Necas Center for Mathematical Modeling

Analysis of a phase-field model fortwo-phase compressible fluids

E. Feireisl, H. Petzeltova, E. Rocca and G. Schimperna

Preprint no. 2010-003

Research Team 2Mathematical Institute of the Academy of Sciences of the Czech Republic

Zitna 25, 115 67 Praha 1http://ncmm.karlin.mff.cuni.cz/

Analysis of a phase-field model for two-phase

compressible fluids

Eduard Feireisl∗ Hana Petzeltova † Elisabetta Rocca‡

Giulio Schimperna §

Institute of Mathematics of the Academy of Sciences of the Czech RepublicZitna 25, 115 67 Praha 1, Czech Republic

Mathematical Department, University of MilanVia Saldini 50, 20133 Milano, Italy

Mathematical Department, University of PaviaVia Ferrata 1, 27100 Pavia, Italy

1 Introduction

A fluid-mechanical theory for two-phase mixtures of fluids faces a well known math-ematical difficulty: the movement of the interfaces is naturally amenable to a La-grangian description, while the bulk fluid flow is usually considered in the Eulerianframework. The phase field methods overcome this problem by postulating the exis-tence of a “diffuse” interface spread over a possibly narrow region covering the “real”sharp interface boundary. A phase variable χ is introduced to demarcate the twospecies and to indicate the location of the interface. A mixing energy is defined interms of χ and its spatial gradient the time evolution of which is described by meansof a convection-diffusion equation.

As the underlying physical problem still conceptually consists of sharp interfaces,the dynamics of the phase variable remains to a considerable extent purely fictitious.Typically, different variants of Cahn-Hilliard, Allen-Cahn or other types of dynamicsare used (see Anderson et al. [4], Feng et al. [15]). In this paper, we consider a variantof a model for a two phase flow undergoing phase changes proposed by Blesgen [5].The resulting problem consists of the Navier-Stokes system

∂t%+ divx(%u) = 0, (1.1)

∗The work of E.F. was supported by Grant 201/08/0315 of GA CR in the framework of researchprogrammes supported by AVCR Institutional Research Plan AV0Z10190503

†The work of H.P. was supported by Grant 201/08/0315 of GA CR in the framework of researchprogrammes supported by AVCR Institutional Research Plan AV0Z10190503

‡The work of E.R. was partially supported by the Necas Center for Mathematical Modelingsponsored by MSMT

§The work of G.S. was partially supported by the Necas Center for Mathematical Modelingsponsored by MSMT

1

∂t(%u) + divx(%u⊗ u) = divxT, (1.2)

governing the evolution of the fluid density % = %(t, x) and the velocity field u =u(t, x), coupled with a modified Allen-Cahn equation

∂t(%χ) + divx(%χu) = −µ(%, χ,∆χ), (1.3)

%µ = −∆χ+ %∂f(%, χ)∂χ

, (1.4)

describing the changes of the phase variable χ = χ(t, x).The rheology of the fluid is described by means of the Cauchy stress tensor T =

T(%, χ,∇xχ,∇xu),

T = S−(∇xχ⊗∇xχ−

|∇xχ|2

2I)− p(%, χ)I, (1.5)

where S is the conventional Newtonian viscous stress,

S(χ,∇xu) = ν(χ)(∇xu +∇t

xu−23divxuI

)+ η(χ)divxuI, (1.6)

and p denotes the thermodynamic pressure related to the potential energy f throughformula

p(%, χ) = %2 ∂f(%, χ)∂%

. (1.7)

Furthermore, following Blesgen [5] we consider the potential energy density in theform

f(%, χ) = W (χ) + χG1(%) + (1− χ)G2(%), (1.8)

withW (χ) = L(χ)− b(χ), (1.9)

Gi(%) = Γ(%) + gi(%), i = 1, 2, (1.10)

whereL : (0, 1) → R is a convex function. (1.11)

The function L may be singular at the endpoints χ = 0, 1 (see hypothesis (2.2) below),in particular, the case of the so-called logarithmic potential (cf., e.g., [7, p. 170])

L(χ) = χ logχ+ (1− χ) log(1− χ) (1.12)

is included.System (1.1 - 1.3) may be supplemented by the boundary conditions

u|∂Ω = 0, ∇xχ · n|∂Ω = 0. (1.13)

For technical reasons, and in contrast with [5], we have assumed that the middle-term in (1.5) is independent of the density in the spirit of a similar model proposedby Anderson et al. [4]. Models based on the compressible Navier-Stokes system werealso developed in the seminal work of Lowengrub and Truskinovsky [20].

The models based on the incompressible Navier-Stokes system have been exten-sively studied (see Abels [1], [2], Desjardins [9], Gurtin et al. [18], Nouri and Poupaud

2

[21], Plotnikov [22], and the references cited therein). Considerably less rigorous re-sults are available for the compressible models. In [3], the authors studied the modelproposed by Anderson et al. [4], based on the compressible Navier-Stokes system,where the phase variable satisfies a Cahn-Hilliard type equation. In comparison with[3], the analysis of the present system, based on the Allen-Cahn dynamics of thephase-field variable, is mathematically much more delicate. The main difficulty is thelower regularity of the phase variable due to much weaker a priori estimates, and,last but not least, the presence of the singular potential L in (1.9).

Our main goal is to develop a rigorous existence theory for problem (1.1 - 1.13)based on the concept of weak solution for the compressible Navier-Stokes systemintroduced by Lions [19]. In particular, the theory can handle any initial data offinite energy and the solutions exist globally in time. Unfortunately, we are not ableto exclude the possibility that solutions might develop a vacuum state in a finitetime, which is one of the major technical difficulties to be overcome. In particular,the existence of suitable weak solutions, for which the phase variable ranges betweenthe physically relevant values 0 and 1, is strongly conditioned by a proper choice ofthe approximation scheme. Moreover, in order to gain higher integrability or evenboundedness of the density, we consider an equation of state containing a singularcomponent in the spirit of Carnahan and Starling [8].

The paper is organized as follows. The basic hypotheses concerning the structuralproperties of the constitutive functions, together with the main existence theorem, arepresented in Section 2. In Section 3, we introduce the basic approximation scheme inorder to construct solutions to our problem. The proof of existence of global-in-timesolutions is rather technical and carried over by means of several steps described inSections 4, 5, 6, respectively.

2 Hypotheses and main result

2.1 Hypotheses

It follows immediately from (1.8 - 1.11) that

f(%, χ) = W (χ) + Γ(%) + χg1(%) + (1− χ)g2(%)

= L(χ)− b(χ) + Γ(%) + χg1(%) + (1− χ)g2(%). (2.1)

In accordance with (1.11), we assume that

L : (0, 1) → (0,∞) is convex, ess limχ→0+

L′(χ) = −∞, ess limχ→1−

L′(χ) = ∞, (2.2)

b ∈ C2c (0, 1), (2.3)

meaning W is a perturbation of a singular potential L.Since the quantity %2∂%f(%, χ) represents the pressure, it is natural to take

gi(%) = ai log(%), ai > 0, i = 1, 2. (2.4)

Thus, by (1.7), we have that

p(%, χ) = %2Γ′(%) + %(a1χ+ a2(1− χ)

), (2.5)

where the latter summand on the right hand side represents the thermodynamicpressure of a mixture of two species. The component Γ, identical for both species,

3

penalizes the density changes for large values of the pressure in the spirit of thehard-sphere model.

To state our hypotheses on Γ, we introduce further notation setting

%2Γ′(%) = P (%) or, equivalently, Γ(%) =∫ %

0

P (z)z2

dz, (2.6)

where we require the latter integral to be finite. Moreover, we suppose that

P ∈ C1[0, r), P (0) = P ′(0) = 0, P ′ ≥ 0, lim inf%→r−

P (%)(r − %)3 = Pr > 0. (2.7)

Thus, P turns out to represent a singular pressure in the spirit of Carnahan andStarling [8] and r stands for the upper threshold of the density.

