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Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces

This work is devoted to the global existence of weak solution for a general isothermal model of capillary fluids derived by C. Rohde, which can be used as a phase transition model. This article is structured in the following way: first of all inspired by the result by P.-L. Lions on the Navier-Stokes compressible system we will show the global stability of weak solutions for our system with isentropic pressure and next with general pressure. Next we will consider perturbations close to a stable equilibrium as in the case of strong solutions.


Presentation of the model
The correct mathematical description of liquid-vapor phase interfaces and their dynamical behavior in compressible fluid flow has a long history. We are concerned with compressible fluids endowed with internal capillarity. One of the first model which takes into consideration the variation of density on the interface between two phases, originates from the XIXth century work by Van der Waals and Korteweg [13]. It was actually derived in his modern form in the 1980s using the second gradient theory, see for instance [12,18]. Korteweg suggests a modification of the Navier-Stokes system to account additionally for phase transition phenomena in introducing a term of capillarity. He assumed that the thickness of the interfaces was not null as in the sharp interface approach. This is called the diffuse interface approach. Korteweg-type models are based on an extended version of nonequilibrium thermodynamics, which assumes that the energy of the fluid not only depends on standard variables but on the gradient of the density. In terms of the free energy, this principle takes the form of a generalized Gibbs relation, see [18]. In the present chapter, we follow a new approach introduced by Coquel, Rohde and their collaborators in [4]. They remark that the local diffuse interface approach requires more regular solution than in the original sharp interface approach. Indeed the interfaces are assumed of non zero thickness, so that the density vary continuously between the two interfaces, whereas in the sharp interface models, the interfaces represent zone of discontinuity for the density. Coquel, Rohde and their collaborators present an alternative model with a capillarity term which avoids spatial derivatives. The model reads: ∂ t ρ + div(ρu) = 0 ∂ t (ρ u) + div(ρu ⊗ u) − µ∆u − (λ + µ)∇divu + ∇(P (ρ)) = κρ∇D [ρ] (ρ t=0 , u t=0 ) = (ρ 0 , u 0 ) with: µ > 0 and λ + 2µ > 0 where ρ denotes the density of the fluid and u ∈ R N the velocity, µ and λ represent the viscosity coefficients, κ is a coefficient of capillarity, P is a general pressure function. We are particulary interested by Van der Waals type pressure: where a, b, R, T * are positive constants, R being the specific gas constant. For fixed values a, b we choose the constant reference temperature T * so small as P to be monotone decreasing in some non-empty interval. Further we impose the conditions: In last section, we consider also more general situations: monotone pressure, other boundary conditions than boundary condition 1.1, at infinity instead of 1.1: u(t, x) → u ∞ , ρ(t, x) → ρ ∞ as |x| → +∞, (1.2) where ρ ∞ is a given nonnegative constant. The term κρ∇D[ρ] corresponds to the capillarity which is supposed to model capillarity effects close to phase transitions in [13]. The classical Korteweg's capillarity term is D[ρ] = ∆ρ. Based on Korteweg's original ideas Coquel, Rohde and their collaborators in [4] and Rohde in [17] choose a nonlocal capillarity term D which penalizes rapid variations in the density field close from the interfaces. They introduce the following capillarity term: where φ is chosen so that: φ(x)dx = 1, φ even, and φ ≥ 0.
This choice of capillarity term allows to get solution with jumps, i.e with sharp interfaces. Before tackling the global stability theory for the system (N SK), let us derive formally the uniform bounds available on (ρ, u).
Notice that Π is convex. Multiplying the equation of momentum conservation by u and integrating by parts over R N , we obtain the following energy estimate: where we have: The only non-standard term is the energy term E global which comes from the product of u with the capillarity term κρ∇(φ * ρ − ρ). Indeed we have: x)([φ * ρ(t, ·))](x) − ρ(t, x))dx, To derive the last equality we use the relation: where we just use integration by parts. In the sequel we will note: We are interested to use the above inequality energy to determine the functional space we must work with. So if we expand E global [ρ(., t)](x) we get: Because by the mass equation we obtain that ρ is bounded in L ∞ (0, T ; L 1 (R N )) if we suppose that ρ 0 ∈ L 1 and we have supposed that φ ∈ L ∞ (R N ), we obtain that ρ (φ * ρ) is bounded in L ∞ (0, T ; L 1 (R N )). So we get that ρ 2 + φ * ρ 2 ∈ L ∞ (0, T ; L 1 (R N )) and as φ ≥ 0 and ρ ≥ 0 we get a control of ρ in L ∞ (0, T ; L 2 (R N )) (a property which turns out to be important to taking advantage of the theory of renormalized solutions, indeed ρ in L ∞ (0, T ; L 2 (R N )) implies that ρ ∈ L 2 loc (R + × R N ) and we will can use the theorem of Diperna-Lions on renormalized solutions, see [14]) In view of (1.4), we can specify initial conditions on ρ /t=0 = ρ 0 and ρu /t=0 = m 0 where we assume that: • m 0 = 0 a.e on ρ 0 = 0, (1.6) We deduce the following a priori bounds which give us the energy space in which we will work: We will use this uniform bound in our result of compactness. Let us emphasize at this point that the above a priori bounds do not provide any control on ∇ρ in contrast with the case of D[ρ] = ∆ρ (see in [6]).

