Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities

In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu [Inventiones mathematicae (2016) and arXiv:1501.06803 (2015)] and by Li-Xin [arXiv:1504.06826 (2015)].More precisely we are able to consider a physical symmetric viscous stress tensor $\sigma=2\mu(\rho)\,{\mathbb{D}}(u)+\bigl(\lambda(\rho){\rm div}u -P(\rho)\bigr)\, {\rm Id}$ where ${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$ with a shear and bulk viscosities (respectively $\mu(\rho)$ and $\lambda(\rho)$) satisfying the BD relation $\lambda(\rho)=2(\mu'(\rho)\rho - \mu(\rho))$ and a pressure law $P(\rho)=a\rho^\gamma$ (with $a>0$ a given constant) for any adiabatic constant $\gamma>1$. The nonlinear shear viscosity $\mu(\rho)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $\mu(\rho)= \mu\rho^\alpha$ with $2/3<\alpha<4$ and $\mu>0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69\`eme ann\'ee, 2016--2017, no 1135.


Introduction
When a fluid is governed by the barotropic compressible Navier-Stokes equations, the existence of global weak solutions, in the sense of J. Leray (see [32]), in space dimension greater than two remained for a long time without answer, because of the weak control of the divergence of the velocity field which may provide the possibility for the density to vanish (vacuum state) even if initially this is not the case.
There exists a huge literature on this question, in the case of constant shear viscosity µ and constant bulk viscosity λ. Before 1993, many authors such as Hoff [24], Jiang-Zhang [26], Kazhikhov-Shelukhin [29], Serre [44], Veigant-Kazhikhov [45] (to cite just some of them) have obtained partial answers: We can cite, for instance, the works in dimension 1 in 1986 by Serre [44], the one by Hoff [24] in 1987, and the one in the spherical case in 2001 by Jiang-Zhang [26]. The first rigorous approach of this problem in its generality is due in 1993 by P.-L. Lions [35] when the pressure law in terms of the density is given by P (ρ) = aρ γ where a and γ are two strictly positive constants. He has presented in 1998 a complete theory for P (ρ) = aρ γ with γ ≥ 3d/(d + 2) (where d is the space dimension) allowing to obtain the result of global existence of weak solutionsà la Leray in dimension d = 2 and 3 and for general initial data belonging to the energy space. His result has been then extended in 2001 to the case P (ρ) = aρ γ with γ > d/2 by Lin-Xin [34] to answer a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by Rousset [43]. More precisely extending and coupling carefully the two-velocities framework by Bresch-Desjardins-Zatorska [12] with the generalization of the quantum Böhm identity found by Bresch-Couderc-Noble-Vila [7] (proving a generalization of the dissipation inequality used by Jüngel [27] for Navier-Stokes-Quantum system and established by Jüngel-Matthes in [28]) and with the renormalized solutions introduced in Lacroix-Violet and Vasseur [31], we can get global existence of entropy-weak solutions to the following Navier-Stokes equations: and where P (ρ) = aρ γ denotes the pressure with the two constants a > 0 and γ > 1, ρ is the density of fluid, u stands for the velocity of fluid, Du = [∇u + ∇ T u]/2 is the strain tensor. As usually, we consider u 0 = m 0 ρ 0 when ρ 0 = 0 and u 0 = 0 elsewhere, |m 0 | 2 ρ 0 = 0 a.e. on {x ∈ Ω : ρ 0 (x) = 0}.
The main result of our paper reads as follows: Theorem 1.1. Let µ(ρ) verify (1.8)-(1.10) and µ and λ verify (1.3). Let us assume the initial data satisfy ( 1.11) with k ∈ (0, 1) given. Let T be given such that 0 < T < +∞, then, for any γ > 1, there exist a renormalized solution to ( Our result may be considered as an improvement of [34] for two reasons: First it takes into account a physical symmetric viscous tensor and secondly, it extends the range of coefficients α and γ. The method is based on the consideration of an approximated system with an extra pressure quantity, appropriate non-linear drag terms and appropriate capillarity terms. This generalizes the Quantum-Navier-Stokes system with quadratic drag terms considered in [46,47]. First we prove that weak solutions of the approximate solution are renormalized solutions of the system, in the sense of [31]. Then we pass to the limit with respect to r 2 , r 1 , r 0 , r, δ to get renormalized solutions of the compressible Navier-Stokes system. The final step concerns the proof that a renormalized solution of the compressible Navier-Stokes system is a global weak solution of the compressible Navier-Stokes system. Note that, thanks to the technique of renormalized solution introduced in [31], it is not necessary to derive the Mellet-Vasseur type inequality in this paper: This allows us to cover the all range γ > 1.

