisfy the necessary condition:

* Corresponding author.

E-mail addresses:jiang@iapcm.ac.cn (S. Jiang), zhouchunhuixj@gmail.com (C. Zhou).

0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2011.02.008

Ω

ρfidx=0 for 1i3. (1.3)

However, if we considerfwith symmetry f_i(x)= −f_i

Y_i(x)

and f_i(x)=f_i Y_j(x)

, ifi=j, i, j=1,2,3, (1.4)

where

Y_i(. . . , x_i, . . .)=(. . . ,−x_i, . . .), (1.5)

thenuwill have the same symmetry andρwith the symmetry ρ(x)=ρ

Yi(x)

fori=1,2,3, (1.6)

and condition (1.3) is satisfied automatically. Moreover,usatisfies

Ω

ui(x) dx=0 for all 1i3. (1.7)

So, in this paper we will consider the external forcefthat has the symmetry (1.4).

In the last decades, the well-posedness of Eqs. (1.1), (1.2) for largef has been investigated by a number of re- searchers. In 1998, under the assumption thatγ >1 in two dimensions andγ >5/3 in three dimensions, Lions [12]

proved the existence of weak solutions to (1.1), (1.2). Roughly speaking, the condition onγ comes from the integra- bility of the density ρinL^p. The higher integrability ofρ has, the smallerγ can be allowed. Iffis potential, then weak solutions are shown to exist for anyγ >3/2, see [14]. Then, Frehse, Goj and Steinhauer [6], Plotnikov and Sokolowski [15] obtained an improved integrability bound for the density by deriving a new weighted estimate of the pressure, assuming a priori theL¹-boundedness ofρu²which, unfortunately, was not shown to hold. Recently, by combining theL^∞-estimate of⁻¹P with the (usual) energy and density bounds, Bˇrezina and Novotný [2] were able to show the existence of weak solutions to the spatially periodic problem with symmetries (1.4)–(1.6) for any γ > (3+√

41)/8 whenfis potential, or for anyγ >1.53 whenf∈L^∞, without assuming the boundedness ofρu² inL¹. More recently, Frehse, Steinhauer and Weigant [8] established the existence of weak solutions to the Dirichlet problem in three dimensions for anyγ >4/3 in the framework of [2]. Also, the existence of a weak solution to (1.1), (1.2) with periodic or mixed boundary conditions was obtained in the two-dimensional isothermal case (γ=1) [7].

The aim of this paper, inspired by the works [8,2], is to improve the existence result in [2], namely, we shall prove the existence of spatially periodic weak solutions to the system (1.1), (1.2) for anyγ >1, thus extending the existence in [2] fromγ >4/3 toγ >1. Roughly speaking, the basic idea in our proof is to employ a careful bootstrap argument to obtain the higher integrability of the density which eventually relaxes the restriction onγ in [2], see the paragraph below Theorem 1.2 for more details on the proof idea.

We mention that for a 3D model of steady compressible heat-conducting flows (i.e., the steady compressible Navier–Stokes–Fourier system), Mucha and Pokorný [13] recently studied the existence of weak solutions under some assumptions on the pressure and heat-conductivity, which unfortunately excludes the case of polytropic idea gases. For the corresponding non-steady system (to (1.1), (1.2)) with large initial data, Lions [12] first proved the global existence of weak solutions in the case ofγ3n/(n+2)(n=2,3: dimension). His result has been improved and generalized recently in [5,10,11] and among others, where the conditionγ >3/2 is required in three dimensions for general initial data.

Before defining a weak solution to (1.1), (1.2), we introduce the notation used throughout this paper.

Notation.LetG be a domain inR³ or the periodic cell. We denote by L^p(G)the Lebesgue spaces, byW^k,p(G) (k∈N) the usual Sobolev spaces, byC^k(G)(resp.C^k(G)) the space ofktimes continuously differentiable functions inG(resp.G). We define

D R³

=

φ(x)∈C^∞ R³

, φ(x)is periodic inx_iof period 2πfor all 1i3 and

D(G)=

φ(x)∃ ˜φ(x)∈D R³

, s.t.φ(x)= ˜φ(x),forx∈G .