Notion of weak solutions
We now explain what we mean by renormalized weak solutions, weak solutions, and bounded energy weak solution of problem (N SK). Multiplying mass equation by b ′ (ρ), we obtained the so-called renormalized equation (see [14]): with growth conditions at infinity: is called a renormalized weak solution of problem (N SK) if we have: • Equation of mass holds in D ′ (R N ).
Definition 1.2 Let the couple (ρ 0 , u 0 ) satisfy; where the quantity Π is defined in (1.3). We have the following definitions: is called a bounded energy weak solution of problem (N SK) if in addition to (1d), (1e), (1f), (1g) we have: • The quantity E 0 is finite and inequality (1.4) with E defined by (1.5) and with E 0 in place of E(ρ(0), ρu(0)) holds a.e in R.

Mathematical results
We wish to prove global stability results for (N SK) with D[ρ] = φ * ρ − ρ in functional spaces very close to energy spaces. In the non capillary case and P (ρ) = aρ γ , P-L. Lions in [15] proved the global existence of weak solutions (ρ, u) to (N SK) with κ = 0 (which become the system of Navier- if N = 2, 3 and initial data (ρ 0 , m 0 ) such that: where we agree that m 0 = 0 on {x ∈ R N / ρ 0 (x) = 0}. More precisely, he obtains the existence of global weak solutions (ρ, u) to (N SK) with κ = 0 such that for all t ∈ (0, +∞): • ρ ∈ L ∞ (0, T ; L γ (R N )) and ρ ∈ C([0, T ], L p (R N )) if 1 ≤ p < γ, Notice that the main difficulty for proving Lions' theorem consists in exhibiting strong compactness properties of the density ρ in L p loc (R + × R N ) spaces required to pass to the limit in the pressure term P (ρ) = aρ γ . Let us mention that Feireisl in [8] generalized the result to γ > N 2 in establishing that we can obtain renormalized solution without imposing that ρ ∈ L 2 loc (R + × R N ) (what needed Lions in dimension N = 2, 3, that's why γ − 1 + 2γ N ≥ 2), for this he introduces the concept of oscillation defect measure evaluating the loss of compactness. We refer to the book of Novotný and Straškraba for more details (see [16]). Let us mention here that the existence of strong solution with D[ρ] = ∆ρ is known since the work by Hattori an Li in [9], [10] in the whole space R N . In [6], Danchin and Desjardins study the well-posedness of the problem for the isothermal case with constant coefficients in critical Besov spaces. We recall too the results from Rohde in [17] who obtains the existence and uniqueness in finite time for two-dimensional initial data in In the present chapter, we aim at showing the global stability of weak solutions in the energy spaces for the system (N SK). This work is composed of four parts, the first one concerns estimates on the density to get a gain of integrability on the density needed to pass to the weak limit in the term of pressure and of capillarity. The second part is the passage to the weak limit in the non-linear terms of the density and the velocity according to Lions' methods. The idea is to use renormalized solution to test the weak limit on convex test functions. In this part we will concentrate on the case of simple pressure of type P (ρ) = aρ γ . We get the following theorem where (ρ n , u n ) n∈N is a suit of global regular approximate solution for the problem (N SK).
In addition we suppose that ρ 0 n converges in L 1 (R N ) to ρ 0 , then up to a subsequence (ρ n , u n ) converges strongly to a weak solution (ρ, u) of the system (N SK) satisfying the initial condition (ρ 0 , u 0 ) as in (1.6). Moreover we have the following convergence: In addition we have: • ρ n u n → ρu in L p (0, T ; L r (R N )) for all 1 ≤ p < +∞ and 1 ≤ r < 2s s+1 , In the third part we will focus on general pressure, and particulary van der Waal's pressure. In the fourth part we concentrate on the case with initial data close to a constantρ, and we will work in Orlicz space, this case is the most adapted for the strong solution because it enables us to control the vacuum so that one can use the property of ellipticity of the momentum equation.
2 Existence of weak solution for a isentropic pressure law

A priori estimates on the density
In this part we are interested by getting a gain of integrability on the density and we consider the case where P (ρ) = aρ γ . This will enable us to pass to the weak limit in the pressure and the Korteweg terms. It is expressed by the following theorem: with M depending only on the initial conditions and on the time T . If γ < N 2 (1 + 1 N ) for N ≥ 4, we have: for any arbitrary compact K with M ′ depending only on the initial conditions, on K and on the time T .