Finally, we suppose that the viscosity coefficients are bounded functions of thephase parameter, more precisely

ν, η ∈ C1[0, 1], ν(χ) ≥ ν > 0, η(χ) ≥ 0 for all χ ∈ [0, 1]. (2.8)

2.2 Weak solutions

We shall say that a trio %,u, χ is a weak solution of problem (1.1 - 1.13) supple-mented with the initial data

%(0, ·) = %0, (%u)(0, ·) = (%u)0, (%χ)(0, ·) = (%χ)0 (2.9)

if:

• the density % is a bounded measurable function, 0 ≤ %(t, x) ≤ r for a.a. (t, x) ∈(0, T )× Ω, u ∈ L2(0, T ;W 1,2

0 (Ω; R3)), and the integral identity∫ T

0

∫Ω

(%∂tϕ+ %u · ∇xϕ) dx dt = −∫

Ω

%0ϕ(0, ·) dx (2.10)

holds for any test function ϕ ∈ C∞c ([0, T )× Ω);

• the phase function χ satisfies

χ ∈ L∞(0, T ;H1(Ω)) ∩ L2(0, T ;H2(Ω)) (2.11)

together with 0 ≤ χ(t, x) ≤ 1 for a.a. (t, x) ∈ (0, T ) × Ω. Moreover, p(%, χ) ∈L1((0, T )× Ω), and the integral identity∫ T

0

∫Ω

(%u · ∂tϕ+ %(u⊗ u) : ∇xϕ) dx dt

=∫ T

0

∫Ω

T : ∇xϕ dx dt−∫

Ω

(%u)0 · ϕ(0, ·) dx, (2.12)

holds for any ϕ ∈ C∞c ([0, T )×Ω; R3), where the Cauchy stress T satisfies (1.5),(1.6) (note that the regularity conditions on u, χ and p guarantee, in particular,that T belongs to L1((0, T )× Ω));

4

• µ ∈ L2((0, T )× Ω), and the integral identity∫ T

0

∫Ω

(%χ∂tϕ+ %χu · ∇xϕ) dx dt =∫ T

0

∫Ω

µϕ dx dt−∫

Ω

(%χ)0ϕ(0, ·) dx

(2.13)holds for any ϕ ∈ C∞c ([0, T )× Ω), where µ satisfies (1.4), with

%1/2W ′(χ) ∈ L2((0, T )× Ω). (2.14)

Finally, we require that the second condition in (1.13) holds.

2.3 Main result

Having collected all the preliminary material, we are in a position to formulate themain result of the present paper.

Theorem 2.1 Let Ω ⊂ R3 be a bounded domain of class C2+ν , ν > 0. Suppose thatthe function f is given by (2.1), where the functions L, b, Γ, g1, g2 satisfy hypotheses(2.2 - 2.7), and that the viscosity coefficients ν, η obey (2.8). Furthermore, let theinitial data satisfy

0 < ess infx∈Ω %0(x) ≤ ess supx∈Ω %0(x) < r,

(%χ)0 = %0χ0, with 0 < ess infx∈Ω χ0(x) ≤ ess supx∈Ω χ0(x) < 1,

∇xχ0 ∈ L2(Ω; R3),

(%u)0 = %0u0, with u0 ∈ L2(Ω; R3).

(2.15)

Then problem (1.1 - 1.13) possesses a weak solution %,u, χ in (0, T ) × Ω in thesense specified in Section 2.2.

The rest of the paper is devoted to the proof of Theorem 2.1.

3 Approximation scheme

The solution %,u, χ will be constructed by means of a multi-level approximationscheme similar to that used in [11, Chapter 7]. To begin, we regularize the initialdata replacing %0 by %0,δ, u0 by u0,δ, and χ0 by χ0,δ where δ ∈ (0, 1/4) tends to 0,and the quantities %0,δ, u0,δ, and χ0,δ are smooth in Ω and satisfy a stronger versionof hypothesis (2.15), namely

0 < δ < ess infx∈Ω %0,δ ≤ ess supx∈Ω %0,δ(x) < r − δ,

0 < δ < ess infx∈Ω χ0,δ ≤ ess supx∈Ω χ0,δ(x) < 1− δ,

‖∇xχ0,δ‖L2(Ω;R3) ≤ c,

‖u0δ‖L2(Ω;R3) ≤ c uniformly for δ → 0.

(3.1)

5

Similarly, we introduce

fδ(%, χ) = Lδ(χ)− b(χ) + Γδ(%) + χg1,δ(%) + (1− χ)g2,δ(%), (3.2)

withLδ ∈ C∞ ∩W 1,∞(R) a convex function, (3.3)

g1,δ, g2,δ ∈ C∞ ∩ L∞[0,∞), g′1,δ, g′2,δ ≥ 0. (3.4)

Thanks to (2.2), we can assume that, for all δ ∈ (0, 1/4),

−L′δ(χ) + b′(χ) + g2,δ(%)− g1,δ(%) < 0 for χ > 1− δ, % ≥ 0, (3.5)

−L′δ(χ) + b′(χ) + g2,δ(%)− g1,δ(%) > 0 for χ < δ, % ≥ 0. (3.6)

Finally, we take

Γδ(%) =∫ %

0

Pδ(z)z2

dz, (3.7)

where Pδ ∈ C1[0,∞) satisfies

P ′δ(%) ≥ δ%+ c∗%γ−1, (3.8)

Pδ(%) ≤ c∗(%γ + 1), (3.9)

with c∗ = c∗(δ) > 0. The exponent γ > 0 (large enough) will be specified below.Moreover, we may assume that

dds

(gi,δ(s) + sg′i,δ(s)

)≥ −c∗

2sγ−2 for i = 1, 2, (3.10)

where c∗ > 0 is the same as in (3.8). Indeed the functions gi,δ can be simplyconstructed by truncation and mollification; then it is not difficult to check thatgi,δ(s) + sg′i,δ(s) are monotone for small values of s. Thus, in order to have (3.10),it is sufficient to truncate gi(s) in a suitable way for large values of s. We shall as-sume that Lδ, Γδ, and gi,δ, tend to L, Γ, and gi , i = 1, 2, respectively, uniformly oncompact subsets of (0, 1), (0, r) and (0,∞) (further details will be given in Section 6below).

At the first level of the approximation procedure, the continuity equation (1.1)is supplemented with an artificial viscosity term, the momentum equation (1.2) isreplaced by its Faedo-Galerkin approximation, while the Allen-Cahn system (1.3),(1.4) is provided with an extra term in order to keep the energy estimates valid.Then, the resulting approximate system reads:

• % is a smooth solution (% ∈ C1([0, T ];Cν(Ω)) ∩ C0([0, T ];Cν+2(Ω)), strictlypositive in [0, T ]×Ω) (cf. [11, Sect. 7.3.1])) to the initial-boundary value problem

∂t%+ divx(%u) = ε∆%, ε > 0, in (0, T )× Ω, (3.11)

∇x% · n|∂Ω = 0, (3.12)

%(0, ·) = %0,δ; (3.13)

6

• u ∈ C1([0, T ];Xn) satisfies the integral identity∫ T

0

∫Ω

(%u · ∂tϕ+ %[u⊗ u] : ∇xϕ

)dx dt =

∫ T

0

∫Ω

ε(ϕ∇xu) · ∇x% dx dt

+∫ T

0

∫Ω

T(∇xχ,∇xu) : ∇xϕ dx dt−∫

Ω

%0,δu0,δ · ϕ(0, ·) dx (3.14)

for any test function ϕ ∈ C1c ([0, T );Xn), where Xn is a (suitably chosen) finite

dimensional subspace of C∞c (Ω; R3);

• χ, µ solve in the classical sense the system

∂t(%χ) + divx(%χu) = −µ+ ε∆% χ, (3.15)

∇xχ · n|∂Ω = 0, (3.16)

(%χ)(0, ·) = %0,δχ0,δ, (3.17)

with

%µ = −∆χ+ %∂fδ(%, χ)

∂χ, (3.18)

where fδ was specified through (3.2 - 3.8).

Given ε > 0, δ > 0 and n finite, the approximate system (3.11 - 3.18) can be solvedon the time interval (0, T ) by means of the Schauder fixed point argument, similarlyto [11, Chapter 7]. Accordingly, the proof of Theorem 2.1 reduces to performingsuccessively the limits n→∞, ε→ 0, and, finally, δ → 0.

4 Limit in the Faedo-Galerkin approximations

4.1 Uniform bounds

Our first goal is to let n → ∞ in the sequence of solutions %n,un, χn∞n=1 to theapproximate problem (3.11 - 3.18). It is a routine matter to check, in accordance withthe hypotheses introduced in (3.1 - 3.8), that all quantities are regular, in particular,both (3.11) and (3.15) are satisfied in the classical sense, and the density %n is boundedbelow away from zero.

Thus, as the first step, we use (3.11) and (3.18) to rewrite (3.15) in the form

%n∂tχn + %nun · ∇xχn =1%n

∆χn − L′δ(χn) + b′(χn) + g2,δ(%n)− g1,δ(%n).