By D(R³)(resp.D(G)), we denote the dual space ofD(R³)(resp.D(G)). For example,D(R³)is the space of periodic distributions inR³(dual toD(R³)). We also introduce the space of symmetric functions:(W_sym^k,p(Ω))³denotes the space of vector functions in(W^k,p(Ω))³which possess the symmetry (1.4), whileL^p_sym(Ω)stands for the space of functions inL^p(Ω)with symmetry (1.6).BR(a):= {x∈R³:|x−a|< R}denotes the open ball centered atawith radiusR.

Now, let us recall the definition of a bounded energy weak solution to (1.1), (1.2).

Definition 1.1(Renormalized bounded energy weak solution).We call(ρ,u)a renormalized bounded energy weak solution to the spatially periodic problem of the system (1.1) and (1.2), if the following is satisfied.

(1) ρ0,ρ∈L^γ(Ω),u∈(H¹(Ω))³,

Ωρ(x) dx=M >0.

(2) (ρ,u)satisfies the energy inequality:

Ω

μ|∇u|²+ ˜μ|divu|² dx

Ω

ρf·udx. (1.8)

(3) The system (1.1), (1.2) holds in the sense ofD(Ω).

(4) The mass equation (1.1) holds in the sense of renormalized solutions, i.e., div b(ρ)u

+ b(ρ)ρ−b(ρ)

divu=0 inD(Ω) (1.9)

for anyb∈C¹(R), such thatb(z)=0 whenzis big enough.

Remark 1.1.In the periodic case, the periodic cellΩ in Definition 1.1 actually can be replaced by any cube inR³ with length 2π.

The main result of the current paper reads as

Theorem 1.2.Letγ >1andf∈L^∞(R³)satisfy(1.4). Then, there exists a renormalized bounded energy weak solution (ρ,u), satisfying(1.6)and(1.4), to the spatially periodic problem of the system(1.1), (1.2).

The proof of Theorem 1.2 is based on the uniform a priori estimates for the approximate solutions and the weak convergence method in the framework of Lions [12]. First, to get higher integrability of the densityρ_δ, we derive the weighted estimates for both pressure P_δ and kinetic energyρ_δ|uδ|² which can be also understood as estimates in a Morrey space. These weighted estimates are inspired by the papers [8,2]. However, the new idea in the current paper is that bothP_δ andρ_δ|uδ|² are bounded simultaneously (cf. Lemma 2.2), while in [8,2] a weighted estimate only for the pressurePδwas shown. This simultaneous boundedness of bothPδandρδ|uδ|²plays a crucial role in the derivation of the higher integrability ofρδfor anyγ >1, and moreover, implies some new uniform estimates for the velocityuδand the pressurePδ(cf. Lemmas 2.4 and 2.6). Then, with the help of the higher integrability ofρδand the new estimates for(uδ, Pδ), we use a bootstrap argument to obtain the a priori uniform estimates for the approximate solutions for anyγ >1 (cf. Theorem 2.1). To pass to the limit and obtain the existence of a weak solution, we cannot directly use the arguments in [12], sinceρδ∈L^p(Ω)(p >5/3) is required in [12] and this is not the case here. In fact, here we only haveρδ∈L^{γ r}(Ω)with somer >1 being very close to 1 whenγ is close to 1. Instead, we exploit the proved uniform boundedness ofρδuδ andρδ|uδ|²inL^r(Ω), and employ a careful analysis based on the classical method of weak convergence to circumvent this difficulty to prove the existence.

This paper is organized as follows. In Section 2, we first construct a sequence of approximate solutions(ρδ,uδ) and then derive the uniform weighted estimates for both pressure Pδ and kinetic energy ρδ|uδ|². In Sections 2.2 and 2.3 we show the additional uniform estimates for the velocityuδ and the pressureP_δ in terms of the quantity A= Pδ|uδ|²+ρ_δ^β|uδ|^2+2βL¹(Ω), 0< β <1. This is crucial in the derivation of the higher integrability ofρδ for anyγ >1. In Section 2.4, by using of a bootstrap argument (see, for example, [2,6]), we prove thatAis uniformly bounded which in turn implies the uniformH¹-boundedness ofuδ, and theL^r-boundedness ofP_δ,ρ_δuδandρδ|uδ|². In Section 3, we prove the main theorem by using the weak convergence method in the framework of Lions [12].