First
Step. Motivated by the work of [31], the first step is to establish the existence of global κ entropy weak solution to the following approximation where the barotorpic pressure law and the extra pressure term are respectively P (ρ) = aρ γ , P δ (ρ) = δρ 10 with δ > 0. (1.13) The matrix S µ is defined in (1.4) and T µ is given in(1.5)-(1.7). The matrix S r is compatible in the following sense: where Remark. Note that the previous system is the generalization of the quantum viscous Navier-Stokes system considered by Lacroix-Violet and Vasseur in [31] (see also the interesting papers by Antonelli-Spirito [3,4] and by Carles-Carrapatoso-Hillairet [17]). Indeed if we consider µ(ρ) = ρ and λ(ρ) = 0, we can write µ(ρ)S r as using Z(ρ) = 2 √ ρ. The Navier-Stokes equations for quantum fluids was also considered by A. Jüngel in [27].
As the first step generalizing [47], we prove the following result.
Sketch of proof for Theorem 1.2. To show Theorem 1.2, we need to build the smooth solution to an approximation associated to (1.12). Here, we adapt the ideas developed in [12] to construct this approximation. More precisely, we consider an augmented version of the system which will be more appropriate to construct approximate solutions. Let us explain the idea. First step: the augmented system. Defining a new velocity field generalizing the one introduced in the BD entropy estimate namely and a drift velocity v = 2∇s(ρ) and s(ρ) defined in (1.6).

Second
Step and main result concerning the compressible Navier-Stokes system. To prove global existence of weak solutions of the compressible Navier-Stokes equations, we follow the strategy introduced in [31,47]. To do so, first we approximate the viscosity µ by a viscosity µ ε 1 such that inf s∈[0,+∞) µ ′ ε 1 (s) ≥ ε 1 > 0. Then we use Theorem 1.2 to construct a κ entropy weak solution to the approximate system (1.12). We then show that this κ entropy weak solution is a renormalized solution of (1.12) in the sense introduced in [31]. More precisely we prove the following theorem: Theorem 1.3. Let µ(ρ) verifies (1.8)-(1.10), λ(ρ) given by (1.3). If r 0 > 0, then we assume also that inf s∈[0,+∞) µ ′ (s) = ǫ 1 > 0. Assume that r 1 is small enough compared to r and r 2 is small enough compared to δ, the initial values verify and (1.24) Then the κ entropy weak solutions is a renormalized solution of (1.12) in the sense of Definition 1.1.
We then pass to the limit with respect to the parameters r, r 0 , r 1 , r 2 and δ to recover a renormalized weak solution of the compressible Navier-Stokes equations and prove our main theorem.
Definitions. Following [31] (based on the work in [47]), we will show the existence of renormalized solutions in u. Then, we will show that this renormalized solution is a weak solution. The renormalization provides weak stability of the advection terms ρu ⊗ u together and ρu ⊗ v. Let us first define the renormalized solution: Definition 1.1. Consider µ > 0, 3λ + 2µ > 0, r 0 ≥ 0, r 1 ≥ 0, r 2 ≥ 0 and r ≥ 0. We say that ( √ ρ, √ ρu) is a renormalized weak solution in u, if it verifies (1.18)-(1.21), and for any function ϕ ∈ W 2,∞ (R d ) with ϕ(s)s ∈ L ∞ (R d ), there exists three measures where the constant C depends only on the solution ( √ ρ, √ ρu), and for any function where S µ is given in (1.4) and T µ is given in (1.7). The matrix S r is compatible in (1.14), (1.15), and (1.16). The vector valued function F is given by (1.25) For every i, j, k between 1 and d: and We define a global weak solution of the approximate system or the compressible Navier-Stokes equation (when r = r 0 = r 1 = r 2 = δ = 0) as follows and S r the capillary quantity in L 2 ((0, T ) × Ω) given by (1.14)-(1.16). Let us denote P (ρ) = aρ γ and P δ (ρ) = δρ 10 . We say that (ρ, u) is a weak solution to (1.12)-(1.15), if it satisfies the a priori estimates (1.18)-(1.21) and for any function with F given through (1.25) and for any ψ ∈ C ∞ c (Ω): Remark. As mentioned in [14], the equation on µ(ρ) is important: By taking ψ = divϕ for all ϕ ∈ C ∞ 0 , we can write the equation satisfied by ∇µ(ρ) namely (1.29) This will justify in some sense the two-velocities formulation introduced in [12] with the extra velocity linked to ∇µ(ρ).