Proof:
We will begin with the case where N ≥ 3 and we treat after the specific case N = 2.
Case N ≥ 3: We apply to the momentum equation the operator (−∆) −1 div in order to concentrate us on the pressure and we get: (2.10) and in multiplying by ρ ε with 0 < ε ≤ min( 1 N , 2 N γ − 1) to estimate ρ γ+ε , we get: (2.11) where we note ξ = λ + µ. We now rewrite the previous equality as follows: Next we integrate (2.12) in time on [0, T ] and in space so we obtain: (2.13) where R i is the classical Riesz transform. Now we want to control the term T 0 R N ρ γ+ε + κ 2 ρ 2+ε dxdt. As ρ is positive, it will enable us to control ρ L γ+ε t,x and ρ L 2+ε . This may be achieved by controlling each term on the right side of (2.13).
We start with the term |divu|ρ ε |(−∆) −1 div(ρu)| where we have: We are in a critical case for the Sobolev embedding ( i.e r ≥ N ) only when N = 3 and γ ≥ 6, that's why for N = 3 and γ ≥ 6. So by Hölder inequalities we get We now treat the case N = 3 and γ ≥ 6 where we choose in this case ε = 2 N γ − 1 to explicit precisely this case: We now want to treat the term: by Hölder inequalities and the fact that we have ρ ∈ L ∞ (L 1 ) and ∇φ ∈ L 1 . After we get that ρ ε (−∆) −1 div(ρ∇(φ * ρ)) ∈ L ∞ (L r 1 ) with: 1 We conclude that ρ ε (−∆) −1 div(ρ∇(φ * ρ)) is L ∞ (L 1 ) in using interpolation when N = 2, 3. Indeed we have ρ∇(φ * ρ) ∈ L ∞ (L 1 ) and in choosing ε = 2 N s − 1 we have: . This is the case when N = 2, 3, and this is the case when N ≥ 4 and γ ≥ N 2 (1 + 1 N ). In the other case we need to work in arbitrary compact. We have after the term (div(u))ρ ε . We recall that ρ ε is in , the bound is obvious because 1 2 + ε ≥ 1 and 1 2 + ε s < 1, we can then conclude by interpolation. On the other hand, this rather simple term presents a technical difficulty when ε ≤ 1 2 since we do not know in that case if divu ρ ε ∈ L 1 (R N × (0, T )). One way to get round the difficulty is to multiply (2.10) by ρ ε 1 {ρ≥1} . Then we obtain an estimate We end with the following term ). In the same way than in the previous inequalities we have ). Indeed we have by Hölder inequalities and the fact that . And we conclude by interpolation. We treat the term ρ ε u i R i R j (ρ u j ) similarly.
We have to treat now the case N = 2 where we have to modify the estimates when we are in critical cases for Sobolev embedding.
Case N = 2: In the case N = 2 most of the proof given above stay exact except for the slightly more delicate terms We start with the term |ρ e divu(−∆) −1 div(ρu)|. In our previous estimate it was possible to use Sobolev embedding on the term (−∆) −1 div(ρu)| only if r ≥ N (see above the notation), so in the case where N = 2 we are in a critical case for the Sobolev embedding when γ ≥ 2. This may be overcome by using that, by virtue of Sobolev embedding, we have: . We use the fact that u is bounded in L 2 (0, T ;Ḣ 1 ) and thus in L 2 (0, T ; BM O). Then by the Coifman-Rochberg-Weiss commutator theorem in [3], we have for almost all t ∈ [0, T ]: So we have: In view of the previous inequalities we get finally: and the L γ+ε (L γ+ε ) bound on ρ is proven since 1 2 + ε < γ + ε.