Then, by hypotheses (3.1), (3.5), (3.6), combined with the classical maximum princi-ple argument, we obtain that

δ ≤ χn(t, x) ≤ 1− δ for all (t, x) ∈ [0, T ]× Ω, (4.1)

where we point out that the bound is independent of both n and ε.Next, we aim to derive a global energy estimate. To obtain this, we first test (3.15)

by µn and (3.18) by ∂tχn. We then notice that, by (3.11),∫Ω

∂t%nχnµn =∫

Ω

%nχn∇xµn · un +∫

Ω

%nµn∇xχn · un + ε

∫Ω

∆%nχnµn. (4.2)

7

Thus, standard computation leads to

ddt

∫Ω

(12|∇xχn|2 + %nfδ(%n, χn)

)dx+ ‖µn‖2L2(Ω)

=∫

Ω

(fδ(%n, χn) + %n∂%fδ(%n, χn)

)∂t%n dx−

∫Ω

%nµn∇xχn · un dx; (4.3)

whence we have to handle the terms on the right hand side.The former can be treated expressing ∂t%n by means of (3.11). As for the latter

term, we use (3.18) that gives rise to

−∫

Ω

%nµn∇xχn · un dx =∫

Ω

(∆χn − %n∂χfδ

)∇xχn · un dx. (4.4)

Then, taking un as a test function in (3.14), multiplying (3.11) by |un|2/2, andadding both relations to (4.3), we check that the term depending on ∆χn in (4.4)cancels out with the corresponding term in the χ-dependent part of the stress tensor.Consequently, integrating by parts the terms depending on fδ, and performing someadditional manipulation, we end up with the approximate total energy balance:

ddt

∫Ω

[12%n|un|2 +

12|∇xχn|2 + %nfδ(%n, χn)

]dx+

∫Ω

[Sn(∇xun) : ∇xun + |µn|2

]dx

− ε

∫Ω

(fδ(%n, χn) + %n∂%fδ(%n, χn)

)∆%n dx = 0, (4.5)

where Sn is the n-approximation of S given by (1.6).In order to obtain suitable uniform bounds independent of n (and, in fact, of ε),

we have to control the last integral. To this end, we first observe that, by (3.7 - 3.8),

−ε∫

Ω

(Γδ + %nΓ′δ)∆%n dx = ε

∫Ω

P ′δ(%n)%n

|∇x%n|2 dx ≥ ε

∫Ω

|∇x%n|2(δ + c∗%

γ−2n

)dx.

(4.6)Next, we notice that, by (4.1) and (3.10),

−ε∫

Ω

(Wδ(χn)+χn

(g1,δ(%n)+%ng

′1,δ(%n)

)+(1−χn)

(g2,δ(%n)+%ng

′2,δ(%n)

))∆%n dx

≥ −cδε‖∇x%n‖L2(Ω;R3)‖∇xχn‖L2(Ω;R3) −εc∗2

∫Ω

%γ−2n |∇x%n|2 dx

≥ −εδ2‖∇x%n‖2L2(Ω;R3)

+ εcδ‖∇xχn‖2L2(Ω;R3)− εc∗

2

∫Ω

%γ−2n |∇x%n|2 dx, (4.7)

where Wδ = Lδ − b (cf. (3.2)).Finally, using Gronwall’s lemma in (4.5), we infer that for a suitable subsequence

(not relabeled) of n∞ there hold:

un → u weakly in L2(0, T ;H10 (Ω; R3)), (4.8)

χn → χ weakly-(*) in L∞(0, T ;H1(Ω)) ∩ L∞((0, T )× Ω), (4.9)

µn → µ weakly in L2(0, T ;L2(Ω)), (4.10)

8

where (4.8) follows from Poincare’s and Korn’s inequalities, and (4.9) also leans on(4.1). Moreover, we have ∥∥%n|un|2

∥∥L∞(0,T ;L1(Ω))

≤ c (4.11)

independently of n.Finally, using (3.7 - 3.8), (4.6), and (4.1), we may infer that

%n → % weakly-(*) in L∞(0, T ;Lγ(Ω)) ∩ L2(0, T ;H1(Ω)). (4.12)

4.2 Limit in the continuity equation

In order to derive further estimates on %n we adopt (3.11) the procedure described in[11, Lemma 7.5]. In what follows, we assume that γ ≥ 6 (cf. (3.8)), observing that,in fact, it is enough to take γ = 6 in most steps.

We rewrite (3.11) as

∂t%n +Aε%n = %n − divx(%nun) = %n − %ndivxun − un · ∇x%n, (4.13)

where we have set Aε = Id−ε∆, and where ∆ denotes the Laplace operator supple-mented with the homogeneous Neumann boundary conditions. We also introduce,for s ∈ R, H2s := D(As

ε), the domain of Asε, and, correspondingly, for v ∈ H2s,

‖v‖2s := ‖Asεv‖L2(Ω). Then, by (4.8), (4.12), and standard interpolation and embed-

dings,

‖divx(%nun)‖L1(0,T ;H−1/2) ≤ c‖divx(%nun)‖L1(0,T ;L3/2(Ω)) ≤ c. (4.14)

Next, thanks to (4.11) and (4.12),

‖%nun‖L∞(0,T ;H−1/4) ≤ c‖%nun‖L∞(0,T ;L12/7(Ω)) ≤ c, (4.15)

whence‖divx(%nun)‖L∞(0,T ;H−5/4) ≤ c. (4.16)

Interpolating between (4.14) and (4.16) it then follows that

‖divx(%nun)‖Lp(0,T ;H−1) ≤ c for some p > 2; (4.17)

whence, applying the Lp-regularity theory to (4.13) (notice that, indeed, H−1 =(H1)∗), we get

‖%n‖Lp(0,T ;H1) ≤ c for some p > 2. (4.18)

Thus, using once more (4.8), we can improve (4.14) to

‖divx(%nun)‖Lp((0,T )×Ω) ≤ c for some p > 1. (4.19)

Applying the Lp-theory to (4.13), we finally have

∂t%n → ∂t%, ∆%n → ∆% weakly in Lp((0, T )× Ω) for some p > 1. (4.20)

Consequently, in accordance with (4.12),

%n → % strongly in Lp((0, T )× Ω) for all p ∈ [1, γ), (4.21)

and%n → % in Cw([0, T ];Lγ(Ω)). (4.22)

Then, by (4.8), we also get

%nun → %u weakly in L2((0, T )× Ω), (4.23)

so that we can take the limit n∞ in (3.11).

9

4.3 Limit in the Allen-Cahn equation

Firstly, we notice that, by (4.8), (4.9) and (4.12) (recall that γ ≥ 6),

‖%nχn‖L∞(0,T ;L6(Ω)) ≤ c, (4.24)

‖%nχnun‖L2(0,T ;L3(Ω)) ≤ c. (4.25)

Moreover, expanding divx(%nχnun) by Leibnitz’ formula and using again (4.18),we easily see that

‖divx(%nχnun)‖Lp((0,T )×Ω) ≤ c for some p > 1. (4.26)

Since the same bound holds for the right hand side of (3.15), relations (4.1) and (4.20)give rise to

‖∂t(%nχn)‖Lp((0,T )×Ω) ≤ c for some p > 1. (4.27)

Now, let us handle the last term in (3.18) (cf. (1.8), (1.10)). We obtain

%n∂fδ(%n, χn)

∂χ= %nW

′δ(χn) + %n

(g1,δ(%n)− g2,δ(%n)

). (4.28)

By (4.21), (3.4) and Lebesgue’s theorem, we then infer that

%n

(g1,δ(%n)− g2,δ(%n)

)→ %

(g1,δ(%)− g2,δ(%)

)strongly in L2((0, T )× Ω). (4.29)

Moreover, by virtue of (4.9), (4.22), (4.1) and (3.3),

%nW′δ(χn) → %W ′

δ(χ) weakly-(*) in L∞(0, T ;L6(Ω)), (4.30)

Finally, due to (4.10) and (4.21), we have

%nµn → %µ weakly in L1((0, T )× Ω). (4.31)

Thus we can take the limit in (3.18) to get

%µ = −∆χ+ %W ′δ(χ) + %

(g1,δ(%)− g2,δ(%)

). (4.32)

Next, we test (3.18) by χn and integrate over (0, T )× Ω:∫ T

0

∫Ω

|∇xχn|2 dx dt =∫ T

0

∫Ω

%nµnχn dx dt−∫ T

0

∫Ω

%nW′δ(χn)χndx dt.

−∫ T

0

∫Ω

%n

(g1,δ(%n)− g2,δ(%n)

)χndx dt. (4.33)

Observe that

∂t

(%nW

′δ(χn)

)= ∂t%n

(W ′

δ(χn)−W ′′δ (χn)χn

)+W ′′

δ (χn)∂t(%nχn); (4.34)

whence, by (4.1), (3.3), (4.20) and (4.27),∥∥∂t(%nW′δ(χn))

∥∥Lp((0,T )×Ω)

≤ c for some p > 1. (4.35)

This implies that (4.30) can be improved to

%nW′δ(χn) → %W ′

δ(χ) in Cw([0, T ];L6(Ω)). (4.36)

10

Using (4.9), we deduce

%nW′δ(χn)χn → %W ′

δ(χ)χ weakly in L2((0, T )× Ω). (4.37)

Thus, we can take the limit n∞ in (4.33). Indeed, the first term on the righthand side can be treated in a similar (and in fact simpler) way. Comparing the resultwith (4.32) integrated in space and time and using also Poincare’s inequality in theform [14, Lemma 3.1], we finally obtain

χn → χ strongly in L2(0, T ;H1(Ω)) (4.38)

and, consequently, W ′δ(χ) = W ′

δ(χ). Thus, we can take the limit of (3.18).Finally, we aim to take the limit of (3.15). Here, we simply notice that, by (4.38),

(4.1) and Lebesgue’s theorem,

χn → χ strongly in Lp((0, T )× Ω) for all p ∈ [1,∞). (4.39)

Thus, by virtue of the strong convergence established in (4.38) and (4.21), it is easyto pass to the limit. In particular, the last term on the right hand side is treated bymeans of (4.38) and (4.20).