The first level of approximation procedure
The goal of this section is to construct a sequence of approximated solutions satisfying the compactness structure to prove Theorem 1.2 namely the existence of weak solutions of the approximation system with capillarity and drag terms. Here we present the first level of approximation procedure.
1. The continuity equation with modified initial data Here ε 3 and ε 4 denote the standard regularizations by mollification with respect to space and time. This is a parabolic equation recalling that in this part Inf [0,+∞) µ ′ (s) > 0. Thus, we can apply the standard theory of parabolic equation to solve it when w is given smooth enough. In fact, the exact same equation was solved in paper [12]. In particular, we are able to get the following bound on the density at this level approximation 2. The momentum equation with drag terms is replaced by its Faedo-Galerkin approximation with the additional regularizing term satisfied for any t > 0 and any test function ψ ∈ C([0, T ], X n ), where λ(ρ) = 2(µ ′ (ρ)ρ − µ(ρ)), and s ′ (ρ) = µ ′ (ρ)/ρ, and X n = span{e i } n i=1 is an orthonormal basis in W 1,2 (Ω) with e i ∈ C ∞ (Ω) for any integers i > 0.

The Faedo-Galerkin approximation for the equation on the drift velocity v reads
satisfied for any t > 0 and any test function The above full approximation is similar to the ones in [12]. We can repeat the same argument as their paper to obtain the local existence of solutions to the Galerkin approximation. In order to extend the local solution to the global one, the uniform bounds are necessary so that the corresponding procedure can be iterated.

2.1.
The energy estimate if the solution is regular enough. For any fixed n > 0, choosing test functions ψ = w, φ = v in (2.3) and (2.4), we find that (ρ, w, v) satisfies the following κ−entropy equality where s ′ = µ ′ (ρ)/ρ and p(ρ) = ρ γ . Compared to the calculations made in [12], we have to take care of the capillary term and then to take care of the drag terms showing that they can be controlled using that s∈[0,T ] µ ′ (s) ≥ ε 1 for the linear drag, using the extra pressure term δρ 10 for the quadratic drag term and using the capillary term rρ∇( K(ρ)∆( ρ 0 K(s)) for the cubic drag term. To do so, let us provide some properties on the capillary term and rewrite the terms coming from the drag quantities.
2.1.1. Some properties on the capillary term. Using the mass equation, the capillary term in the entropy estimates reads (2.6) In fact, we write term I 1 as follows By (1.22), we have (2.7) Control of norms using I 2 . Let us first recall that since As the second term in the right-hand side is positive, lower bound on the quantity will provide the same lower bound on I 2 .
Let us now precise the norms which are controlled by (2.8). To do so, we need to rely on the following lemma on the density. In this lemma, we prove a more general entropy dissipation inequality than the one introduced by Jüngel in [27] and more general than those by Jüngel-Matthes in [28].
i) Assume ρ > 0 and ρ ∈ L 2 (0, T ; H 2 (Ω)) then there exists ε(k) > 0, such that we have the following estimate where C is a universal positive constant. ii) Consider a sequence of smooth densities ρ n > 0 such that Z(ρ n ) and Z 1 (ρ n ) converge strongly in L 1 ((0, T ) × Ω) respectively to Z(ρ) and Z 1 (ρ) and The case of Z = 2 √ ρ for the inequality was proved in [27], which is critical to derive the uniform bound on approximated velocity in L 2 (0, T ; L 2 (Ω)) in [46,47]. The above lemma will play a similar role in this paper.
Proof. Let us first prove the part i). Note that Z ′ (ρ) = √ µ(ρ) ρ µ ′ (ρ), we get the following calculation: (2.9) By integration by parts, the cross product term reads as follows (2.10) To this end, we are able to control I 1 directly, where C is a universal positive constant. We calculate I 2 to have Relying on (2.9)-(2.12), we have This ends the proof of part i). Concerning part ii), it suffices to pass to the limit in the inequality proved previously using the lower semi continuity on the left-hand side.

Note that
(2.16) From (1.9), for any ρ ≥ 1, we have for any time t > 0. c) Cubic drag term. The non-linear cubic drag term gives |w − 2κ∇s(ρ)| 2 (w − 2κ∇s(ρ)) · (2κ∇s(ρ)) dx dt. (2.18) The novelty now is to show that we control the second drag term of the right-hand side using the Korteweg-type information on the left-hand side Remark that the first term in the right-hand side may be absorbed using the first term in (2.18). Let us now prove that if r 1 small enough, the second term in the right-hand side may be absorbed by the term coming from the capillary quantity in the energy. From It remains to check that This concludes assuming r 1 small enough compared to r.