Compactness results for compressible Navier-Stokes equations of
Korteweg type in the case of isentropic pressure In the sequel we are not going to treat in details the case with N ≥ 4 and γ < N 2 (1 + 1 N ), we just remark that the proof is the same as in the case N = 2, it suffices to localize because we can only apply the theorem 2.2 on the gain of integrability on any compact K. So let follow the theorem 2.2 and assume that γ > We can observe that in this case q > s = max(γ, 2). We will see that it will be very useful in the sequel to justify the passage to the weak limit in some terms to get a gain of integrability on the density. Indeed the key point to proving the existence of weak solutions is the passage to the limit in the term of pressure and in the term of capillarity ρ∇(φ * ρ − ρ). First, we assume that a sequence (ρ n , u n ) n∈N of approximate weak solutions has been constructed by a mollifying process, which have suitable regularity to justify the formal estimates like the energy estimate and the theorem 2.2. (ρ n , u n ) n∈N has the initial data of the theorem 1.1 with uniform bounds, i.e: Moreover (ρ n , u n ) n∈N verifies the energy inequalities (1.4) and the previous theorem 2.2, we have then: And we have in passing to the weak limit in the previous bound in extracting subsequence if necessary: • u n → u weakly in L 2 (0, T,Ḣ 1 (R N )), Notation 1 We will always write in the sequel B(ρ) to mean the weak limit of the sequence B(ρ n ) bounded in appropriate space that we will precise.
We recall that the main difficulty will be to pass to the limit in the pressure term and the capillary term. The idea of the proof will be to test the convergence of the sequence (ρ n ) n∈N on convex functions B in order to use their properties of lower semi-continuity with respect to the weak topology in L 1 (R N ). In this goal we will use the theory of renormalized solutions introduced by Diperna and Lions in [?]. So we will obtain strong convergence of ρ n in appropriate spaces.

Idea of the proof
We here give a sketchy proof of the theorem 1.1. In this spirit we can rewrite mass conservation of the regular solution (ρ n , u n ) n∈N on the form: Supposing that B(ρ n ) is bounded in appropriate space we can pass to the weak limit where we have in the energy space ρ n ⇀ ρ and u n ⇀ u, so we get: . Next in seeing the mass equation for approximate solutions (ρ n , u n ) n∈N , and in passing directly to the limit via the weak convergence arguing like P-L. Lions in [15] p 13 we get: After we will just have to verify the passage to the limit for the product ρu. Next we will use the theorem on the renormalized solutions of Diperna-Lions in [14] on (2.15) in recalling that ρ ∈ L ∞ (L 2 ). So we get: Next we subtract (2.14) to (2.16), so we obtain: Consequently, in order to estimate the difference B(ρ)−B(ρ) which tests the convergence of ρ n , we need to estimate the difference b(ρ)div(u) − b(ρ)div(u). We choose then B a concave function and we have: The goal will be now to prove the reverse inequality in order to justify that B(ρ n ) tends to B(ρ) a.e. So now we aim at estimating the difference b(ρ)divu − b(ρ)divu. This may be achieved by introducing the effective viscous pressure P ef f = P − (2µ + λ)divu after D. Hoff in [?], which satisfies some important properties of weak convergence.
In fact owing to the capillarity term we adapt Hoff's concept to our equation in setting: Proof of theorem 1.1 We begin with the case N ≥ 3, and next we will complete the proof by the case N = 2 in specifying the changes to bring. Before getting into the heart of the proof, we first recall that we obtain easily the convergence in distribution sense of ρ n u n to ρu and ρ n (u n ) i (u n ) j to ρu i u j . We refer to the classical result by Lions (see [15]) or the book of Novotný and Straškraba [16].
We have seen in the idea of the proof 2.3 that our goal is to compare B(ρ) and B(ρ) for certain concave functions B . From the mass equation we have obtained: So before comparing B(ρ) and B(ρ), we have to investigate the expression b(ρ)div(u) − b(ρ)div(u). By virtue of theorem 2.2 which gives a gain of integrability we can take the function B(x) = x ε , as we control for ε small enough ρ s+ε . Our goal now is to exhibit the effective pressure P ef f , and to multiply it by ρ ε to extract divu b(ρ) . We will see in the sequel how to compare it with b(ρ)div(u). So we focus on the convergence of the pressure and capillarity terms.