4.4 Limit in the momentum equation

We choose ϕ ∈ C1c ([0, T );Xm) for fixed m ∈ N and examine the identity (3.14) for

n ≥ m. Our aim is to let n∞. First, we consider the stress tensor T given by (1.5).It is clear that the components specified in (1.6) admit limits thanks to hypothesis(2.8) and relations (4.8) and (4.39). Next, the terms depending only on χ in (1.5) aretreated by means of (4.38). Finally, to take the limit of the pressure term containedin T, we observe that, thanks to (4.6),

%n → % weakly in Lγ(0, T ;L3γ(Ω)); (4.40)

whence we can conclude by (4.22), interpolation and Lebesgue’s theorem (cf. assump-tion (3.9)).

Now, let us notice that, by means of (4.15), the integral relation (3.14), andAscoli’s theorem (cf., e.g., [11, Cor. 2.1]), we have

%nun → %u in Cw([0, T ];L12/7(Ω)); (4.41)

whence, by (4.8),

%n(un ⊗ un) → %(u⊗ u) weakly in L2(0, T ;L4/3(Ω)). (4.42)

Finally, to treat the first term on the right hand side of (3.14) we need to provethat

%n → % strongly in L2(0, T ;H1(Ω)). (4.43)

To obtain this, we proceed similarly to the Allen-Cahn equation. Namely, we test(3.11) by %n and integrate over (0, T )× Ω. We then notice that∫

Ω

divx(%nun)%n dx =∫

Ω

%2n

2divxun dx, (4.44)

11

and the same holds for the limit functions %, u. Thus, using (4.22) to treat the termdepending on the initial datum, we may infer that

lim supn∞

∫ T

0

∫Ω

|∇x%n|2 dx dt ≤∫ T

0

∫Ω

|∇x%|2 dx dt, (4.45)

which implies (4.43). Thus, the limit of (3.14) holds for any ϕ ∈ C1c ([0, T ];Xm) and,

due to arbitrariness of m and density of ∪Xm, holds also for ϕ ∈ C1c ([0, T ) × Ω), as

desired.

4.5 Limit in the energy inequality

In order to perform the passage ε 0, we need to prove that the solution constructedabove still satisfies a suitable version of inequality (4.5). Notice that this cannot beachieved by using test functions in the limit equations as at this level the solutionsare no longer regular. Instead we take the limit in (4.5) for n∞. Integrating (4.5)over (0, τ), τ ∈ (0, T ], we obtain∫

Ω

(12%n(τ)|un(τ)|2 +

12|∇xχn(τ)|2 + %n(τ)fδ(%n(τ), χn(τ))

)dx

+∫ τ

0

∫Ω

Sn(∇xun) : ∇xun dx dt+∫ τ

0

∫Ω

|µn|2 dx dt

+ε∫ τ

0

∫Ω

(%−1

n P ′δ(%n) + χnh′1,δ(%n) + (1− χn)h′2,δ(%n)

)|∇x%n|2 dx dt

+ε∫ τ

0

∫Ω

(W ′

δ(χn) + h1,δ(%n)− h2,δ(%n))∇xχn · ∇x%n dx dt

=∫

Ω

(12%0,δ|u0,δ|2 +

12|∇xχ0,δ|2 + %0,δfδ(%0,δ, χ0,δ)

)dx, (4.46)

where hi,δ(s) = gi,δ(s) + sg′i,δ(s), i = 1, 2. Our aim is to take the lim inf as n∞ inthe above inequality.

First notice, by (4.38) and (4.43), that

|∇x%n|2 → |∇x%|2, ∇x%n∇xχn → ∇x% · ∇xχ strongly in L1((0, T )× Ω), (4.47)

and, on the other hand,

χnh′1,δ(%n) + (1− χn)h′2,δ(%n) +W ′

δ(χn) + h1,δ(%n)− h2,δ(%n) (4.48)

converges to the corresponding limit weakly-(*) in L∞((0, T )× Ω). Moreover,∫ T

0

∫Ω

%−1n P ′δ(%n)|∇x%n|2 dx dt =

∫ T

0

∫Ω

|∇xQδ(%n)|2dx dt, (4.49)

where (Q′δ)2 = P ′δ/%. Moreover, we check easily that Qδ(%n) → Qδ(%) weakly in

L2(0, T ;H1(Ω)).Concerning the other terms, we observe that∫ T

0

∫Ω

ν(χn)∇xun : ∇xun dx dt = ‖ψn‖2L2((0,T )×Ω), (4.50)

12

where

ψn = ν1/2(χn)∇xun → ψ = ν1/2(χ)∇xu weakly in L2((0, T )× Ω; R3×3). (4.51)

Thus, one can compute the lim inf of all terms on the left hand side of (4.46)except those on the first line, evaluated pointwise in time. To deal with these, onehas to perform one more integration in time of the energy equality. Thus, we obtainthat ∫ t+τ

t

∫Ω

(12%|u|2 +

12|∇xχ|2 + %fδ(%, χ)

)dx ds

≤ lim infn∞

∫ t+τ

t

∫Ω

(12%n|un|2 +

12|∇xχn|2 + %nfδ(%n, χn)

)dx ds (4.52)

for all τ > 0. Then, thanks to semi-continuity of norms with respect to the weak orweak-(*) convergence, we obtain the limit form of the energy estimate. Dividing byτ and letting τ 0, the limit energy inequality is then recovered in the original formand for a.a. value t of the time variable.

5 Artificial viscosity limit

Our aim is to let ε 0. In accordance with the preceding step, we can assume to havea family of approximate solutions uε, %ε, µε, χεε>0 satisfying (3.11 - 3.18). At thisstage, the regularity properties of uε, %ε, µε, χε are those established in the previousstep. In addition, as stated in Section 4.5, we also know that it is possible to performthe limit n∞ in (4.46).

5.1 Limit in the continuity equation

We rewrite (the ε-version) of (4.13) as

∂t%ε + εA%ε = ε%ε − divx(%εuε). (5.1)

Here, similarly to Section 4.2, A = Id−∆, with the homogeneous Neumann boundaryconditions, and, for s ∈ R, H2s := D(As) with the natural norms. By means of theε-analogues of (4.8) and (4.12), it is not difficult to conclude that

‖divx(%εuε)‖L2(0,T ;H−1) + ‖divx(%εuε)‖L∞(0,T ;H−5/4) ≤ c. (5.2)

Here and hereafter, the constants c are independent of ε. Standard parabolic estimates(namely, testing (5.1) by ε%ε) yield

‖%ε‖H1(0,T ;H−1) + ε1/2‖%ε‖L∞(0,T ;L2(Ω)) + ε‖%ε‖L2(0,T ;H1) ≤ c. (5.3)

Using the energy inequality we have

%ε → % in Cw([0, T ];L6(Ω)), (5.4)

uε → u weakly in L2(0, T ;H10 (Ω; R3)), (5.5)

%εuε → %u weakly in L2(0, T ;L3(Ω;R3)), (5.6)

in particular, we can pass to the limit in (5.1).

13

Since %ε, uε satisfy (5.1), we can use the regularization procedure introducedby DiPerna and Lions [10] to show that %δ, uδ represent a renormalized solution ofequation (1.1). Namely, there holds∫ T

0

∫Ω

(b(%δ)∂tϕ+ b(%δ)uδ · ∇xϕ+ (b(%δ)− b′(%δ)%δ) divxuδϕ

)dx dt (5.7)

= −∫

Ω

b(%0,δ)ϕ(0, ·) dx,

for any ϕ ∈ C∞c ([0, T ) × Ω) and any b ∈ W 1,∞[0,∞). It is easy to see, at leastformally, that such a formula can be deduced by testing the limit equation of (5.1)by b′(%δ)ϕ for any ϕ ∈ C∞c ([0, T )× Ω) and integrating by parts.