2. Uniform estimates of the approximate solutions 2.1. The approximate system

To prove Theorem 1.2, we first work with the standard approximation by introducing an artificial pressure term:

P_δ(ρ):=aρ^γ +δρ⁶,

where 0< δ1. Here we chooseρ⁶just for technical reason, and in fact we can takeρ^αfor anyα6 instead ofρ⁶. We consider the following approximate problem inΩ:

div(ρδuδ)=0, (2.1)

−μuδ− ˜μ∇divuδ+div(ρδuδ⊗uδ)+ ∇P_δ(ρ_δ)=ρ_δf. (2.2) According to [2], there is at least a weak solution(ρ_δ,uδ)to the problem (2.1), (2.2) with the following properties (γ=max(γ ,6)):

(1) ρ_δ∈L^2γ_sym(Ω), uδ∈

W_sym^1,2(Ω)3

,

Ω

ρ_δdx=M; (2.3)

(2) div b(ρδ)uδ

+ b(ρδ)ρδ−b(ρδ)

divuδ=0 inD(Ω); (2.4)

(3)

Ω

μ|∇uδ|²+ ˜μ|divuδ|² dx

Ω

ρ_δf·uδdx, (2.5)

wherebis the same as in (1.9).

In the rest of this section we will show some uniform-in-δestimates for(ρ_δ, u_δ)which will be used in passing to the limit asδ→0 in the next section to get a weak solution of the system (1.1), (1.2).

If we define

A=P_δ|uδ|²+ρ_δ^β|uδ|^2+2β

L¹(Ω), 0< β <1, (2.6)

then, we have

Theorem 2.1.ForAdefined by(2.6), it holds for any1< r <2−1/γ that A+ uδH¹(Ω)+ PδL^r(Ω)+ρδ|uδ|²

L^r(Ω)+ ρδuδL^r(Ω)C, (2.7)

δ

Ω

ρ_δ⁶dx→0 asδ→0, (2.8)

where the constantCdepends only onfL^∞(Ω),μ,μ,˜ M,γ andβ (but not onδ).

The proof of Theorem 2.1 is broken up into several lemmas given in Sections 2.2–2.4. We start with the following potential estimate.

2.2. A potential estimate

In this section we first derive the weighted estimate forPδandρδ|uδ|²which can also be understood as an estimate in a Morrey space, and then use this estimate to get an estimate forAby the classical theory of elliptic equations of second order.

Lemma 2.2.Let(ρδ,uδ)be the solutions of the approximate problem(2.1), (2.2). Then the following estimate holds.

B₁(x₀)

P_δ+(ρ_δ|uδ|²)^β

|x−x₀| dxC

1+ P_δL¹(Ω)+ρ_δ|uδ|²

L¹(Ω)+ uδH¹(Ω)

(2.9)

for allβ∈(0,1)andx₀∈Ω, where the constant C depends only onfL^∞(Ω),μ,μ, M, γ˜ andβ, but not onx₀andδ.

Proof. Forx₀∈Ω, we defineφ=(φ¹, φ², φ³)with φⁱ(x)=(x−x₀)ⁱ

|x−x₀|^βη

|x−x₀|

inb(x₀, π ), i=1,2,3,

where 0< β1,b(x₀, π )= {x=(x¹, x², x³)∈R³: |xⁱ−x₀ⁱ|< π, i=1,2,3}is a periodic cell, andη∈C₀^∞(R)is a cut-off function satisfying 0η(t )1,|Dη|2 and

η(t )=

1 |t|1, 0 |t|2.

If we extendφtoR³periodically inxi with period 2π for all 1i3, thenφ∈H_loc¹ (R³)can be a test function.