Control of the term divu b(ρ)
So we take the div on the momentum equation satisfied by the regular solution. We get: (2.20) After we multiply (2.20) by ρ ε n with ε that we choose enough small with ε ∈ (0, 1): (2.21) So if we rewrite (2.21), we have: Next we have: (2.22) or finally: (2.23) Now like in Lions [?] we want to pass to the limit in the distribution sense in (2.23) in order to estimate divu(ρ) ε .
Passage to the weak limit in (2.23) In this goal we use the following lemma of P-L Lions in [15] to express the weak limit of the non-linear terms.
So we use the above lemma to pass to the weak limit in the four following non-linear terms of (2.23): . So we choose the different g i n and h i n as follows: To show that u n (ρ n ) ε converges in distribution sense to uρ ε we apply easily lemma 1 with h n = u n and g n = ρ ε n . We now want to examine each term and apply the above lemma to pass to the limit in the weak sense. We start with the first term T 1 n . We have that ρ ε n u n ∈ L ∞ (L q ) ∩ L 2 (L r ) with 1 q = ε 2s + 1 2 In addition the hypothesis (2.24) is immediately verified (use the momentum equation).
We now want to verify the hypothesis (2.25), so we have h 1 n belongs to This result enables us to verify the hypothesis (2.25) by Sobolev embedding.
Finally we have to study the last non linear following term that we treat similarly as P-L.Lions in [15]: We can express this term A n as follows: where R ij = (−∆) −1 ∂ 2 ij with R i the classical Riesz transform. Next, we use a result by Coifman, Meyer on this type of commutator (see [5]) to take advantage of the regularity of [u j n , R ij ](ρ n u i n ). Theorem 2.3 The following map is continuous for any N ≥ 2: with: 1 r 3 = 1 r 1 + 1 r 2 . To pass to the weak limit in A n we will use the previous lemma. We start with the case with s > 3. This quantity belongs to the space L 1 (W 1,q ) provided that Du n ∈ L 2 (L 2 ) and ρu j ∈ L 2 (L r ) where 1 r = N −2 2N + 1 s = 1 2 − 1 N + 1 s in which case 1 q = 1 r + 1 2 = 1 − 1 N + 1 s ≤ 1. for After we can use the above lemma applied to h n = [R ij , u j n ](ρ n u i n ) and g n = ρ ε n . We can show easily in using again lemma 1 that h n converges in distribution sense to [R ij , u j ](ρ n u i ). So we can take: q 1 = 1, p 1 = +∞ and q 2 ∈ (q, qN N −q ), p 2 = 1 − 1 q 2 , this one because we can use interpolation and we can localize as we want limit in distribution sense. In the case where s ≤ 3, a simple interpolation argument can be used to accommodate the general case. It suffices to fix L r 2 (R N ) in the application (2.26) and use a result of Riesz-Thorin. Finally according to the equation (2.21), and after passing to the limit we get: Inequality between the terms ρ ε divu and divu ρ ε Now we are interested in estimating the term ρ ε divu in order to describe the quantity ρ ε divu − divu ρ ε before considering the quantity ρ ε divu − divu ρ ε . We pass to the weak limit directly in (2.20) and we get in using again the lemma ??: (2.28) Now we just multiply (2.28) with ρ ε and we can see that each term has a distribution sense. So we get in proceeding in the same way as before: (2.29) In subtracting (2.29) from (2.27), we get: Next we observe that by convexity: So we get: Comparison between ρ and ρ ε 1 ε As on since (ρ n , u n ) are regular solutions we get the equality (2.16) applied to B(x) = x ε . So we get: And after we pass to the weak limit in (2.31) and we get: When we pass to the weak limit in combining with (2.30) we get: Now we wish to conclude about the pointwise convergence of ρ n in proving that (ρ ε ) 1 ε = ρ and to finish we will use the following theorem (see [8] p 34) applied to B(x) = x 1 ε which is convex.

Then:
v n (y) → v(y) a.e. extracting a subsequence as the case may be.
Now we want to use a type of Diperna-Lions theorem on inequality (2.33). Our goal is to renormalize this inequality with the function B(x) = x 1 ε so that one can compare ρ and ρ ε 1 ε . Although (2.33) doesn't correspond exactly to the mass equation, we can use the same technics to renormalize the solution provided that ρ ∈ L ∞ (L 2 ) which is the case. In our case it is very important that ρ ∈ L ∞ (L 2 ), indeed it avoids to have supplementary conditions on the index γ like for the compressible Navier-Stokes system in [15]. We recall of Diperna-Lions theorem on renormalized solution for the mass equation.
Theorem 2.5 Suppose that ρ ∈ L ∞ (L 2 ) and β ∈ C 1 ([0, ∞); R). We have then: We now want to adapt this theorem for our equation (??) with β(x) = x 1 ε , so we may regularize by ω α (with ω α = 1 α N ω( · α ) where ω ∈ C ∞ 0 (R N ), supp ω ∈ B 1 and ωdx = 1) and find for all β ∈ C ∞ 0 ([0, +∞)): where we have: After we pass to the limit when α → 0 and we see that R α tends to 0 in using lemma on regularization in [14] p 43. This looks like a rather innocent manipulation but it's at this point that we require to control ρ in L 2 (0, T ; R N ). And in our case we don't need to impose γ > N 2 for N = 2, 3. Hence: We 2], and we obtain: For proving that, we notice that by convexity (ρ) ε 1 ε ≤ ρ, so we get : We have concluded by dominated convergence. We have concluded by dominated convergence.
At this stage we subtract the mass equation to (2.34) and we get in setting r = ρ − (ρ) ε We now want to integrate and to use the fact that r ≥ 0 to get that r = 0 a.a. To justify the integration we test our inequality against a cut-off function of the form ϕ( · R ) where ϕ ∈ C ∞ 0 (R N ), ϕ = 1 on B(0, 1), Suppϕ ⊂ B(0, 2) and R > 1. We test the equation (2.35) against ϕ R and we get: Next we notice that: It implies that: We have then: Now we want to verify that r(0, ·) = 0 to conclude because. Indeed we will obtain that: then r = 0. We know that ρ n is uniformly bounded in L ∞ (L 1 ∩L s (R N )), then ρ ε n is relatively compact in C([0, T ]; L p − w) with 1 < p < s (where L p − w denote the space L p endowed with weak topology). Moreover (ρ ε 0 ) n converges to ρ ε 0 , we deduce then r(0) = 0 a.a. Now as r = 0 we conclude in using the theorem 2.4 ρ n converges a.a to ρ and that ρ n converges to ρ in L p ([0, T ] × B R ) for all p ∈ [1, q) and in L p 1 (0, T, L p 2 (B R )) for all p 1 ∈ [1, +∞), p 2 ∈ [1, s) and for all R ∈ (0, +∞).