5.2 Limit in the Allen-Cahn system and in the momentumequation

Our aim is to let ε 0 in the system

∂t(%εχε) + divx(%εχεuε) = −µε + ε∆%εχε, (5.8)

%εµε = −∆χε + %ε∂fδ(%ε, χε)

∂χε. (5.9)

Let us first notice that, by the energy inequality, the ε-analogue of (4.1), (5.4), andPoincare’s inequality (once more in the form [14, Lemma 3.1]), we have

µε → µ weakly in L2((0, T )× Ω), (5.10)

χε → χ weakly-(*) in L∞(0, T ;H1(Ω)) ∩ L∞((0, T )× Ω). (5.11)

Thus, using once more (5.4), we also obtain

%εχε → %χ weakly-(*) in L∞(0, T ;L6(Ω)). (5.12)

Moreover, by (5.5),‖%εχεuε‖L2(0,T ;L3(Ω)) ≤ c. (5.13)

The energy inequality, hypothesis (3.4), relations (4.1) and (5.4), and a comparisonof (5.9) give rise to

‖%εµε‖L2(0,T ;L3/2(Ω)) + ‖∆χε‖L2(0,T ;L3/2(Ω)) ≤ c. (5.14)

Let us pick φ ∈ C1c ([0, T );H2

0 (Ω)) and notice that

ε

∫ T

0

∫Ω

∆%εχεφ dx dt = −ε∫ T

0

∫Ω

φ∇x%ε ·∇xχε dx dt−ε∫ T

0

∫Ω

χε∇x%ε ·∇xφ dx dt.

(5.15)Thus, testing (5.8) by φ, integrating over (0, T ) × Ω, and making use of relations(5.13), (5.10), and (5.3), we easily see that∥∥∂t(%εχε)

∥∥L2(0,T ;H−1(Ω))+L2(0,T ;L1(Ω))

≤ c, (5.16)

whence (5.12) is improved to

%εχε → %χ in Cw([0, T ];L6(Ω)) (5.17)

14

and, consequently, by (5.5), we also have

%εχεuε → %χu weakly in L2(0, T ;L3(Ω; R3)). (5.18)

In order to pass to the limit in (5.8), we use φ as a test function and observe thatthe right hand side of (5.15) can be transformed into

ε

∫ T

0

∫Ω

φ%ε∆χε dx dt+ 2ε∫ T

0

∫Ω

%ε∇xχε · ∇xφ dx dt+ ε

∫ T

0

∫Ω

%εχε∆φ dx dt,

(5.19)where all terms clearly go to 0 for φ as above.

Next, we take the limit in (5.9), which is more involved. First, we observe that

W ′δ(χε) →W ′

δ(χ) weakly-(*) in L∞(0, T ;H1(Ω)). (5.20)

Thus, due to (5.4),

%εW′δ(χε) → %W ′

δ(χ) weakly-(*) in L∞(0, T ;L3(Ω)), (5.21)

and, thanks to (5.17),

%εχεW′δ(χε) → %χW ′

δ(χ) weakly-(*) in L∞(0, T ;L3(Ω)). (5.22)

At this stage, we follow step by step the procedure developed in [3, Subsec. 2.6].Accordingly, we focus on the principal steps only. First, we claim that, by (5.11) and(5.17), ∫ T

0

∫Ω

%χ2ε dx dt→

∫ T

0

∫Ω

%χ2 dx dt. (5.23)

As a matter of fact, we can simply check that

χ2ε → χ2 weakly in L2(0, T ;H1(Ω)), (5.24)

whence

%χ2ε − %χ2 = (%χ2

ε − %εχ2ε) + (%εχ

2ε − %εχεχ) + (%εχεχ− %χ2), (5.25)

where the last three summands on the right hand side go to 0 due to (5.11), (5.17),and (5.24). Thus, it follows that

χε → χ strongly in Lp(Q+T ) for all p ∈ [1,∞), (5.26)

where Q+T (Q0

T ) denotes the subset of (0, T )×Ω where % > 0 (% = 0). Moreover, since% is non-negative, it is clear that

%ε → % strongly in Lp(Q0T ) for all p ∈ [1, 6). (5.27)

Now, we may rewrite (5.9) as

%εµε = −∆χε + %εW′δ(χε) + hδ(%ε), (5.28)

where we have set hδ(s) = s(g1,δ(s)− g2,δ(s)). At this level, taking the limit (in thesense of distributions) yields

%µ = −∆χ+ %W ′δ(χ) + hδ(%). (5.29)

15

Now, as in Section 4.3, we test (5.28) by χε and integrate over (0, T )×Ω. It is clear,thanks to (5.20 - 5.22), that the desired conclusion

χε → χ strongly in L2(0, T ;H1(Ω)) (5.30)

follows as soon as we can prove that∫ T

0

∫Ω

(%µχ− hδ(%)χ

)dx dt =

∫ T

0

∫Ω

(%µχ− hδ(%)χ

)dx dt. (5.31)

Note that, by (5.27),∫∫Q0

T

hδ(%)χ dx dt =∫∫

Q0T

hδ(%)χ dx dt =∫∫

Q0T

hδ(%)χ dx dt = 0, (5.32)

whereas, thanks to (5.26),∫∫Q+

T

hδ(%)χ dx dt =∫∫

Q+T

hδ(%)χ dx dt. (5.33)

Analogously, ∫∫Q0

T

%µχ dx dt =∫∫

Q0T

%µχ dx dt = 0 (5.34)

and, on the other hand,∫∫Q0

T

%µχ dx dt =∫∫

Q0T

%µχ dx dt = 0. (5.35)

Moreover, still by (5.26),∫∫Q+

T

%µχ dx dt =∫∫

Q+T

%µχ dx dt. (5.36)

Thus, collecting (5.32 - 5.36), we get (5.31), and, consequently, (5.30). Moreover,we arrive at the relation

%µ = −∆χ+ %W ′δ(χ) + hδ(%). (5.37)

In order to identify the remaining two terms, we need strong convergence of %ε, whoseproof will be discussed later.

Finally, we pass to the limit in the momentum equation, that now reads∫ T

0

∫Ω

(%εuε · ∂tϕ+ %ε[uε ⊗ uε] : ∇xϕ

)dx dt

=∫ T

0

∫Ω

T(∇xχε,∇xuε) : ∇xϕ dx dt−∫

Ω

%0,δu0,δ · ϕ(0, ·) dx, (5.38)

for ϕ as in (2.12). Actually, it follows from (5.38) that ∂t(%εuε) is uniformly boundedin some negative order Sobolev space. Thus, using (5.4) and (5.5), we conclude asbefore that

%εuε → %u in Cw([0, T ];L12/7(Ω)) (5.39)

as well as%ε(uε ⊗ uε) → %(u⊗ u) weakly in L2(0, T ;L4/3(Ω)). (5.40)

Next, thanks to (5.5) and (5.30), the tensor T can be treated as in Section 4.4, withthe exception of the pressure term pδ(%ε, χε), whose treatment also requires the strongconvergence of %ε.

16

5.3 Conclusion of the proof

Our main task is to obtain strong L1-convergence of the densities %ε. For the sake ofclarity, we just give the highlights of this procedure, referring to the next section wherethe same arguments will be repeated (in fact in an even more delicate situation).

• As a first step, we proceed as in Section 6.2.1 below, i.e., we use the analogueof the test function in (6.24). By just adapting the notation, we then arrive atthe analogue of (6.26). In the present situation, thanks to (3.8 - 3.9), this givesin particular

p(%ε, χε) → p(%, χ) weakly in L(γ+1)/γ((0, T )× Ω). (5.41)

• Second, we have to perform Lions’ argument as in Section 6.5. Following stepby step that procedure, we then arrive at a pointwise (a.e.) convergence %ε → %,which permits in particular to identify the remaining limits in (5.37) and (5.41).The only difference with respect to estimate (6.45) consists in the presence ofother two terms

−ε∫ T

0

ψ

∫Ω

ξ%εuε · ∇x∆−1(divx(1Ω∇x%ε)) dx dt

and

ε

∫ T

0

ψ

∫Ω

ξ∇x%ε∇xuε · ∇x∆−1(1Ω%ε) dx dt ,

which tend to zero as ε 0 due to (5.15), (5.39), and (5.45).

Having the strong convergence of %ε at our disposal, it is now clear that we can takethe limit ε 0 in all equations. To conclude, it remains to pass to the limit in theenergy inequality (cf. (4.46)). To do this, we can use the argument similar to thatin Section 4.5. The main difference is that now we also have to test (5.1) by %ε anddeduce that

12

ddt

∫Ω

%2ε dx+ ε

∫Ω

|∇x%ε|2 dx ≤ −12

∫Ω

%2ε|divxuε| dx. (5.42)

This immediately leads to

ε1/2‖%ε‖L2(0,T ;H1(Ω)) ≤ c. (5.43)

More precisely, integrating (5.42) in time, we obtain

ε

∫ t

0

∫Ω

|∇x%ε|2 dx ds ≤ −12

∫Ω

%2ε(t) dx+

12

∫Ω

%20,δ dx (5.44)

−12

∫ T

0

∫Ω

%2εdivxuε dx ds,

so that, taking the lim sup as ε 0, using semicontinuity of norms w.r.t. weakconvergence, and comparing the result with the limit momentum equation (5.7) (inthe renormalized form (5.7) with b(%) = %), we may infer that

ε1/2%ε → 0 strongly in L2(0, T ;H1(Ω)) (5.45)

Thus, the limit energy inequality can be computed as in Section 4.5.