Testing (2.2) with thisφ, we find that

B₂(x₀)

Pδdivφ dx+

B₂(x₀)

ρδuⁱ_δu^j_δ∂jφⁱdx

=μ

B2(x0)

∇uδ: ∇φ dx+ ˜μ

B2(x0)

divuδdivφ dx+

B2(x0)

ρ_δfφ dx. (2.10)

Since

divφ(x)= 3−β

|x−x₀|^βη+(x−x₀)· ∇η

|x−x₀|^β (2.11)

and

B2(x0)

ρδuⁱ_δu^j_δ∂jφⁱdx

=

B₂(x₀)

ρ_δ|uδ|²

|x−x₀|^βη dx−β

B₂(x₀)

ρ_δ[uδ·(x−x₀)]²

|x−x₀|^2+β η dx+

B₂(x₀)

ρ_δuⁱ_δu^j_δ(x−x₀)ⁱ

|x−x₀|^β∂_jη (1−β)

B2(x0)

ρ_δ|uδ|²

|x−x0|^βη dx+

B2(x0)

ρ_δuⁱ_δu^j_δ(x−x₀)ⁱ

|x−x0|^β∂_jη dx, (2.12)

we substitute (2.11) and (2.12) into (2.10) to obtain, after a straightforward calculation, that

B₁(x₀)

P_δ

|x−x₀|^β dx+(1−β)

B₁(x₀)

ρ_δ|uδ|²

|x−x₀|^β dxC

1+ P_δL¹(Ω)+ρ_δ|uδ|²

L¹(Ω)+ uδH¹(Ω)

, from which it follows that

B1(x0)

P_δ

|x−x0|dxC

1+ P_δL¹(Ω)+ρ_δ|uδ|²

L¹(Ω)+ uδH¹(Ω)

forβ=1,

and

B₁(x₀)

ρ_δ|uδ|²

|x−x₀|^β dx C 1−β

1+ P_δL¹(Ω)+ρ_δ|uδ|²

L¹(Ω)+ uδH¹(Ω)

(2.13)

for any 0< β <1.

Using Hölder’s inequality, we easily see that for any 0< β <1,

B1(x0)

(ρ_δ|uδ|²)^β

|x−x0| dx C 1−β

1+ P_δL¹(Ω)+ρ_δ|uδ|²

L¹(Ω)+ uδH¹(Ω)

. (2.14)

Thus, (2.9) follows from (2.13) and (2.14) immediately. 2

The following lemma is similar to the potential estimate in Section 3 of [8], and for the convenience of the reader, we give its proof in Appendix A.

Lemma 2.3.LetAbe defined by(2.6), then we have ACuδ²_H1(Ω)

1+ PδL¹(Ω)+ρδ|uδ|²

L¹(Ω)+ uδH¹(Ω)

, (2.15)

where the constant C depends onfL^∞(Ω),μ,μ,˜ M,γ andβ, but not onδ.

2.3. Uniform estimates

The next lemma shows that theH¹-norm ofuδcan be bounded by some power ofA.

Lemma 2.4.Let(ρ_δ,uδ)be the solution of the approximate problem(2.1), (2.2). Then, uδ_H¹_(Ω)CA

γ−β

4(γ β+γ−2β), (2.16)

where the constantCdepends only onfL^∞(Ω),μ,μ,˜ M,γ andΩ;βis the same as in Lemma2.2.

Proof. First, we use the energy inequality (2.5) to obtain μ

Ω

|∇uδ|²dx+ ˜μ

Ω

|divuδ|²dx

Ω

ρ_δf·uδdxfL^∞(Ω)ρ_δuδL¹(Ω). (2.17) By Hölder’s and Sobolev’s inequalities, we see that

ρ_δuδL¹(Ω)=

Ω

P_δu²_δ2(γ β+γ−2β)^1−β

ρ_δ^βu^2β_δ ⁺²2(γ β+γ−2β)^γ−1 ρ

2γ β+γ−3β 2(γ β+γ−2β)

δ

CA^{2(γ βγ−β+γ−2β)}, (2.18)

where we have used the fact that

Ωρδ=M. Substituting (2.18) into (2.17), utilizing (1.7) and Poincaré’s inequality, we obtain (2.16). This completes the proof. 2

The following lemma is due to Bogovskii [1] and will be needed in Lemma 2.6.

Lemma 2.5.Suppose that the bounded domain Ω⊂Rⁿ is starlike with respect to some ball contained in it, and that 1< p <∞andn2. Then, there exists a constantC >0, depending only onn, p andΩ, such that for any f ∈L^p(Ω)with

Ωf (x) dx=0, there is a vector fieldω∈W_p¹(Ω)satisfying:

divω=f inΩ,

ω=0 on∂Ω, (2.19)

and

ωW_p¹(Ω)CfL^p(Ω). (2.20)

Now, if we take f =P_δ^s−1− 1

|Ω|

Ω

P_δ^s−1dx with 1< s <2

in (2.19), then by virtue of Lemma 2.5, there is at least one solutionωδof (2.19) satisfying (2.20) and the estimate:

w_δ

W^{1, ss}−1(Ω)C

p^s_δ⁻¹− 1

|Ω|

Ω

P_δ^s−1dx L

s s−1(Ω)

C(s, Ω)P_δ^ss⁻¹. (2.21)

With the help of Lemma 2.5, we can show now

Lemma 2.6.Let(ρ_δ,uδ)be the solution of the approximate problem(2.1), (2.2). Then fors∈(1, β+1−β/γ], we have

P_δ^s_L^s_(Ω)C 1+A

γ s−β γ β+γ−2β

, (2.22)

where the constantCdepends only onfL^∞(Ω),μ,λ,M,γandΩ.

Proof. We use the functionω_δ in (2.21) to test the momentum equation (2.2), and by a direct computation similar to Lemma 2.3 in [6] we obtain

P_δ^s_L^s_(Ω)C

1+ uδ^s_W1,2(Ω)+ρ_δ|uδ|^2s

L^s(Ω)

, (2.23)

where the last term can be bounded as follows, using Hölder’s and Sobolev’s inequalities.

ρ_δ|uδ|^2s

L^s(Ω)=

Ω

ρ_δ^s|uδ|^2s=

Ω

P_δ|uδ|^{2 2s−β−1}

γ β+γ−2β

ρ^β|uδ|^{2β+2γ s+1−2s}

γ β+γ−2βρ

γ β+γ−β−γ s γ β+γ−2β

δ

CPδ|uδ|^{2γ β2s+−γβ−−2β1}

L¹(Ω) ρ_δ^β|uδ|^{2β+2γ βγ s++γ1−−2s2β}

L¹(Ω)

CA

γ s−β

γ β+γ−2β. (2.24)

The estimate (2.24), together with (2.23) and (2.16), gives then P_δ^s_L^s_(Ω)C

1+A

s(γ−β)

4(γ β+γ−2β) +A^{γ βγ s−β+γ−2β}

C

1+A^{γ βγ s−β+γ−2β} , which proves the lemma. 2

2.4. Proof of Theorem 2.1

In this section we will first show that the quantityAis uniformly bounded (with respect toδ), then with the help of this uniform boundedness, we can easily get (2.7) and (2.8).

Recalling that Lemma 2.6 is true for anys∈(1, β+1−β/γ], we writes=1+ε, whereεwill be chosen small enough later on, and use (A.1), (2.16), (2.22) and (2.24) to infer that

ACuδ²_H1(Ω)

1+ uδH¹(Ω)+ρδ|uδ|²

L¹(Ω)+ PδL¹(Ω)

CA^{2(γ βγ−β+γ−2β)}

1+A^{4(γ βγ−β+γ−2β)} +A^{(γ βγ s−β+γ−2β)·}

1 1+ε

C

1+A

3(γ−β)

2(γ β+γ−2β)+O(ε)

. (2.25)

Since (2.25) remains valid for anyβ∈(0,1), if we writeβ=1−σ with 0< σ <1, then γ > 1−σ

1−2σ ⇒ 3(γ −β)

2(γ β+γ−2β)<1,

whereεandσ can be arbitrary small. So, by our choice of the parametersεandσ, the exponent _2(γβ^3(γ₊⁻_γ^β)_−2β) +O(ε) can be made less than 1, and therefore we conclude by (2.25) that

AC,

which immediately implies that uδH¹(Ω)C,

pδL^r(Ω)+ρδ|uδ|²

L^r(Ω)C, ρδuδ^r_L^r_(Ω)=

Ω

ρ^r_δu^2r_δ ¹

2 ρ^r_δ¹

2 Cρ^r_δu^2r_δ ¹²

L¹(Ω)ρ_δ^r12

L¹(Ω)C, δ

Ω

ρ_δ⁶dxCδ

γ (s−1) 6+γ (s−1)

Ω

δρ_δ^{6+γ (s−1)}dx

_{6+γ (s}⁶₋₁₎

Cδ

γ (s−1) 6+γ (s−1).

From the above four inequalities, we obtain (2.7) and (2.8). This completes the proof of Theorem 2.1.

3. Proof of Theorem 1.2

In this section we will take to the limit as δ→0 for the approximate problem (2.1) and (2.2) to obtain a weak solution of (1.1), (1.2) for anyγ >1. As mentioned in the introduction, we cannot directly use the arguments in [12]

to take to the limit, because we just haveρ_δ∈L^{γ r}(Ω)withrbeing very close to 1 whenγ is close to 1, while in [12]

ρ_δ∈L^p(Ω)(p >5/3) is required. To overcome this difficulty, we exploit the estimates established in Theorem 2.1 and Lemma 3.1 to show the weak compactness of the effective viscous flux (see Sections 3.2, 3.3). Then, by the standard procedure of the weak convergence method in Section 3.4, we can finish the proof of Theorem 1.2 (cf. [3, 12]).

3.1. Preliminaries

In this section we give the necessary preliminary lemmas which will be used in the proof of Theorem 1.2.

Lemma 3.1.Let1< p₁, p₂, p <∞,pp₁andΩbe a bounded domain inR³. Suppose that

f_n f weakly inL^p1(Ω), (3.1)

g_n→g strongly inL^p2(Ω), (3.2)

and

fngnare uniformly bounded inL^p(Ω). (3.3)

Then there is a subsequence off_ng_n(still denoted byf_ng_n), such that f_ng_n f g weakly inL^p(Ω).

Remark 3.1.We point out here that _p¹

1 +_p¹

2 1 is not needed in Lemma 3.1.

Proof of Lemma 3.1. Sincef_ng_nis uniformly bounded inL^p(Ω), there exists a subsequence off_ng_n(still denoted byf_ng_n) andF ∈L^p(Ω), such that

f_ng_n F weakly inL^p(Ω). (3.4)

On the other hand, (3.2) implies that for the subsequencegnin (3.4), there is a subsequence (still denoted bygn), such that

g_n→g a.e. inΩ.

By Egoroff’s theorem, for anym∈N, there exits a subsetE_m⊂Ωwith|E_m|<1/(2m), such that g_n→g uniformly inΩ\E_m.

Sinceg∈L^p2(Ω), we see thatgis finite a.e. inΩ, that is, for allk∈N,

klim→∞x∈Ω,g(x)> k=0.

Hence, for any givenm, there exists a sufficiently largeK_m∈N, such that E_m^Km:=

x∈Ω, g(x)> Km

satisfy E_m^Km< 1 2m.

Now, we denoteΩ_m:=Ω\(E_m∪E_m^Km), then|Ω\Ω_m|<1/m,|g|K_mandg_n→g inΩ_m. Thus, for any φ∈L^p(Ω), 1/p+1/p=1, one has

nlim→∞

Ωm

(f_ng_n−f g)φ dx= lim

n→∞

Ωm

(f_n−f )gφ dx+ lim

n→∞

Ωm

f_n(g_n−g)φ dx=0. (3.5)

For anyφ∈L^p(Ω), if we define φ(x)˜ =

φ(x), x∈Ω_m, 0, x∈Ω\Ω_m,

we still haveφ˜∈L^p(Ω). Thus, in view of (3.4), we have

nlim→∞

Ω

(f_ng_n−F )φ dx˜ =0= lim

n→∞

Ωm

(f_ng_n−F )φ dx. (3.6)

From (3.5) and (3.6) we easily get that f g=F, a.e. inΩ_m.