Conclusion
We wish now conclude and get the convergence of our theorem in the total space. We aim at proving here the convergence of We have just to show the convergence of ρ n to ρ in C([0, T ], L 1 (R N )). To this end, we introduce d n = √ ρ n which clearly converges to √ ρ in ρ for all R ∈ (0, +∞). We next remark that ρ ∈ C([0, T ], L 1 (R N )) and thus d ∈ C([0, T ], L 2 (R N ). Indeed, using once more the regularization lemma in [14] we obtain the existence of a bounded ρ α ∈ C([0, T ], L 1 (R N )) smooth in x for all t satisfying: with r α = div(uρ α ) − div(ρu) * w α (where w is defined as in the previous part).
From these facts, it is straightforward to deduce that: and thus: Since ρ ∈ C([0, T ], L p (B R ) − w) (for all R ∈ (0, +∞), 1 < p < s), we may then deduce that ρ α converge to ρ in C([0, T ], L 1 (R N )). Next, we observe that we can justify as we did above that d n and d satisfy: Therefore once more, Thus in order to conclude, we just have to show that whenever t n ∈ [0, T ], t n → t, . This is the case since we deduce from the mass equation, integrating this equation over R N and justifying the integration exactly like previously that: We then conclude by uniform continuity that ρ n (t n ) − ρ n (t) L 1 tends to 0.
First of all, the main difficulty is the fact that we no longer have global L p bounds on u n . That's why most of the proof is in fact local and we know that u n is bounded in L 2 (0, T ; L p (B R )) for all p ∈ [1, +∞), R ∈ (0, +∞). As we need to localize the argument, we get the following limit: let ϕ ∈ C ∞ 0 (R N ), 0 ≤ ϕ ≤ 1, suppϕ ⊂ K for an arbitrary compact set K ∈ R N . We apply the operator (∆) −1 div to the momentum equation that we have localized and we pass directly to the weak limit: (2.37) with: Now we multiply (2.37) with ρ ε and we verify that each term has a sense. So we get in proceeding in the same way as before, we can verify that (ρ n ) ε (−∆) −1 R n converges in distribution sense to ρ ε (−∆) −1 R for small enough ε. We get as in the previous case for N ≥ 3: We then deduce the following inequalities as in the previous proof: We see that the only point left to check is the justification of the integration over R 2 of terms like div(ρ ε 1 ε u) or div(ρu) and more precisely that the integral vanishes. This is in fact straightforward provided we use the bounds on ρ ∈ L ∞ (L 1 (R N )) and ρ|u| 2 ∈ L ∞ (L 1 (R N )) and so ρu ∈ L ∞ (L 1 (R N )). Then, letting ϕ ∈ C ∞ 0 (R 2 ), 0 ≤ ϕ ≤ 1, ϕ = 1 on B(0, 1) and ϕ = 0 on c B(0, 2). We set ϕ R (·) = ϕ( · R ) for R ≥ 1, we have similarly as in the previous case: We can then conclude as in the previous proof.
Proof of the convergence assertion on ρ n u n We now want to show the convergence of ρ n u n to have informations on strong convergence of u n modulo the vacuum. We recall in this part some classical inequalities to get the convergence of ρ n u n , for more details see Lions in [15] We use once more a mollifier where k ∈ C +∞ 0 (R N ) and we let g α = g * k α for an arbitrary function g. We first observe that we have for all N 2 < p < s: We have in using Hölder inequalities with the measure k α (x − y)dy: Hence for all t ≥ 0 converge to 0 as α goes to 0 + uniformly in n. In addition, the convergence on ρ n assure that sup |z|≤α ρ n (·+ z) − ρ n L p converge to 0 as α goes to 0 + uniformly in n. Therefore in conclusion, (ρ n u n ) α − ρ n u n converge to 0 in L 2 (0, T ; L 1 ) as α goes to 0 + uniformly in n. Next (ρ n u n ) α is smooth in x, uniformly in n and in t ∈ [0, T ]. Therefore, remarking that ∂ ∂t (ρ n u n ) α is bounded in a L 2 (0, T ; H m ) for any m ≥ 0, we deduce that (ρ n u n ) α converge to (ρu) α as n goes to +∞ in L 1 ((0, T ) × R N ) for each α. Then using the bound on ρ n u n in L ∞ (L 2s s+1 ), we deduce that ρ n u n converge to ρu in L 1 ((0, T ) × R N ) and we can conclude by interpolation. The last convergence is a consequence of the strong convergence of ρ n and ρ n u n .