17

6 Artificial pressure limit

Our ultimate goal is to let δ 0. To this end, consider a family %δ,uδ, χδδ>0 ofthe approximate solutions constructed in the previous part. Accordingly, we choosethe initial data in (3.1) in such a way that

ess infΩ %0 ≤ %0,δ(x) ≤ ess supΩ %0 < r, x ∈ Ω, %0,δ → %0,

ess infΩ χ0 ≤ χ0,δ(x) ≤ ess supΩ χ0, x ∈ Ω, χ0,δ → χ0 a.e. in Ω,

u0,δ → u0 in L2(Ω; R3)

(6.1)

as δ 0.In addition, it is a routine matter to construct a family of convex functions Lδ ∈

C∞(R) such thatLδ(χ) L(χ) for any χ ∈ (0, 1), (6.2)

andL′δ(χ) ≥ L′(1− δ) for all χ > 1− δ,

L′δ(χ) ≤ L′(δ) for all χ < δ(6.3)

for a suitable sequence δ 0. Consequently, in accordance with hypothesis (2.2), wecan find the functions g1,δ, g2,δ such that (3.4 - 3.6) hold, and, in addition,

gi,δ(%) ai log(%) for % ≥ 1,

gi,δ(%) ai log(%) for 0 ≤ % ≤ 1

as δ 0, i = 1, 2, (6.4)

g′i,δ(%) →ai

%in C(0,∞) as δ 0, i = 1, 2. (6.5)

Finally, we take

Pδ(%) = δ%2 +

P (%) if 0 ≤ % ≤ r − δ,

P (r − δ) + ([%− r − 1]+)γ if % ≥ r − δ.(6.6)

6.1 Uniform bounds

To begin, we recall (cf. (4.1)) that the functions χδ satisfy the uniform bound

δ ≤ χδ(t, x) ≤ 1− δ for a.a. t ∈ (0, T )× Ω. (6.7)

Moreover, the energy inequality∫ T

0

∫Ω

[12

(%δ|uδ|2 + |∇xχδ|2

)+ %δfδ(%δ, χδ)

]dx ∂tψ dt (6.8)

−∫ T

0

∫Ω

[Sδ(∇xuδ) : ∇xuδ + |µδ|2

]dx ψ dt

≥ −∫

Ω

[12

(%0,δ|u0,δ|2 + |∇xχ0,δ|2

)+ %δfδ(%0,δ, χ0,δ)

]dx

18

holds for any ψ ∈ C∞c [0, T ), ψ ≥ 0, ψ(0) = 1.It follows from hypothesis (2.15), (3.1), and (6.1) that∣∣∣∣∫

Ω

[12

(%0,δ|u0,δ|2 + |∇xχ0,δ|2

)+ %δfδ(%0,δ, χ0,δ)

]dx

∣∣∣∣ ≤ c,

where the constant is independent of δ. Consequently, we deduce the following uni-form estimates:

√%δuδδ>0 bounded in L∞(0, T ;L2(Ω; R3)), (6.9)

χδδ>0 bounded in L∞(0, T ;H1(Ω)), (6.10)

%δΓδ(%δ)δ>0 bounded in L∞(0, T ;L1(Ω)). (6.11)

In particular,%δδ>0 bounded in L∞(0, T ;Lγ(Ω)). (6.12)

Moreover,µδδ>0 is bounded in L2((0, T )× Ω), (6.13)

and, as a direct consequence of Korn’s inequality and hypothesis (2.8),

uδδ>0 is bounded in L2(0, T ;H1(Ω; R3)). (6.14)

Now, it follows from (2.10) and the uniform bounds (6.12), (6.14) that

%δ → % in Cweak([0, T ];Lγ(Ω)), (6.15)

uδ → u weakly in L2(0, T ;H10 (Ω; R3)), (6.16)

at least for suitable subsequences. Similarly, (6.10) yields

χδ → χ weakly-(*) in L∞(0, T ;H1(Ω)), (6.17)

where, as consequence of (6.11), (4.1),

0 ≤ %(t, x) ≤ r for a.a. t, x, (6.18)

0 ≤ χ(t, x) ≤ 1 for a.a. t, x. (6.19)

As a matter of fact, (6.18) follows as the functional on L2((0, T ) × Ω) associated tothe function Zδ(r) = rΓδ(r) is “essentially” convex and converges to Z(r) = rΓ(r) inthe sense of Mosco. Thus, (6.18) follows from (6.15), (6.11) and the lim inf-inequality∫ T

0

∫Ω

Z(%) dx dt ≤ lim infδ0

∫ T

0

∫Ω

Zδ(%δ) dx dt <∞. (6.20)

The next step is to multiply (3.18) by F (L′δ(χδ)) and integrate by parts to obtain∫Ω

(F ′(L′δ(χδ))L′′δ (χδ)|∇xχδ|2 + %δL

′δ(χδ)F (L′δ(χδ))

)dx

=∫

Ω

(%δµδF (L′δ(χδ)) + %δb

′(χδ)F (L′δ(χδ)) + %δ(g2,δ(%δ)− g1,δ(%δ))F (L′δ(χδ)))

dx,

19

where F is a non-decreasing function on R. Taking F (z)z ≈ |z|β+1 for large valuesof |z|, we can use the uniform bounds established in (6.12), (6.13) in order to deducethat ∫ T

0

∫Ω

%δ|L′δ(χδ)|β+1 dx dt ≤ c uniformly with respect to δ 0, (6.21)

where0 < β = β(γ) 1 provided γ →∞ in (6.6).

Thus going back to (3.18) we may infer that

∆χδδ>0 is bounded in Lα((0, T )× Ω) where 1 < α 2 for γ →∞. (6.22)

Relations (6.10), (6.22), together with the standard elliptic estimates and a simpleinterpolation argument, yield

∇xχδδ>0 bounded in Lq((0, T )× Ω; R3) for a certain q > 2 (6.23)

provided the exponent γ in (6.6) was chosen large enough.

6.2 Refined pressure estimates

One of the principal difficulties in the proof of Theorem 2.1 stems from the fact thatthe uniform bounds established in the previous part do not imply, in general, anyuniform estimates on the pressure pδ(%δ, χδ), not even in the space L1((0, T )× Ω).

6.2.1 Integrability of the pressure term

Following the strategy of [13], we use the quantities

ϕ = ψ(t)B[b(%δ)−

1|Ω|

∫Ω

b(%δ) dx], ψ ∈ C∞c (0, T ), (6.24)

as test functions in the weak formulation of the momentum equation (2.12), wherethe symbol B ≈ div−1

x denotes the integral operator introduced by Bogovskii [6]. Theoperator B assigns to each g ∈ Lp(Ω) with

∫Ωg dx = 0 a solution B[g] ∈W 1,p

0 (Ω; R3)of the problem

divx B[g] = g in Ω, B[g]|∂Ω = 0.

It can be shown that B specified in [6] is a bounded linear operator acting on Lq(Ω)with values in W 1,q

0 (Ω; R3) for any 1 < q <∞ (see Galdi [16, Chapter 3]), and, more-over, B can be extended as a bounded linear operator on the dual space [W 1,q(Ω)]∗

with values in Lq′(Ω; R3) for any 1 < q <∞ (see Geissert, Heck, and Hieber [17]).We recall that %δ and uδ, for any ϕ ∈ C∞c ([0, T ) × Ω) and any b ∈ W 1,∞[0,∞),

satisfy the renormalized equation (5.7).With (5.7) at hand, we can use the quantities ϕ specified in (6.24) as test functions

in (2.12) to obtain∫ T

0

ψ

∫Ω

pδ(%δ, χδ)[b(%δ)−

1|Ω|

∫Ω

b(%δ)(y) dy]

dx dt =6∑

i=1

Ii,δ, (6.25)

20

where

I1,δ = −∫ T

0

ψ

∫Ω

((∇xχδ ⊗∇xχδ −

12|∇xχδ|2I

): ∇xB

[b(%δ)−

1|Ω|

∫Ω

b(%δ) dy])

dx dt,

I2,δ =∫ T

0

ψ

∫Ω

Sδ : ∇xB[b(%δ)−

1|Ω|

∫Ω

b(%δ) dy]

dx dt,

I3,δ = −∫ T

0

ψ

∫Ω

%δ (uδ ⊗ uδ) : ∇xB[b(%δ)−

1|Ω|

∫Ω

b(%δ) dy]

dx dt,

I4,δ = −∫ T

0

∂tψ

∫Ω

%δuδ · B[b(%δ)−

1|Ω|

∫Ω

b(%δ) dy]

dx dt,

I5,δ =∫ T

0

ψ

∫Ω

%δuδ · B [divx(b(%δ)uδ)] dx dt,

and

I6,δ =∫ T

0

ψ

∫Ω

%δuδ · B[(b′(%δ)%δ − b(%δ)) divxuδ

− 1|Ω|

∫Ω

(b′(%δ)%δ − b(%δ)) divxuδ dy]

dx dt.

Taking b(%) = %, and using the uniform bounds established in (6.9 - 6.23), togetherwith boundedness of the operator B in Lq and [W 1,q]∗, we deduce, exactly as in [13],that all integrals Ii,δ, i = 1, . . . , 6 are bounded uniformly for δ 0 as long as theexponent γ in (6.6) is large enough. Consequently, we obtain∣∣∣ ∫ T

0

∫Ω

pδ(%δ, χδ)[%δ −

1|Ω|

∫Ω

%δ(y) dy]

dx dt∣∣∣ ≤ c,

with c independent of δ 0.Now, it follows from (6.1) that

1|Ω|

∫Ω

%δ(t, ·) dx =1|Ω|

∫Ω

%0,δ dx = mδ < r,

therefore ∫ T

0

∫Ω

pδ(%δ, χδ)(%δ −

1|Ω|

∫Ω

%δ(y) dy)

dx dt = J1,δ + J2,δ,

with

J1,δ =∫%δ<(mδ+r)/2

pδ(%δ, χδ)(%δ −

1|Ω|

∫Ω

%δ(y) dy)

dx dt

J2,δ =∫%δ≥(mδ+r)/2

pδ(%δ, χδ)(%δ −

1|Ω|

∫Ω

%δ(y) dy)

dx dt

≥ r −mδ

2

∫%δ≥(mδ+r)/2

pδ(%δ, χδ) dx dt.

Since Ω is a bounded domain, the integrals J1,δ are evidently bounded, and we mayconclude that

pδ(%δ, χδ)

δ>0,

pδ(%δ, χδ)%δ

δ>0

are bounded in L1((0, T )× Ω), (6.26)

uniformly w.r.t. δ 0.

21

6.2.2 Equi-integrability of the pressure term

Estimate (6.26) is still not sufficient for passing to the limit in the pressure term. Inorder to establish at least weak convergence of the pressure, we need equi-integrabilityof the family pδδ>0. To this end, we make use of hypothesis (2.7).

Analogously as in the previous section, we take the quantities

ϕ(t, x) = ψ(t)B[ηδ(%δ)−

1|Ω|

∫Ω

ηδ(%δ) dx]

(6.27)

with ψ ∈ C∞c (0, T ),

ηδ = ηδ(%) =

log(r − %) if % ≤ r − δ,

log(δ) otherwise,

as test functions in the momentum equation (2.12).As P satisfies hypothesis (2.7), and Pδ is given by (6.6), there are constants c1 > 0,

c2 such thatΓδ(%) ≥

c1(r − %)2

− c2 for all 0 ≤ % ≤ r − δ,

in particular,

Γδ(%δ) ≥ c1(q)|ηδ(%δ)|q − c2(q) for any 1 ≤ q <∞, (6.28)

and, similarly,Γδ(%δ) ≥ c1|η′δ(%δ)|2 − c2, (6.29)

where Γδ is given through (3.7).Thus, exactly as in the previous section, we deduce a uniform bound∫ T

0

∫Ω

|pδ(%δ)ηδ(%δ)| dx dt ≤ c, c independent of δ 0, (6.30)

as soon as we are able to control the integrals on the right-hand side of formula (6.25).We check easily that the most difficult term reads

I6,δ =∫ T

0

∫Ω

ψ%δuδ · B[(η′δ(%δ)%δ − ηδ(%δ)

)divxuδ (6.31)

− 1|Ω|

∫Ω

(η′δ(%δ)%δ − ηδ(%δ)

)divxuδ dy

]dx dt.

In accordance with the uniform bounds established in (6.9), (6.11), (6.14), we canuse (6.28), (6.29) in order to obtain that∥∥∥B[(

η′δ(%δ)%δ − ηδ(%δ))divxuδ (6.32)

− 1|Ω|

∫Ω

(η′δ(%δ)%δ − ηδ(%δ)

)divxuδ dx

]∥∥∥L2(0,T ;Lq(Ω;R3))

≤ c(q)

for any q < 3/2.

Indeed we have used the embedding L1(Ω) → [W 1,q]∗(Ω) for q > 3, together with theresult of Geissert et al. [17] on boundedness of the operator B : [W 1,q]∗ → Lq′ .

22

On the other hand, seeing that H1(Ω) → L6(Ω), we can take γ > 6 in (6.6) inorder to control the momentum %δuδ by help of the energy estimates (6.9 - 6.14).Consequently, the integrals I6,δ are bounded uniformly for δ 0. Thus we haveshown (6.30).

Estimate (6.30) implies equi-integrability of the family pδ(%δ, χδ)δ>0 in theLebesgue space L1((0, T )× Ω); whence we may conclude that

pδ(%δ, χδ) → p(%, χ) weakly in L1((0, T )× Ω). (6.33)

6.3 Convergence of convective terms

At this stage, we are ready to perform the limit δ 0 in the approximate equationsand to identify the limit system. We start with the convective terms that can behandled in a rather uniform way repeating essentially the arguments of Sections 4.4,5.2.

To begin, relations (6.9), (6.15), (6.16) give rise to

%δuδ → %u weakly-(*) in L∞(0, T ;Lp(Ω; R3)) for a certain p > 6/5

provided γ > 0 is large enough. This can be strengthened to

%δuδ → %u in Cw([0, T ];Lp(Ω; R3)) (6.34)

by means of (2.12). Thus the same argument leads finally to

%δuδ ⊗ uδ → %u⊗ u weakly in Lp((0, T )× Ω; R3×3) for a certain p > 1. (6.35)

In the same fashion, relation (6.17) yields

%δχδ → %χ in Cw([0, T ];Lγ(Ω)), (6.36)

and%δχδuδ → %χu weakly in L2((0, T )× Ω; R3). (6.37)

Finally, the same argument used to deduce (5.23) yields∫ T

0

∫Ω

%|χδ|2 dx dt→∫ T

0

∫Ω

%|χ|2 dx dt,

in other words

χδ → χ a.a. in the set (t, x) ∈ (0, T )× Ω | %(t, x) > 0. (6.38)

6.4 Strong convergence of the extra stress

Following the method developed in [3] we show strong convergence of ∇xχδδ>0.The first part of the argument goes along the lines of Section 5.2. First, we let δ 0in (3.18) to obtain ∫ T

0

∫Ω

%µϕ dx dt =∫ T

0

∫Ω

∇xχ · ∇xϕ dx dt

+∫ T

0

∫Ω

(%L′δ(χ)ϕ− %b′(χ)ϕ+ %(g2,δ(%)− g1,δ(%))ϕ

)dx dt

23

for any sufficiently regular test function ϕ, where the bars denote weak limits inL1. Note that %δL

′δ(χδ)δ>0 is bounded in Lp((0, T ) × Ω) for a certain p > 1 as a

consequence of (6.12), (6.21). In particular, taking ϕ = χ we get∫ T

0

∫Ω

%µχ dx dt =∫ T

0

∫Ω

∇xχ · ∇xχ dx dt (6.39)

+∫ T

0

∫Ω

(%L′δ(χ)χ− %b′(χ)χ+ %(g2,δ(%)− g1,δ(%))χ

)dx dt

On the other hand, we also have∫ T

0

∫Ω

%µχ dx dt = limδ→0

∫ T

0

∫Ω

∇xχδ · ∇xχδ dx dt (6.40)

+∫ T

0

∫Ω

(%L′δ(χ)χ− %b′(χ)χ+ %(g2,δ(%)− g1,δ(%))χ

)dx dt.

Now, it follows from (6.38) and the uniform bounds (6.12), (6.13), and (6.17) that

%µχ = %µχ, %b′(χ)χ = %b′(χ)χ, and %(g2,δ(%)− g1,δ(%))χ = %(g2,δ(%)− g1,δ(%))χ.

At this stage, the above procedure must be modified as the terms L′δ(χδ) are nolonger uniformly bounded. We observe, however, that

limδ→0

∫ T

0

∫Ω

%δL′δ(χδ)χδ dx dt =

∫ T

0

∫Ω

%L′δ(χ)χ dx dt.