Compactness Lemmas.
In this subsection, we provide general compactness lemmas which will be used several times in this paper.
Let us now focus on the convergence of First let us recall that Let us now prove that Recall first that α 1 > 2 3 , we just have to consider ρ n ≥ 1. We write We can use the fact that ρ (4γ/3) + n ∈ L 1 ((0, T )×Ω) uniformly to conclude on (2.25). Thanks to we have the weak convergence of (2.24) in L 1 ((0, T ) × Ω).
We now investigate limits on u independent of the parameters. We need to differentiate the case with hyper-viscosity ε 2 > 0, from the case without. In the case with hyperviscosity, the estimate depends on ε 1 because of the drag force r 1 , while the estimate in the case ε 2 = 0 is independent of all the other parameters. This is why we will consider the limit ε 2 converges to 0 first. Lemma 2.3. Assume that ε 1 > 0 is fixed. Then, there exists a constant C > 0 depending on ε 1 and C in , but independent of all the other parameters (as long as they are bounded), such that for any initial values (ρ 0 , Assume now that ε 2 = 0. Let Φ : R + → R be a smooth function, positive for ρ > 0, such that Assume that the initial values (ρ 0 , √ ρ 0 u 0 ) verify (1.24) for a fixed C in > 0. Then, there exists a constant C > 0 independent of ε 1 , r 0 , r 1 , r 2 , δ (as long as they are bounded), such that Proof. We split the proof into the two cases.
Case 1: Assume that ε 1 > 0. From the equation on ρu and the a priori estimates, we find directly that We have µ(ρ) ≥ ε 1 ρ, and from (1.18), we have the a priori estimate Case 2: Assume now that ε 2 = 0. Multiplying the equation on (ρu) by Φ(ρ)/ρ, we get, as for the renormalization, that Note that Lemma 2.4. Assume either that ε 2,n = 0, or ε 1,n = ε 1 > 0. Let (ρ n , √ ρ n u n ) be a sequence of solutions for a family of bounded parameters with uniformly bounded initial values verifying (1.24) with a fixed C in . Assume that there exists α > 0, and a smooth function h : R + ×R 3 → R such that ρ α n is uniformly bounded in L p ((0, T )×Ω) and h(ρ n , u n ) is uniformly bounded in L q ((0, T ) × Ω), with Then, up to a subsequence, ρ n converges to a function ρ strongly in L 1 , √ ρ n u n converges weakly to a function q in L 2 . We define u = q/ √ ρ whenever ρ = 0, and u = 0 on the vacuum where ρ = 0. Then ρ α n h(ρ n , u n ) converges strongly in L 1 to ρ α h(ρ, u).
Proof. Thanks to the uniform bound on the kinetic energy ρ n |u n | 2 , and to Lemma 2.2, up to a subsequence, ρ n converges strongly in L 1 ((0, T ) × Ω) to a function ρ, and √ ρ n u n converges weakly in L 2 ((0, T ) × Ω) to a function q.
Some compactness when the parameters are fixed. For any positive fixed δ, r 0 , r 1 , r 2 and r, to recover a weak solution to (1.12), we only need to handle the compactness of the terms and ρ n µ ′ (ρ n ) |u n | 2 u n .
Indeed due to the term r 0 ρ n |u n |u n and the fact that inf s∈[0,+∞) µ ′ (s) > ε 1 > 0, one obtains the compactness for all other terms in the same way as in [12,37].
Recalling the assumptions on µ(s) and the relation λ(s) = 2(µ ′ (s)s − µ(s)), we have This means that the coefficients k(ρ n ) and j(ρ n ) are comparable to µ(ρ n ). Using the compactness of the density ρ n and the informations on µ(ρ n ) given in Corollary 2.2, we conclude the compactness of A 2 doing as for A 1 .
Cubic non-linear drag term. We will use Lemma 2.4 to show the compactness of ρ n µ ′ (ρ n ) |u n | 2 u n .
With above compactness of this section, we are able to pass to the limits for recovering a weak solution. In fact, to recover a weak solution to (1.12), we have to pass to the limits as the order of ε 4 → 0, n → ∞, ε 3 → 0 and ε → 0 respectively. In particular, when passing to the limit ε 3 tends to zero, we also need to handle the identification of v with 2∇s(ρ). Following the same argument in [12], one shows that v and 2∇s(ρ) satisfy the same moment equation. By the regularity and compactness of solutions, we can show the uniqueness of solutions. By the uniqueness, we have v = 2∇s(ρ). This ends the proof of Theorem 1.2.