Letm→ ∞, we arrive at f g=F, a.e. inΩ, which proves the lemma. 2

Lemma 3.2(Div–Curl lemma). Let1< p₁, p₂, q₁, q₂<∞(1/p1+1/p2=1/p <1), andΩ be a domain inR³. Assume that

fnf weakly in

L^p1(Ω)3

, gng weakly in

L^p2(Ω)3

,

and

divfn→divf strongly inW^−1,q1(Ω), curlgn→curlg strongly in

W^−1,q2(Ω)3

.

Then

fn·gnf·g weakly inL^p(Ω).

The Div–Curl lemma is a classical statement of the compensated compactness due to Murat (1978) and Tartar (1975). Lemma 3.2 is one of its general formulation. For the proof of Lemma 3.2 we refer to, for example, the reference [14]. Actually, in the current paper we will apply Lemma 3.2 with

divfn=0, curlgn=0.

At the end of this section we introduce the operatorA=(A1,A2,A3)inR³by Aj=∂_j⁻¹=F⁻¹

−iξ_j

|ξ|²

, j=1,2,3, divA=id,

where⁻¹is the inverse of the Laplace operator, and the Riesz operator defined by Rij[v] =∂_jAi[v], Rij=Rj i, i, j=1,2,3.

Lemma 3.3.LetΩbe a domain inR³and the operatorAis defined as above, then we have

Aiv_W^1,s_(Ω)C(s, Ω)v_L^s_(R³₎, 1< s <∞, i=1,2,3. (3.7)

For the proof of Lemma 3.3 one can see, for example, the reference [16].

3.2. Vanishing limit asδ→0

In this section we will study the limit for the problem (2.1), (2.2) asδ→0.

By Theorem 2.1, and the compact embeddingW^1,2(Ω) →→L^p(Ω),p∈ [1,6), we have the following limits:

δρ_δ⁶→0 inD(Ω), uδu weakly in

W^1,2(Ω)3

, uδ→u strongly in

L^p(Ω)3

, 1p <6, ρδ ρ weakly inL^{γ r}(Ω),

ρ^γ_δ ρ^γ weakly inL^r(Ω). (3.8)

It is easy to see that Theorem 2.1, together with Lemma 3.1 and (3.8), yields that

ρ_δuδ ρu and ρ_δuδ⊗uδ ρu⊗u inL^r(Ω). (3.9)

Lettingδ→0 in (2.1) and (2.2), applying (3.8) and (3.9), we find that the weak limit(ρ,u)of(ρ_δ,uδ)satisfies

div(ρu)=0 inD(Ω), (3.10)

div(ρu⊗u)+a∇ρ^γ−μu− ˜μ∇divu=ρf inD(Ω); (3.11)

div b(ρ_δ)uδ

+ b(ρ_δ)ρ_δ−b(ρ_δ)

divuδ=0 inD(Ω), (3.12)

Ω

μ|∇u|²+ ˜μ|divu|² dx

Ω

ρf·udx. (3.13)

So, to show that(ρ,u)is a weak solution of (1.1), (1.2), we have to prove

ρ^γ =ρ^γ a.e. onΩ. (3.14)

We will employ the weak convergence method in the framework of Lions (cf. [12,5]) to get (3.14).

Ifγ r2, then by use of the DiPerna–Lions transport theory,(ρ,u)satisfies the renormalized continuity equation (1.9) inD(Ω).

If 1< γ r <2, we will use the following cut-off function, due to Feireisl, Novotný and Petzeltová (cf. [5]), to prove that(ρ,u)is a renormalized solution by replacingρbyT_k(ρ)and letting thenk→0.

T_k(z)=kT z

k

, k=1,2, . . . , where

T (z)=

z, z1

2, z3 ∈C^∞(R), concave, z∈R. (3.15)

Since(ρ_δ,uδ)is the renormalized solution (recallingρ_δ∈L²(Ω)), we can takeb(z)=T_k(z)in the definition of renormalized solutions to get

div

T_k(ρ_δ)uδ

+ T_k(ρ_δ)ρ_δ−T_k(ρ_δ)

divuδ=0 inD(Ω).