Indeed,∫ T

0

∫Ω

%δL′δ(χδ)χδ dx dt =

∫∫%>0

%δL′δ(χδ)χδ dx dt+

∫∫%=0

%δL′δ(χδ)χδ dx dt,

where, as a consequence of (6.38),∫∫%>0

%δL′δ(χδ)χδ dx dt→

∫∫%>0

%L′δ(χ)χ dx dt,

while, in accordance with (6.12), (6.21),∫∫%=0

|%δL′δ(χδ)χδ|dxdt ≤ ‖%1/p

δ L′δ(χδ)χδ‖Lp((0,T )×Ω)‖%δ‖1/p′

L1(%=0) → 0

as δ 0.In view of the previous arguments, we then arrive at

limδ→0

∫ T

0

∫Ω

|∇xχδ|2 dx dt =∫ T

0

∫Ω

|∇xχ|2 dx dt;

in other words,

∇xχδ → ∇xχ (strongly) in L2((0, T )× Ω; R3). (6.41)

To conclude this part, we remark that regularity properties (2.11) and (2.14) can berecovered a posteriori by testing (the limit of) (3.18) by a suitable truncation of W ′(χ)and then letting the truncation parameter go to 0 in a standard way (the use of atruncation seems necessary since W ′(χ) a priori could not have a sufficient regularityto be used as a test function).

24

6.5 Strong convergence of the density

In order to complete the proof of Theorem 2.1, we have to show that

p(%, χ) = p(%, χ)

in (6.33). To this end, we need strong (a.a. pointwise) convergence of %δδ>0.To begin, we use the renormalized continuity equation (5.7) to deduce (cf. [11, p.

140])∫Ω

(% log(%)− % log(%)

)(τ, ·) dx+

∫ τ

0

∫Ω

(%divxu− %divxu

)dx dt = 0 (6.42)

for any τ ≥ 0, where, as always, we have used the bar to denote weak limits in L1 ofsequences of composed functions.

On the other hand, as the densities %δ are bounded in L∞(0, T ;Lγ(Ω)), withγ large enough, we can use directly the procedure of Lions [19], together with thenecessary modifications introduced in [12] to handle the variable viscosity coefficients,to establish the “weak continuity” of the effective viscous pressure, in particular,∫ T

0

∫Ω

ξ

(43ν(χ) + η(χ)

) (%divxu− %divxu

)dx dt (6.43)

≥ lim infδ→0

∫ τ

0

∫Ω

ξ(pδ(%δ, χδ)%δ − p(%, χ)%

)dx dt for any ξ ∈ C∞c (Ω), ξ ≥ 0.

The proof of (6.43) is tedious but nowadays well understood (see [12]). The basicidea is to use multipliers of the form

ϕ(t, x) = ψ(t)ξ(x)∇x∆−1[1Ω%δ], ϕ(t, x) = ψ(t)ξ(x)∇x∆−1[1Ω%],

ψ ∈ C∞c (0, T ), ξ ∈ C∞c (Ω),(6.44)

as test functions in the momentum equation (2.12) and its asymptotic limit for δ → 0,respectively. Here, the symbol ∆−1 stands for the inverse of the Laplacian consideredon the whole space R3.

After a straightforward manipulation, we obtain∫ T

0

∂tψ

∫Ω

ξ%δuδ · ∇x∆−1[1Ω%δ] dx dt (6.45)

−∫ T

0

ψ

∫Ω

ξ%δuδ · ∇x∆−1divx[%δuδ] dx dt

+∫ T

0

ψ

∫Ω

%δ(uδ ⊗ uδ) : (∇xξ ⊗∇x∆−1[1Ω%δ]) dx dt

+∫ T

0

ψ

∫Ω

ξ%δ(uδ ⊗ uδ) : ∇x∇x∆−1[1Ω%δ] dx dt

+∫ T

0

ψ

∫Ω

pδ(%δ, χδ)∇xξ · ∇x∆−1[1Ω%δ] dx dt+∫ T

0

ψ

∫Ω

ξpδ(%δ, χδ)%δ dx dt

=∫ T

0

ψ

∫Ω

ν(χδ)[(∇xuδ +∇t

xuδ −23divxuδI

)∇xξ

]· ∇x∆−1[1Ω%δ] dx dt

25

+∫ T

0

ψ

∫Ω

ν(χδ)ξ(∇xuδ +∇t

xuδ −23divxuδI

): ∇x∇x∆−1[1Ω%δ] dx dt

+∫ T

0

ψ

∫Ω

η(χδ)divxuδ

(∇xξ · ∇x∆−1[1Ω%δ] + ξdivx∇x∆−1[1Ω%δ]

)dx dt

+∫ T

0

ψ

∫Ω

[(|∇xχδ|2

2I−∇xχδ ⊗∇xχδ

)∇xξ

]· ∇x∆−1[1Ω%δ] dx dt

+∫ T

0

ψ

∫Ω

ξ

(|∇xχδ|2

2I−∇xχδ ⊗∇xχδ

): ∇x∇x∆−1[1Ω%δ] dx dt,

and∫ T

0

∂tψ

∫Ω

ξ%u · ∇x∆−1[1Ω%] dx dt−∫ T

0

ψ

∫Ω

ξ%u · ∇x∆−1divx[%u] dx dt (6.46)

+∫ T

0

ψ

∫Ω

%(u⊗ u) : (∇xξ ⊗∇x∆−1[1Ω%]) dx dt

+∫ T

0

ψ

∫Ω

ξ%(u⊗ u) : ∇x∇x∆−1[1Ω%] dx dt

+∫ T

0

ψ

∫Ω

p(%, χ)∇xξ · ∇x∆−1[1Ω%] dx dt+∫ T

0

ψ

∫Ω

ξp(%, χ)% dx dt

=∫ T

0

ψ

∫Ω

ν(χ)[(∇xu +∇t

xu−23divxuI

)∇xξ

]· ∇x∆−1[1Ω%] dx dt

+∫ T

0

ψ

∫Ω

ν(χ)ξ(∇xu +∇t

xu−23divxuI

): ∇x∇x∆−1[1Ω%] dx dt

+∫ T

0

ψ

∫Ω

η(χ)divxu(∇xξ · ∇x∆−1[1Ω%] + ξdivx∇x∆−1[1Ω%]

)dx dt

+∫ T

0

ψ

∫Ω

[(|∇xχ|2

2I−∇xχ⊗∇xχ

)∇xξ

]· ∇x∆−1[1Ω%] dx dt

+∫ T

0

ψ

∫Ω

ξ

(|∇xχ|2

2I−∇xχ⊗∇xχ

): ∇x∇x∆−1[1Ω%] dx dt.

Now, relation (6.43) can be deduced by letting δ 0 in (6.45) and comparingthe resulting expression with (6.46). This hardly non-trivial step is the heart of theexistence theory for the barotropic Navier-Stokes system developed by Lions [19] andextended to variable viscosity coefficients in [12]. The reader may consult [3, Section3.3] for an adaptation of this method to the present problem. Let us only point outthat the main ingredient is the so-called commutator lemma:

Lemma 6.1 [see [12, Lemma 4.2]]Let w ∈ W 1,p(R3) and V ∈ L2(R3; R3), where p > 6/5. Then there exists ω =

ω(p) > 0 and q(p) > 1 such that∥∥∇x∆−1divx[wV]− w∇x∆−1divx[V]∥∥

W ω,q(R3;R3)≤ c(p)‖w‖W 1,p(R3)‖V‖L2(R3;R3).

26

Consequently, in order to show strong convergence of the densities, it is enoughto observe that

lim infδ→0

∫ τ

0

∫Ω

ξ(pδ(%δ, χδ)%δ−p(%, χ)%

)dx dt ≥ 0 for any ξ ∈ C∞c (Ω), ξ ≥ 0. (6.47)

Indeed, in case (6.47) holds, it follows that∫Ω% log(%)dx →

∫Ω% log(%)dx a.a. in

(0, T ) and so %δ → % a.a. in (0, T )× Ω.In view of (6.4) and strong convergence of χδδ>0 established in (6.41), relation

(6.47) follows as soon as we observe that

lim infδ→0

∫ τ

0

∫Ω

ξ(Pδ(%δ)%δ − P (%)%

)dx dt ≥ 0 for any ξ ∈ C∞c (Ω), ξ ≥ 0. (6.48)

To see (6.48), it is enough to realize that, due to (6.6), % 7→ Pδ(%) are non-decreasing (for % ≥ 0) for any fixed δ > 0, and Pα ≥ Pβ provided α ≤ β; therefore

lim infδ→0

∫ T

0

∫Ω

ξ(Pδ(%δ)%δ−P (%)%

)dx dt ≥ lim inf

δ→0

∫ T

0

∫Ω

ξ(Pβ(%δ)%δ−P (%)%

)dx dt

≥∫ T

0

∫Ω

ξ(Pβ(%)− P (%)

)% dx dt for any fixed β,

where the last inequality follows from monotonicity of Pβ .However,

Pβ(%)− P (%) → 0 in L1((0, T )× Ω) for β → 0,

as a direct consequence of the equi-integrability property of the pressure establishedin (6.30).

Summing up relations (6.42 -6.47) we conclude that

%δ → % a.a. in (0, T )× Ω and p(%, χ) = p(%, χ),

which completes the proof of Theorem 2.1.

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