From weak solutions to renormalized solutions to the approximation
This section is dedicated to show that a weak solution is a renormalized solution for our last level of approximation namely to show Theorem 1.3. First, we introduce a new function with η a smooth nonnegative even function compactly supported in the space time ball of radius 1, and with integral equal to 1. In this section, we will rely on the following two lemmas to proceed our ideas. Let ∂ be a partial derivative in one direction (space or time) in these two lemmas. The first one is the commutator lemma of DiPerna and Lions, see [35].
for some C ≥ 0 independent of ε, f and g, r is determined by 1 r = 1 p + 1 q . In addition, for some C ≥ 0 independent of ε, f and g, r is determined by 1 r = 1 p + 1 q . In addition, We also need another very standard lemma as follows.
− H(f ) L s loc (Ω × R + ) = 0, for any 1 ≤ s < ∞. We define a nonnegative cut-off functions φ m for any fixed positive m as follows.
The following estimates are necessary. We state them in the lemma as follows.
For the other terms in the momentum equation, we can follow the same way as above method for (3.6) to have Thanks to (3.6), we have The goal of this subsection is to derive the formulation of renormalized solution following the idea in [31]. We choose the function ψϕ ′ ([u m ] ε ) ε as a test function in (3.7). As the same argument of Lemma 3.5, we can show that Tr( µ(ρ)S µ + rS r ))∇φ m (ρ) + φ m (ρ)F (ρ, u) dx dt as ε goes to zero. Putting these two limits together, we have Now we should pass to the limit in (3.8) as m goes to infinity. To this end, we should keep the following convergences in mind: φ m (ρ) converges to 1, for almost every(t, x) ∈ R + × Ω, u m converges to u, for almost every(t, x) ∈ R + × Ω, |ρφ ′ m (ρ)| ≤ 2, and converges to 0 for almost every(t, x) ∈ R + × Ω. (3.9) We can find that Note that converges to zero for almost every (t, x). Thus, the Dominated convergence theorem yields that A 2m converges to zero as m → ∞. Meanwhile, the Dominated convergence theorem also gives us A 1m converges to T µ in L 2 t,x . Hence, with (3.9) at hand, letting m → ∞ in (3.8), one obtains that Tr((S µ + rS r )Id) + ψϕ ′ (u)F (ρ, u) dx dt = 0.

renormalized solutions and weak solutions
The main goal of this section is the proof of Theorem 1.1 that obtains the existence of renormalized solutions of the Navier-Stokes equations without the additional terms, thus the existence of weak solutions of the Navier-Stokes equations.
4.1. Renormalized solutions. In this subsection, we will show the existence of renormalized solutions. To this end, we need the following lemma of stability.
With the help of Lemma 2.2, we can pass to the limit on pressure, thus we can recover the renormalized solutions.

4.2.
Recover weak solutions from renormalized solutions. In this part, we can recover the weak solutions from the renormalized solutions constructed in Lemma 4.2. Now we show that Lemma 4.2 is valid without the condition ε 1 > 0. For such a µ, we construct a sequence µ n converging to µ in C 0 (R + ) and such that ε 1n = inf µ ′ n > 0. Lemma 4.1 shows that, up to a subsequence, ρ n → ρ in C 0 (0, T ; L p (Ω)) and ρ n u n → ρu in L ∞ (0, T ; L p+1 2p (Ω)) for any 1 ≤ p < γ, where (ρ, √ ρu) is a renormalized solution to (1.1). Now, we want to show that this renormalized solution is also a weak solution in the sense of Definition 1.2. To this end, we introduce a non-negative smooth function Φ : R → R such that it has a compact support and Φ(s) = 1 for any −1 ≤ s ≤ 1. LetΦ(s) = s 0 Φ(r) dr, we define ϕ n (y) = nΦ( y 1 n )Φ( y 2 n )....Φ( y N n ) for any y = (y 1 , y 2 , ...., y N ) ∈ R N . Note that ϕ n is bounded in W 2,∞ (R N ) for any fixed n > 0, ϕ n (y) converges everywhere to y 1 as n goes to infinity, ϕ ′ n is uniformly bounded in n and converges everywhere to unit vector (1, 0, ....0), and