Existence of weak solution with general pressure
In the sequel we concentrate us only on the cases N = 2, 3. We now want to extend our previous result to more general and physical pressure laws. In particular we are now interested by two cases, the first one concerns monotonous pressure law (close in a certain sense that we will precise to ρ γ pressure) , the second one is the case of a slightly modified Van der Waals pressure. The technics of proof will be very similar to the previous proof, only technical points change.

Monotonous pressure
In this section, we shall investigate an extension of the preceding results to the case of a general monotonous pressure P , i.e P is assumed to be a C 1 non-decreasing function on [0, +∞) vanishing at 0. We want here to mention in the general situation our new energy inequality, we recall the inequality (1.4): where we define Π by: ∂ ∂t ( Π(t) t ) = P (t) t 2 for all t > 0. There are two cases worth considering: first of all if P (t) is such that 1 0 P (s) s 2 ds < +∞ then we can choose Π(ρ) = ρ ρ 0 P (s) In the other case, i.e lim t→0 P (t) t = c > 0, we can choose Π(ρ) = ρ 1 ρ p(s) s 2 ds and Π behaves like ρ log ρ as ρ goes to 0. We now consider a sequence of solutions (ρ n , u n ) and we make the same assumptions on this sequence as in the previous section except that we need to modify the assumptions on ρ n . We assume always that ρ n is bounded in C([0, T ], L 1 (R N )), P (ρ n ) is bounded in L ∞ (0, T, L 1 (R N )).
and we also assume that we have: (ρ ε n P (ρ n )) n≥1 is bounded in L 1 (K × (0, T )) for some ε > 0, where K is an arbitrary compact set included in R N .

Proof:
In this situation, the proof of theorem 3.6 is based on the same compactness argument as in the theorem 1.1. In particular, there is essentially one observation which allows us to adapt the proof of theorem 1.1. Namely we stills obtain the following identity for the effective viscous flux: with β(ρ) = ρ ε for ε small enough. We have then: Now we can recall a lemma coming from P.-L.Lions in [15]: Lemma 2 Let p 1 , p 2 ∈ C([0, ∞)) be non-decreasing functions. We assume that p 1 (ρ n ), p 2 (ρ n ) are relatively weakly compact in L 1 (K × (0, T )) for any compact set K ⊂ R N . Then, we have: We get finally as in the proof of theorem 1.1 in using lemma 2: All remaining argumentation of the proof of theorem 1.1 can be performed to conclude.

Pressure of Van der Waals type
In this section we are interested by pressure of type Van der Waals which consequently are not necessarily non-decreasing. That's why in the following proof we will proceed slightly differently. So we consider the pressure law: and we extend the function P to be strictly increasing function on ρ ≥ b − θ.
We have then: 1. P ′ is bounded from below, that is: 2. P is a strictly increasing function for ρ large enough.
Under the above conditions, it is easy to see that the pressure can be written as: with P 1 a non-decreasing function of ρ, and

Remark 1
The a priori energy estimate give us the bound of ρ in L ∞ (L 2 ), then we have: Then we can controll in measure the set where P is different from the Van der Waals measure, and it's a set of finite measure.
We obtain the following theorem.
Theorem 3.7 If in addition to the above assumptions, we assume that ρ 0 n converges in L 1 (R N ) to ρ 0 then (ρ, u) is a weak solution of the system (N SK) satisfying the initial condition and we have:

Proof:
Most of the proof of theorem 3.7 is similar as theorem 1.1. We will use a approximated sequel T k (introduced by Feireisl in [?]) of ρ by some concave bounded function.