Lettingδ→0 in the above identity and making use of (3.8), we deduce that div

T_k(ρ)u

+ T_k(ρ)ρ−T_k(ρ)

divu=0 inD(Ω), (3.16)

where and in what follows, we denote byf (ρ)the weak limit off (ρ_δ).

3.3. Effective viscous flux

The importance of the effective viscous flux for the existence of weak solutions has been addressed by a number of authors (see, for example, [12,5,3]). In this section we introduce the effective viscous flux and prove its weak compactness which will play a key role in the existence proof.

We define the effective viscous flux by

H˜_δ:=aρ_δ^γ−(μ+ ˜μ)divuδH˜ :=aρ^γ−(μ+ ˜μ)divu, asδ→0, and we have the following lemma.

Lemma 3.4.For anyφ∈C₀^∞(Ω), we have

δlim→0

Ω

φ(x)H˜_δT_k(ρ_δ) dx=

Ω

φ(x)H T˜ _k(ρ) dx. (3.17)

Remark 3.2.By use of the density argument, one can actually takeφ(x)≡1.

Proof of Lemma 3.4. First, testing (2.2) by Φδ ≡(Φ1δ, Φ2δ, Φ3δ)=φ(x)A[ξ(x)Tk(ρδ)] and (1.2) by Φ ≡ (Φ₁, Φ₂, Φ₃)=φ(x)A[ξ(x)T_k(ρ)]withφ, ξ∈C₀^∞(Ω), we obtain

Ω

φ(x)ξ(x)H˜_δT_k(ρ_δ) dx

= −

Ω

ρδuⁱ_δu^j_δ∂iφAj ξ Tk(ρδ)

−

Ω

ρδuⁱ_δu^j_δφRij ξ Tk(ρδ)

−

Ω

aρ_δ^γ+δρ_δ⁶

∇φ·A ξ(x)T_k(ρ_δ) dx+ ˜μ

Ω

divuδ∇φ·Aξ(x)T_k(ρ_δ) dx

+μ

Ω

uδ· ∇φξ(x)Tk(ρδ)+ ∇φ· ∇uⁱ_δAi ξ(x)Tk(ρδ)

+uⁱ_δ∂jφRij ξ(x)Tk(ρδ) dx

−

Ω

ρ_δfφ(x)Aj ξ T_k(ρ_δ)

(3.18)

and

Ω

φ(x)ξ(x)H T˜ _k(ρ) dx

= −

Ω

ρuⁱu^j∂_iφAj ξ T_k(ρ)

−

Ω

ρuⁱu^jφRij ξ T_k(ρ)

−

Ω

aρ^γ∇φ·A ξ(x)T_k(ρ) dx+ ˜μ

Ω

divu∇φ·Aξ(x)T_k(ρ) dx

+μ

Ω

u· ∇φξ(x)T_k(ρ)+ ∇φ· ∇uⁱAi ξ(x)T_k(ρ)

+uⁱ∂_jφRij ξ(x)T_k(ρ) dx

−

Ω

ρfφ(x)Aj ξ T_k(ρ)

. (3.19)

Next, we will pass to the limit in (3.18) asδ→0. For anyk, it is easy to see that T_k(ρ_δ)

L^∞(Ω)2k, uniformly inδ,

from which, (3.7) and Sobolev’s embedding theorem it follows that Rij ξ T_k(ρ_δ)

Rij ξ T_k(ρ)

inL^p(Ω),for any 1< p <∞, A ξ Tk(ρδ)

→A ξ Tk(ρ)

inC⁰(Ω). (3.20)

By (3.8), (3.9) and (3.20), we deduce that

δlim→0

Ω

ρ_δuⁱ_δu^j_δ∂_iφAj ξ T_k(ρ_δ) dx=

Ω

ρuⁱu^j∂_iφAj ξ T_k(ρ) dx,