Definition 3.3
We define the function T ∈ C ∞ (R N ) as follows: T (z) concave on [0, +∞), And T k is the cut-off function: In following the proof of theorem 1.1, we get: where L k (ρ) = ρ log(ρ) for 0 ≤ ρ ≤ k, and 0 ≤ L k (ρ) ≤ ρ log(ρ) otherwise. So we get in integrating in time on [t 1 , t 2 ]: We can show that: For poving the previous inequality, we see that T k (ρ) − T k (ρ) L 1 ((0,T )×R N ) → 0 for k → +∞. We then conclude by interpolation with T k (ρ) − T k (ρ) ∈ L q (0, T ) × R N ) with q > 2. By Hölder inequality we obtain that: We have then: (3.39) We set: We can show as in the previous proof of theorem 1.1 that: for any compact K ⊂ R N .

Weak solutions with data close to a stable equilibrium
We consider in this section one situation which is rather different from the three cases considered in the preceeding sections. This situation is relevant for practical applications and realistic flow and they involve conditions at infinity different from those studied. We wish to investigate the system (N SK) with hypothesis close from these of the theorem for strong solutions. We want then to study the system with a density close from a stable equilibrium in the goal to can choose initial data which avoid the vacuum. We look now for a solution (ρ, u) defined on R × R N of the system (N SK) (where P (ρ) = aρ γ ) with ρ ≥ 0 on R × R N . In addition we require (ρ, u) to satisfy the following limit conditions: Such an analysis requires the use of the Orlicz spaces. We define the Orlicz space L q p (R N ) as follows: For the following properties, we can give this definition too.

Definition 4.4
We define Ψ as a convex function on [0, +∞) which is equal (or equivalent) to x p for x small and to x q for x large.
We can check that L q p (R N ) is a linear space. Now we endow L q p (R N ) with a norm so that L q p (R N ) is a separable Banach space.
We recall now some useful properties of the Orlicz space: Corollary 1 We have: 1. Embedding: 2. Topology: f n → n 0 in L q p (R N ) if and only if Ψ(f n ) → n 0 and that: We recall now some useful properties of the Orlicz space: Proposition 4.1 We then have: • Let F be a continuous function on R such that F (0) = 0, F is differentiable at 0 and F (t)|t| −θ → α = 0 at t → +∞. Then if q ≥ θ, Our goal is now to get energy estimate. We have to face a new difficulty. Indeed ρ, ρ|u| 2 , ρ γ need not belong to L 1 . We first want to explain how it is possible to obtain natural a priori bounds which correspond to energy-like identities. Next we write the following formal identities: We may then integrate in space the equality (4.40) and we get: Notation 2 In the sequel we will note: Now we can recall a theorem (see [15]) on the Orlicz space concerning this quantity: In following this theorem and our energy estimate we get that ρ −ρ ∈ L ∞ (0, T ; L γ 2 (R N )) for all T ∈ R. Moreover we have: Then in using the fact that ρ −ρ ∈ L ∞ (0, T ; L γ 2 (R N )) ֒→ L 2 (R N ) + L γ (R N ) and interpolation on ∇φ, we get that ρ −ρ ∈ L ∞ (0, T ; L 2 (R N )).
We may now turn to our compactness result. First of all, we consider sequences of solutions (ρ n , u n ) of the system corresponding to some initial conditions (ρ 0 n , u 0 n ). In using the above energy inequalities, we assume that j γ (ρ 0 n ), E global [ρ 0 n −ρ] and ρ 0 n |u 0 n | 2 are bounded in L ∞ (L 1 (R N )) and that ρ 0 n −ρ converges weakly in L γ 2 (R N ) to some ρ 0 −ρ.
Extracting subsequences if necessary, we may assume that ρ n , u n converge weakly respectively in L 2 ((0, T ) × B R ), L 2 (0, T ; H 1 (B R )) to ρ, u for all R, T ∈ (0, +∞). We also extract subsequences for which ρ n u n , ρ n u n ⊗ u n converge weakly as previously.

Remark 2
We notice that the conditions at infinity are implicitly contained in the fact that (ρ n −ρ) 2 and ρ n |u n | 2 ∈ L 1 (R N ).
We then have the following theorem.
From these equations, we deduce as in the proof of the previous theorem, that ϕ 2 ρ ∈ C([0, +∞), L 1 (R N )), ϕ √ ρ ∈ C([0, +∞), L 2 (R N )) and that ϕ √ ρ n converges weakly in L 2 (R N ), uniformly in t ∈ [0, T ]. Therefore, in order to conclude, we just have to show that we have: whenever t n ∈ [0, T ], t n → nt , and this is straightforward since we have, in view of the above equation: