Yue-Jun Peng
email: yue-jun.peng@uca.fr

RELAXED EULER SYSTEMS AND CONVERGENCE TO NAVIER-STOKES EQUATIONS

Keywords: Compressible and incompressible Navier-Stokes equations, Newtonian fluid, relaxed Euler systems, local and global convergence AMS Subject Classification (2000) : 35L45, 35L60, 35L65, 35Q30, 35Q31

We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon matrices. These systems are written in the form of balance laws and admit strictly convex entropies, so that they are symmetrizable hyperbolic. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global-in-time convergence is also shown for the initial data near constant equilibrium states of the systems. These convergence results are established not only for the approximate systems with vector variables but also for those with tensor variables.

Introduction

Euler and Navier-Stokes equations are two fundamental models in fluid mechanics. There is a huge number of studies on mathematical analysis around these equations. We refer to [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF][START_REF] Lions | Incompressible Models[END_REF][START_REF] Majda | Vorticity and Incompressible Flow[END_REF][START_REF] Feireisl | Dynamics of Viscous Compressible Fluids[END_REF][START_REF] Masmoudi | Examples of Singular Limits in Hydrodynamics, Handbook Diff. Equations: evolutionary equations[END_REF] and references therein for mathematical results. It is well known that Euler equations can be derived from Navier-Stokes equations as viscosity coefficients tend to zero. In this paper, we consider the approximation of isentropic Navier-Stokes equations by Euler type equations with relaxation which are referred to as relaxed Euler systems. This approximation problem is studied in both compressible and incompressible cases in whole space R d (d = 1, 2, 3 in physical situations).

We start with the compressible case. Let t ≥ 0 be the time variable and x = (x 1 , • • • , x d ) ∈ R d be the space variable. We denote by ν > 0 the shear viscosity and λ > 0 the Bulk viscosity. They are supposed to be constants. We consider the isentropic Navier-Stokes equations Here ρ > 0 is the density, u = (u 1 , • • • , u d ) T ∈ R d is the velocity, p is the pressure function and ∇u = (∂ x j u i ) 1≤i,j≤d . We see that π, σ(u) and ∇u are tensor variables. In (1.1)-(1.3), I d is the unit matrix of order d, the symbols T and ⊗ stand for the transpose and the tensor product, respectively. Throughout this paper, we assume that p is sufficiently smooth and p (ρ) > 0 for all ρ > 0.

For the Navier-stokes equations (1.1)-(1.2), the construction of relaxed Euler systems depends on the way how the term div π of second-order derivatives of u is decomposed into first-order derivative terms. Clearly, there are a lot of ways to do it. Among them a natural one is to replace (1.2) by the Maxwell's constitutive relation [START_REF] Maxwell | On the dynamical theory of gases[END_REF] (1.4)

ε∂ t π + νσ(u) + λ(div u)I d = -π,
where ε > 0 is a relaxation time. Let us denote by tr(π) the trace of π :

tr(π) = d i=1
π ii , π = (π ij ) 1≤i,j≤d .

Since tr(σ(u)) = 0, (1.4) yields ε∂ t tr(π) + tr(π) = -λd(div u), which shows that tr(π) depends on u and in general tr(π) = 0. Combining (1.1) and (1.4), we obtain a first-order system with relaxation :

(1.5)

     ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + div π = 0, ε∂ t π + νσ(u) + λ(div u)I d = -π, in R + × R d .
Formally, as ε → 0, we recover the Navier-Stokes equations (1.1)-(1.2). To our knowledge, so far system (1.5), in particular on its zero relaxation limit towards (1.1)-(1.2), has not been studied in the literature. Remark that the tensor π defined in (1.2) is symmetric but the approximate tensor π defined in (1.4) is not always symmetric. By (1.4), the approximate tensor π(t, •) is symmetric for all time t > 0 if and only if it is symmetric at t = 0. Nevertheless, we may consider a slightly more general approximate system by replacing π by π = (π + π T )/2 in the second equation of (1.5). It is easy to see that π still satisfies (1.4). Consequently, we obtain an approximate system (1.5) with a symmetric tensor π instead of π, even if π(0, •) is not symmetric. Thus, the symmetry of π is not a restriction condition. For this reason, throughout this paper, we suppose that the tensor π defined in (1.4) is symmetric.

In a recent paper [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF] for d = 3, by splitting νσ(u) + λ(div u)I d into νσ(u) and λ(div u)I d , the author proposed a similar first-order system with relaxation :

(1.6)                      ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + 1 √ ε 1 div τ 1 + 1 √ ε 2 ∇τ 2 = 0, ∂ t τ 1 + ν √ ε 1 σ(u) = - τ 1 ε 1 , ∂ t τ 2 + λ √ ε 2 div u = - τ 2 ε 2 , in R + × R d ,
where ε 1 > 0 and ε 2 > 0 are relaxation times, τ 1 is a tensor variable and τ 2 is a scalar variable.

The last two equations in (1.6) were called revised Maxwell's constitutive relations in [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF].

Comparing to (1.5), system (1.6) admits a special property on the trace of τ 1 . Indeed, since tr(σ(u)) = 0, tr(τ 1 ) satisfies a linear differential equation of the form ε 1 ∂ t tr(τ 1 ) + tr(τ 1 ) = 0, which implies that tr(τ 1 (t, •)) = 0 for all t > 0 as soon as tr(τ 1 (0, •)) = 0. Under condition tr(τ 1 (0, •)) = 0, the author of [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF] built a strictly convex entropy which implies that (1.6) is a symmetrizable hyperbolic system. He also proved that the smooth solution of (1.6) converges to that of (1.1) in Sobolev spaces in a uniform time interval as the relaxation times go to zero. However, the case where tr(τ 1 (0, •)) = 0 and the convergence for large time have not been investigated.

In order to see that (1.6) is an approximate system of (1.1) for small parameters ε 1 and ε 2 , we introduce

π 1 = τ 1 √ ε 1 , π 2 = τ 2 √ ε 2 .
Then (1.6) is rewritten as

(1.7)            ∂ t ρ + div(ρu) = 0,
∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + div π 1 + ∇π 2 = 0,

ε 1 ∂ t π 1 + νσ(u) = -π 1 , ε 2 ∂ t π 2 + λ div u = -π 2 , in R + × R d .
Formally, as (ε 1 , ε 2 ) → 0, from the last two equations in (1.7), we have

π 1 = -νσ(u), π 2 = -λ div u.
Substituting these two relations into the first two equations in (1.7), we obtain easily the compressible Navier-Stokes equations.

In what follows, systems (1.5) and (1.7) are referred to as relaxed Euler systems with tensor variables.

The first goal of this paper is to introduce a different approach to construct relaxed Euler systems. This approach is motivated by the theory of symmetrizable hyperbolic systems [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF] and the Cattaneo law for heat diffusion [START_REF] Maxwell | On the dynamical theory of gases[END_REF][START_REF] Cattaneo | Sulla conduzione del calore[END_REF][START_REF] Cattaneo | Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée[END_REF]. In this approach, we only use vector variables instead of tensor variables. This allows to write the approximate systems in the standard form of balance laws. For this purpose, we decompose the diffusion term ∆u into first-order derivative terms of u by introducing d full-rank matrices of order d × r with r ≥ d. We prove the existence with explicit examples of these matrices in the cases where r ≥ d 2 and r = d. In the latter case, Hurwitz-Radon matrices are concerned. See [START_REF] Radon | Lineare scharen orthogonaler matrizen[END_REF][START_REF] Hurwitz | Über die Komposition der quadratischen Formen[END_REF][START_REF] Adams | On matrices whose real linear combinations are non-singular[END_REF] for these matrices and relations with quadratic forms, and [START_REF] Lawson | Spin Geometry[END_REF] for relations with the spin geometry and Clifford algebras.

Y.J.Peng

More precisely, we consider the compressible Navier-Stokes equations under the form

(1.8) ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) = ν∆u + µ∇(div u), in R + × R d ,
where the viscosity coefficients ν and µ are supposed to satisfy (1.9) ν + µ > 0 for d = 1 and ν > 0, µ ≥ 0 for d ≥ 2.

Now we show that system (1.1) with (1.2)-(1.3) is included in (1.8) with (1.9). Indeed, since div(∇u) = ∆u, div(∇u) T = ∇(div u),

we have

div νσ(u) + λ(div u)I d ) = λ∆u, d = 1, ν∆u + µ∇(div u), d ≥ 2, where µ = ν + λ - 2ν d ≥ λ, for d ≥ 2.
Therefore, the conditions in (1.9) are satisfied if ν > 0 and λ > 0. It is clear that system (1.8) in one space dimension is an easy case, because

ν∆u + µ∇(div u) = (ν + µ)∂ xx u, x ∈ R.
For simplifying the presentation of the problem, in what follows, we only consider (1.8) for d ≥ 2. Let r ≥ d be an integer and M i (1 ≤ i ≤ d) be real constant matrices of order d × r which satisfy (1.10)

M i M T i = I d and M i M T j + M j M T i = 0 for 1 ≤ i, j ≤ d, j = i.
The existence of these matrices is given in Section 2 (see Proposition 2.1). Let v I ∈ R r and v II ∈ R be auxiliary variables. We introduce the following first-order system of balance laws with relaxation :

(1.11)                        ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + √ ν d j=1 M j ∂ x j v I + √ µ ∇v II = 0, ε 1 ∂ t v I + √ ν d j=1 M T j ∂ x j u = -v I , ε 2 ∂ t v II + √ µ div u = -v II , in R + × R d ,
where ε 1 > 0 and ε 2 > 0 are relaxation times. System (1.11) is referred to as relaxed Euler system with vector variables. Formally, as (ε 1 , ε 2 ) → 0, the last two equations in (1.11) yield

v I = - √ ν d j=1 M T j ∂ x j u, v II = - √ µ div u, which implies that √ µ∇v II = -µ∇(div u).
Moreover, by (1.10), we have

√ ν d j=1 M j ∂ x j v I = -ν∆u.
Substituting these relations into the first two equations in (1.11), we get (1.8). This shows that the compressible Navier-Stokes equations (1.8) are the formal limit system of (1.11) as (ε 1 , ε 2 ) → 0. We point out that the first-order system (1.11) is not included in the class of systems studied in [START_REF] Peng | Parabolic limits with differential constraints of first-order quasilinear hyperbolic systems[END_REF][START_REF] Peng | Uniform global existence and parabolic limit for partially dissipative hyperbolic systems[END_REF]. The compressible Euler equations are part of (1.11) and such a structure allows us to observe easily the symmetrizable hyperbolicity of the system. In the absence of a theory on the global existence of weak solutions for the nonlinear hyperbolic system in several space dimensions, we consider smooth solutions in Sobolev spaces to the Cauchy problem for (1.11). For this purpose, we first construct a strictly convex entropy of the system. Remark that the existence of a strictly convex entropy is very important in our problem because it provides a symmetrizer of the system to obtain energy estimates in Sobolev spaces. It also provides an L 2 energy equality which is the first step in the study of the global existence of solutions.

We prove that system (1.11) converges to the Navier-Stokes equations not only in a uniform time interval but also globally for all time when the initial data are near constant equilibrium states. In this paper, the convergence is related to a limit as the relaxation times go to zero. By the local convergence, we mean that the smooth solution of an approximate system converges to a smooth solution of the limit system in a uniform time interval with a precise error estimate in

C([0, T ]; H m (R d ))
, where H m (R d ) is a usual Sobolev space. The global convergence means that, for all time, the sequence of smooth solutions of an approximate system admits a convergent subsequence (in strong or weak topology) whose limit is a global smooth solution of the limit system. The global convergence is based on the uniform global existence of solutions with respect to the relaxation times together with compactness arguments.

In the proof of the local convergence, we need to deal with initial layers for (v I , v II ) by introducing correction terms. For simplifying the presentation, we take ε 1 = ε 2 as in [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF]. The proof is based on energy estimates by choosing an appropriate symmetrizer of the system. In the result of the global convergence, we do not require any relation between ε 1 and ε 2 , but the initial data are supposed to be in a uniform neighborhood of constant equilibrium states. The proof is based on three main steps. The first step concerns an L 2 estimate which follows from the entropy equality with a strictly convex entropy. The second step is to prove a usual higher order estimate with a dissipation estimate for (v I , v II ) by using the symmetrizer mentioned above. The last step concerns a dissipation estimate for (∇ρ, ∇u), which depends strongly on condition (1.10) (see in Section 4).

The second goal of this paper is to justify the convergence of the relaxed Euler systems with tensor variables to the Navier-Stokes equations. The local convergence for (1.5) can be obtained in a similar way to that for the relaxed Euler systems with vector variables. The proof of the global convergence is given for both (1.5) and (1.7), for which we construct strictly convex entropies with entropy-flux. In the result of the global convergence for (1.7), ε 1 and ε 2 are independent. Besides the usual smallness conditions on the initial data, we also need a smallness condition on ε 2 tr(π 1 )/ √ ε 1 at the initial time and this condition disappears when ε 2 = O(ε 1 ) or tr(π 1 ) = 0 at t = 0 (see Theorem 6.2). We mention that in [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF], a strictly convex entropy and the local convergence of (1.5) were established under a restriction condition tr(π 1 ) = 0 at t = 0, which is removed in our result. This is a natural treatment since the expression of the entropy and entropy-flux should be independent of the initial data.

In the incompressible case with d ≥ 2, the Navier-Stokes equations read (1.12)

∂ t u + (u • ∇)u + ∇p = ν∆u, div u = 0.
where • stands for the inner product in R d . We propose the relaxed Euler systems with vector variables

(1.13)                  ∂ t u + (u • ∇)u + ∇p + √ ν d j=1 M j ∂ x j v = 0, ε∂ t v + √ ν d j=1 M T j ∂ x j u = -v, div u = 0,
and that with tensor variables

(1.14)      ∂ t u + (u • ∇)u + ∇p + div π = 0, ε∂ t π + ν ∇u + (∇u) T = -π, div u = 0.
We prove that both systems (1.13) and (1.14) converge to (1.12) in the same framework as above.

Finally, we remark that the idea of this kind of approximations of a second-order partial differential equation by first-order hyperbolic systems is not recent. It comes back to the study by Maxwell and Cattaneo [26,[START_REF] Cattaneo | Sulla conduzione del calore[END_REF][START_REF] Cattaneo | Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée[END_REF]. Here is a simple example. The heat equation ∂ t u -∆u = 0 can be expressed by the first law of thermodynamics ∂ t u + div q = 0 together with Fourier law q = -∇u. Cattaneo proposed a revised law ε∂ t q + ∇u = -q, called now Cattaneo law or Maxwell-Cattaneo law, where ε > 0 is a relaxation time. This forms a linear hyperbolic system with relaxation ∂ t u + div q = 0, ε∂ t q + ∇u = -q, and we recover the heat equation as ε → 0. This idea was developed later in the approximation of nonlinear second-order systems of partial differential equations by first-order hyperbolic systems. We refer, for instance, to [START_REF] Saut | Some remarks on the limit of viscoelastic fluids as the relaxation time tends to zero[END_REF][START_REF] Molinet | Newtonian limit for weakly viscoelastic fluid flows of Oldroyd type[END_REF][START_REF] Bresch | Newtonian limit for weakly viscoelastic fluid flows[END_REF] for the approximation of the incompressible Navier-Stokes equations by using Oldroyd-type constitutive laws with relaxation, to [START_REF] Fernández | On the stability of damped Timoshenko systems: Cattaneo versus Fourier law[END_REF][START_REF] Said-Houari | Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same[END_REF] for the approximation of the Timoshenko-Fourier system by the Timoshenko-Cattaneo system, and to [START_REF] Hu | Compressible Navier-Stokes equations with hyperbolic heat conduction[END_REF][START_REF] Hu | Compressible Navier-Stokes equations with revised Maxwell's law[END_REF] for the local convergence of hyperbolic-parabolic systems to the full compressible Navier-Stokes equations. See also [START_REF] Brenier | On a relaxation approximation of the incompressible Navier-Stokes equations[END_REF][START_REF] Paicu | Une perturbation hyperbolique des équations de Navier-Stokes[END_REF][START_REF] Miranville | On a phase-field system based on the Cattaneo law[END_REF][START_REF] Said-Houari | The asymptotic behavior of the Bresse-Cattaneo system[END_REF] on this topic. This paper is organized as follows. In the next section, we construct the relaxed Euler systems with vector variables based on decompositions of ∆u. We study the symmetrizable hyperbolicity of the systems and recall results on the local existence of smooth solutions. In Section 3 we prove the local convergence of the systems. Section 4 is devoted to the result and the proof of the global convergence of the systems. In Section 5, we consider the relaxed Euler systems in the incompressible case and prove their convergence to the incompressible Navier-Stokes equations. In the last section, we study the relaxed Euler systems with tensor variables and prove similar results to those for the relaxed Euler systems with vector variables.

Relaxed Euler systems

2.1. Decomposition of ∆u.

The construction of relaxed Euler systems is based on the decomposition of ∆ into two first-order differential operators. Let u : R d -→ R d be a smooth function. In the usual decomposition ∆u = div(∇u), ∇u is a tensor for d ≥ 2. In order to use the vector variables instead of the tensor variables, we employ different decompositions of ∆u as follows.

Let r ≥ d be an integer and M i (1 ≤ i ≤ d) be real constant matrices of order d × r which satisfy (2.1)

M i M T i = I d and M i M T j + M j M T i = 0 for 1 ≤ i, j ≤ d, j = i. The decomposition of ∆u is (2.2) ∆u = d i,j=1 M i M T j ∂ 2 x i x j u = d i=1 M i ∂ x i d j=1 M T j ∂ x j u .
M i = 0 d×(i-1)d , O i , 0 d×(r-id) , 1 ≤ i ≤ d.
Since M i M T j = δ ij I d , these matrices fulfill all conditions in (2.1). In particular, when r = d 2 and O i = I d for all 1 ≤ i ≤ d, easy calculations give

d j=1 M T j ∂ x j u =   ∂ x 1 u . . . ∂ x d u   def = ∇ b u.
Moreover, let

v I =   v 1 . . . v d   , with v i (t, x) ∈ R d , 1 ≤ i ≤ d.
Then

M i v I = v i and d i=1 M i ∂ x i v I = d i=1 ∂ x i v i .
The latter is the divergence of v I by blocks. In this case, the third equation in (1.11) becomes

ε 1 ∂ t v I + √ ν∇ b u = -v I , or equivalently, ε 1 ∂ t v i + √ ν∂ x i u = -v i , 1 ≤ i ≤ d.
Case 2 : r = d. This is an interesting case where each M i is a square matrix and also of the minimum size. The conditions in (2.1) mean that M i is orthogonal and M i M T j is anti-symmetric for all 1 ≤ i, j ≤ d and j = i. When d ≥ 3 is odd, it is clear that there don't exist such square matrices. Indeed, the second condition in (2.1) implies that det(M i )det(M j ) = 0, j = i, which is contradictory to the fact that M i is an orthogonal matrix.

When d is even, it is possible to build M i as follows. Let us fix an orthogonal matrix M d , denoted by O.

Define A i = M i O T for 1 ≤ i ≤ d -1. When A i is given, we obtain M i through M i = A i O for all 1 ≤ i ≤ d -1 and M d = O. It follows that A i is anti-symmetric and (2.1) is equivalent to (2.3) A i A T i = I d , A 2 i = -I d , A i A T j + A j A T i = 0 for all 1 ≤ i, j ≤ d -1, j = i. Let s ≥ 1 be an integer. Recall that square matrices A 1 , A 2 , • • • , A s of order d are Hurwitz-Radon matrices if (2.3) is satisfied for all 1 ≤ i, j ≤ s, j = i.
Following the results in [START_REF] Radon | Lineare scharen orthogonaler matrizen[END_REF][START_REF] Hurwitz | Über die Komposition der quadratischen Formen[END_REF], for d given, the Hurwitz-Radon matrices can be constructed up to a Radon number s = ρ(d). Moreover, these matrices can be written with entries 0 and ±1 alone [START_REF] Eckmann | Hurwitz-Radon matrices revisited: from effective solution of the Hurwitz matrix equations to Bott periodicity[END_REF]. More precisely, all integer d ≥ 1 can be expressed as

d = (2c + 1)2 4a+b , a, b, c ∈ N, 0 ≤ b ≤ 3.
Then the Radon number ρ(d) is given by [START_REF] Radon | Lineare scharen orthogonaler matrizen[END_REF]

ρ(d) = 8a + 2 b -1.
When 4a + b = 0, we have a = b = 0. Then d is odd and ρ(d) = 0. Thus, 4a + b ≥ 1 is a necessary condition to obtain a minimum positive number of ρ(d). An explicit construction of Hurwitz-Radon matrices can be found in [START_REF] Eckmann | Hurwitz-Radon matrices revisited: from effective solution of the Hurwitz matrix equations to Bott periodicity[END_REF].

Here we are interested in determining

d such that ρ(d) ≥ d -1. This allows to obtain A 1 , A 2 , • • • , A d-1 which provide M 1 , M 2 , • • • , M d .
By the comparison of the expressions of d and ρ(d) above, we have

d -1 -ρ(d) = (c2 4a+b + 2 4a-1+b -8a) + (c2 4a + 2 4a-1 -1)2 b .
Hence, it is easy to see that d -1 > ρ(d) when a ≥ 1 or c ≥ 1. Nevertheless, in the other case, namely, a = c = 0, we have exactly ρ

(d) = d -1 = 2 b -1. Thus, A 1 , A 2 , • • • , A d-1 can be constructed for all dimension d of the form d = 2 b with b = 0, 1, 2, 3, namely, d = 1, 2, 4, 8.
We conclude the results from the above discussion in the following proposition.

Proposition 2.1. Let d ≥ 1 be a given integer. There exist integer r ≥ d and matrices

M i (1 ≤ i ≤ d) of order d × r satisfying (2.1) in the following two cases : i) for all r ≥ d 2 , ii) for r = d with d = 1, 2, 4, 8.

Now we give examples of matrices

M i (1 ≤ i ≤ d) satisfying (2.1) for r = d with d = 2, d = 4. Let Q be the matrix defined by Q = 0 1 -1 0 .
It is easy to see that Q is orthogonal and anti-symmetric. For d = 2, it suffices to take

M 1 = Q, M 2 = I 2 .
For d = 4, we define these matrices by blocs based on Q and I 2 ,

M 1 = Q 0 0 -Q , M 2 = 0 I 2 -I 2 0 , M 3 = 0 Q Q 0 , M 4 = I 4 ,
where M 1 , M 2 and M 3 are orthogonal and anti-symmetric.

Relaxed Euler systems and hyperbolicity.

As mentioned in the introduction, the relaxed Euler systems with vector variables are

(2.4)                        ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + √ ν d j=1 M j ∂ x j v I + √ µ ∇v II = 0, ε 1 ∂ t v I + √ ν d j=1 M T j ∂ x j u = -v I , ε 2 ∂ t v II + √ µ div u = -v II , in R + × R d .
Now we study the hyperbolicity of the systems. Let D 0 (ε) be a diagonal matrix of order r + 1 and N j be a matrix of order d × (r + 1) defined by

D 0 (ε) = diag(ε 1 I r , ε 2 ), N j = √ νM j , √ µe j ,
where (e

1 , • • • , e d ) is the standard basis of R d . Since div u = d j=1 e T j ∂ x j u, ∇v II = d j=1 e j ∂ x j v II , system (2.4) is equivalent to (2.5)                  ∂ t ρ + div(ρu) = 0, ρ∂ t u + (ρu • ∇)u + ∇p(ρ) + d j=1 N j ∂ x j v = 0, D 0 (ε)∂ t v + d j=1 N T j ∂ x j u = -v, v = v I v II .
A part of (2.5) is the compressible Euler equations which is a symmetrizable hyperbolic system. We can choose a diagonal matrix to be its symmetrizer [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF]. Since D 0 (ε) is also a diagonal matrix, by the position of N j and N T j , we see easily that (2.5) is a symmetrizable hyperbolic system too.

For ρ > 0, let h be the enthalpy function defined by h (ρ) = p (ρ)/ρ. System (2.4) can be further written as

(2.6) D 1 (ε)∂ t U + d j=1 A j (ρ, u)∂ x j U = S(v), U =   ρ u v   ∈ R 2+d+r ,
where

D 1 (ε) = diag(I d+1 , ε 1 I r , ε 2 ), S(v) = -   0 0 v   , A j (ρ, u) =     u j ρe T j 0 h (ρ)e j u j I d 1 ρ N j 0 N T j 0     , u =   u 1 . . . u d   .
Let (E 0 , F 0 ) be the pair of entropy-entropy flux for the Euler equations, given by (2.7)

     E 0 (ρ, u) = 1 2 ρ|u| 2 + H(ρ), F 0 (ρ, u) = 1 2 ρ|u| 2 u + ρh(ρ)u,
where H (ρ) = h(ρ) and | • | is the usual Euclidean norm. It is known that E 0 is strictly convex with respect to the conservative variable (ρ, ρu) of the Euler equations for ρ > 0. We define functions E and F by

(2.8) E(U ) = E 0 (ρ, u) + ε 1 2 |v I | 2 + ε 2 2 |v II | 2 , F (U ) = F 0 (ρ, u) + a N (u, v),
where

a N (u, v) : R d × R r+1 -→ R d is a bilinear application defined by a N (u, v) =    u T N 1 v . . . u T N d v    .
We check easily that a smooth solution U of (2.5) satisfies the energy equality (2.9)

∂ t E(U ) + div F (U ) + |v| 2 = 0.
Therefore, (E, F ) is a pair of entropy-entropy flux. Since E is a strictly convex function with respect to the conservative variable (ρ, ρu, v) for ρ > 0, E is a strictly convex entropy of the system. By results in [START_REF] Godunov | An interesting class of quasi-linear systems (Russian)[END_REF][START_REF] Friedrichs | Systems of conservation equations with a convex extension[END_REF][START_REF] Boillat | Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques[END_REF], this implies again that system (2.5) is symmetrizable hyperbolic.

Local existence of solutions.

Let us denote ε = (ε 1 , ε 2 ). We consider the Cauchy problem for the relaxed Euler system :

(2.10)

       D 1 (ε)∂ t U ε + d j=1 A j (ρ ε , u ε )∂ x j U ε = S(v ε ), t = 0 : (ρ ε , u ε , v ε ) = (ρ ε 0 , u ε 0 , v ε 0 ), U ε =   ρ ε u ε v ε   ,
where the initial data may depend on ε. For m ∈ N, we denote by H m the Sobolev space

H m (R d ) and by • m its usual norm. Let m > d 2 + 1 be an integer. The equilibrium state we take is (1, 0, 0) for (ρ, u, v). We assume (ρ ε 0 -1, u ε 0 , v I,ε 0 , v II,ε 0 ) ∈ H m with inf x∈R d ρ ε 0 (x) > 0.
By the local existence of smooth solutions for symmetrizable hyperbolic systems (see [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF][START_REF] Kato | The Cauchy problem for quasilinear symmetric hyperbolic systems[END_REF][START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF]), there exist a maximal time T ε 0 > 0 possibly depending on ε and a unique smooth solution U ε to (2.10), defined on time interval [0, T ε 0 ). This solution satisfies

(ρ ε -1, u ε , v I,ε , v II,ε ) ∈ C([0, T ε 0 ); H m ) ∩ C 1 [0, T ε 0 ); H m-1 .
In the case of the Cauchy problem for the compressible Navier-Stokes equations (2.11)

     ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) = ν∆u + µ∇(div u), t = 0 : (ρ, u) = ρ 0 , u 0 ,
the local existence of solutions is analogous (see [START_REF] Kato | Quasi-linear equations of evolution, with applications to partial differential equations[END_REF][START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat conductive gases[END_REF]). Assume (ρ 0 -1, u 0 ) ∈ H m with inf x∈R d ρ 0 (x) > 0. There exist a time T 0 > 0 and a unique smooth solution (ρ, u) to (2.11), defined on time interval [0, T 0 ]. This solution satisfies

(ρ -1, u) ∈ C([0, T 0 ]; H m ) ∩ C 1 [0, T 0 ]; H m-1 , ∇u ∈ L 2 (0, T 0 ; H m ), inf (t,x)∈[0,T 0 ]×R d ρ(t, x) > 0.
For the local convergence, where ε 1 = ε 2 , still denoted by ε, we have D 0 (ε) = εI d . We define

(2.12) v = - d j=1 N T j ∂ x j u, v0 = - d j=1 N T j ∂ x j u 0 , v ε 0 = v I,ε 0 v II,ε 0 .
Let v 0 be the smooth function satisfying lim ε→0 v ε 0 = v 0 in a strong topology. If v0 = v 0 , because of initial layer formations near t = 0, it is impossible that (v ε ) ε>0 converges to v uniformly in a time interval [0, T ] with T > 0. To treat this difficulty, we introduce a correction variable v L depending on (s, x) with s = t/ε. In view of the equation for v ε in (2.5), we define v L by

∂ s v L = -v L , which gives v L (s, x) = v L (0, x)e -s .
We hope that (v ε -v L ) ε>0 converges to v uniformly in a time interval [0, T ]. In particular, this implies that

(v ε -v L )(0, •) ε>0 converges to v(0, •), hence v L (0, x) = v 0 (x) -v0 (x).
From now on, we denote

(2.13) v ε (t, x) = v 0 (x) -v0 (x) e -t/ε , which satisfies ε∂ t v ε = -v ε .
It follows from (2.12) that

ν∆u + µ∇(div u) = - d j=1 N j ∂ x j (v + v ε ) + d j=1 N j ∂ x j v ε and ε∂ t (v + v ε ) + d j=1 N T j ∂ x j u = -(v + v ε ) + ε∂ t v.
Combining these two equations with (2.11) yields

(2.14)        D 1 (ε)∂ t U ε + d j=1 A j (ρ, u)∂ x j U ε = S(v + v ε -ε∂ t v) - 1 ρ R(∇v ε ), t = 0 : (ρ, u, v + v ε ) = ρ 0 , u 0 , v 0 , U ε =   ρ u v + v ε   , where R(∇v ε ) = -      0 d j=1 N j ∂ x j v ε 0      .

Local convergence

In the results stated below, we suppose that ν > 0, µ ≥ 0, and M i (1 ≤ i ≤ d) are any real constant matrices of order d × r satisfying (2.1). From this section, we denote by c > 0 and c i > 0 (i ∈ N) generic constants independent of any time and ε. In the proof of theorems, we also denote by •, • and • the inner product and the usual norm in L 2 (R d ), respectively. We often use the continuous embedding from

H m (R d ) to W 1,∞ (R d ) for m > d 2 + 1.
Theorem 3.1. (Local convergence for the relaxed Euler system)

Let

ε 1 = ε 2 def = ε > 0. Let m > d 2 + 1 be an integer. Let (ρ ε 0 -1, u ε 0 , v I,ε 0 , v II,ε 0 ) ∈ H m , (ρ 0 -1, u 0 ) ∈ H m+2 and v 0 ∈ H m+1 . We assume inf x∈R d ρ 0 (x) > 0 and (3.1) ρ ε 0 -ρ 0 m + u ε 0 -u 0 m + √ ε v ε 0 -v 0 m ≤ c 1 ε, where c 1 > 0 is a constant independent of ε.
Let (ρ, u) be the unique solution to (2.11) on [0, T 0 ]. Then there exists a constant ε 0 ∈ (0, 1] depending on T 0 , such that for all ε ∈ (0, ε 0 ], the unique solution

(ρ ε , u ε , v ε ) to (2.10) is defined on [0, T 0 ]. Moreover, (3.2) ρ ε (t) -ρ(t) 2 m + u ε (t) -u(t) 2 m + t 0 v ε (t ) -v(t ) -v ε (t ) 2 m dt ≤ cε 2 , ∀ t ∈ [0, T 0 ], (3.3
) v ε (t) -v(t) -v ε (t) m ≤ c √ ε, ∀ t ∈ [0, T 0 ],
where v and v ε are defined in (2.12)-(2.13).

Proof. Let us introduce

z ε = v ε -(v + v ε ), W ε 1 = ρ ε -ρ u ε -u , W ε = W ε 1 z ε , W ε 0 =   ρ ε 0 -ρ 0 u ε 0 -u 0 v ε 0 -v 0   ,
where v ε is defined in (2.13). Obviously,

z ε (0, •) = v ε 0 -v 0 . Let T ε = min(T ε 0 , T 0 ) ∈ (0, T 0 ].
[0, T ε ) × R d . Noting that W ε = U ε -U ε , subtracting (2.10) and (2.14) yields (3.4)                D 1 (ε)∂ t W ε + d j=1 A j (ρ ε , u ε )∂ x j W ε = d j=1 A j (ρ, u) -A j (ρ ε , u ε ) ∂ x j U ε + S(z ε + ε∂ t v) + 1 ρ R(∇v ε ), t = 0 : W ε = W ε 0 , for (t, x) ∈ [0, T ε ) × R d . Since S is linear, we have S(z ε + ε∂ t v) = S(z ε ) + εS(∂ t v). Let α ∈ N d with |α| ≤ m. we denote ∂ α x = ∂ |α| ∂x α 1 1 • • • ∂x α d d with |α| = α 1 + • • • + α d .
We also denote

A 0 (ρ) = diag (h (ρ), ρI d , I r+1 ) , Ãj (ρ, u) = A 0 (ρ)A j (ρ, u), Then A 0 (ρ)D 1 (ε) = diag (h (ρ), ρI d , εI r+1 ) , Ãj (ρ, u) =    h (ρ)u j p (ρ)e T j 0 p (ρ)e j ρu j I d N j 0 N T j 0    .
Since A 0 (ρ)D 1 (ε) is symmetric positive definite and Ãj (ρ, u) is symmetric for all 1 ≤ j ≤ d, A 0 (ρ)D 1 (ε) is a symmetrizer of system (2.6). Applying ∂ α x to (3.4), we get

D 1 (ε)∂ t (∂ α x W ε ) + d j=1 A j (ρ ε , u ε )∂ x j (∂ α x W ε ) (3.5) = S(∂ α x z ε ) + εS(∂ α x ∂ t v) + ∂ α x [ρ -1 R(∇v ε )] + d j=1 I α j ,
where

I α j = A j (ρ ε , u ε )∂ α x (∂ x j W ε ) -∂ α x A j (ρ ε , u ε )∂ x j W ε + ∂ α x A j (ρ, u) -A j (ρ ε , u ε ) ∂ x j U ε .
Taking the inner product of (3.5

) with 2A 0 (ρ ε )∂ α x W ε in L 2 (R d
) and using the fact that both matrices A 0 (ρ ε )D 1 (ε) and Ãj (ρ ε , u ε ) are symmetric, we obtain the classical energy equality [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF] :

d dt A 0 (ρ ε )D 1 (ε)∂ α x W ε , ∂ α x W ε = 2 A 0 (ρ ε )∂ α x W ε , S(∂ α x z ε ) + 2ε A 0 (ρ ε )∂ α x W ε , S(∂ α x ∂ t v) + 2 A 0 (ρ ε )∂ α x W ε , ∂ α x [ρ -1 R(∇v ε )] + 2 d j=1 A 0 (ρ ε )I α j , ∂ α x W ε + div A(ρ ε , u ε )∂ α x W ε , ∂ α x W ε , (3.6) where (3.7) div A(ρ, u) = ∂ t A 0 (ρ)D 1 (ε) + d j=1 ∂ x j Ãj (ρ, u).

Since inf

x∈R d ρ 0 (x) > 0, by (3.1), we may first suppose that, for sufficiently small ε,

(3.8) W ε (t) m ≤ c, ρ ε (t) ≥ c 0 > 0, ∀ t ∈ [0, T ε ).
In view of the expression of A 0 , D 1 (ε) and S, it is straightforward that

c 1 ∂ α x W ε 1 2 + ε ∂ α x z ε 2 ≤ A 0 (ρ ε )D 1 (ε)∂ α x W ε , ∂ α x W ε (3.9) ≤ c ∂ α x W ε 1 2 + ε ∂ α x z ε 2 , (3.10) 2 A 0 (ρ ε )∂ α x W ε , S(∂ α x z ε ) = -2 ∂ α x z ε 2 , and 2ε A 0 (ρ ε )∂ α x W ε , S(∂ α x ∂ t v) = -2ε ∂ α x z ε , ∂ α x ∂ t v .
By the Young inequality, we have

(3.11) 2 ε A 0 (ρ ε )∂ α x W ε , S(∂ α x ∂ t v) ≤ ∂ α x z ε 2 + cε 2 ∂ t v 2 m .
Next, by the Cauchy-Schwarz inequality together with (2.13) , we have

2 A 0 (ρ ε )∂ α x W ε , ∂ α x [ρ -1 R(∇v ε )] = -2 d j=1 ρ ε ∂ α x (u ε -u), N j ∂ α x (ρ -1 ∂ x j v ε ) ≤ c W ε 1 m v 0 -v0 m+1 e -t ε . (3.12)
For the term containing I α j , we observe that the last r + 1 lines of A j (ρ, u) are constant. This implies that the last r + 1 lines of A j (ρ, u) -A j (ρ ε , u ε ) vanish. By the Moser-type calculus inequalities (see Proposition 2.1 in [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF], p.43), we have

A 0 (ρ ε )∂ α x A j (ρ, u) -A j (ρ ε , u ε ) ∂ x j U ε , ∂ α x W ε ≤ c W ε 1 2 m . Similarly, A 0 (ρ ε ) A j (ρ, u)∂ α x (∂ x j W ε ) -∂ α x A j (ρ ε , u ε )∂ x j W ε , ∂ α x W ε ≤ c W ε 1 2 m . Therefore, (3.13) 2 d j=1 A 0 (ρ ε )I α j , ∂ α x W ε ≤ c W ε 1 2 m .
For the last term in (3.6), from the definition of A 0 , D 1 and Ãj , div

A(ρ ε , u ε ) can be expressed as div A(ρ ε , u ε ) = diag(A ε 11 , 0), A ε 11 = O ∇(ρ ε , u ε ) , where A ε 11 is a square matrix of order d + 1. It follows that (3.14) div A(ρ ε , u ε )∂ α x W ε , ∂ α x W ε ≤ c ∂ α x W ε 1 2 .
Thus, we conclude from (3.6) and (3.10)-(3.14) that

d dt A 0 (ρ ε )D 1 (ε)∂ α x W ε , ∂ α x W ε + 2 ∂ α z ε 2 ≤ cε 2 ∂ t v 2 m + c W ε 1 2 m + c W ε 1 m v 0 -v0 m+1 e -t ε .
Integrating this inequality over [0, t] with t ∈ [0, T ε ) and adding the inequalities for all α with |α| ≤ m, together with (3.9), it yields

W ε 1 (t) 2 m + ε z ε (t) 2 m + t 0 z ε (t ) 2 m dt ≤ W ε 1 (0) 2 m + ε z ε (0) 2 m + cε 2 t 0 ∂ t v(t ) 2 m dt + c t 0 W ε 1 (t ) 2 m dt + c t 0 W ε 1 (t ) m v 0 -v0 m+1 e -t ε dt .
From (3.1), we have

W ε 1 (0) 2 m + ε z ε (0) 2 m ≤ cε 2 .
On the other hand, using (2.12) and

-∂ t u = (u • ∇)u + ∇h(ρ) - 1 ρ ν∆u + µ∇(div u) ,
we also have

∂ t v = d j=1 N T j ∂ x j (u • ∇)u + ∇h(ρ) - 1 ρ ν∆u + µ∇(div u) .
Hence,

∂ t v m ≤ c ∇ρ m+1 + ∇u m+2 .
Since (ρ 0 -1, u 0 ) ∈ H m+2 and v 0 ∈ H m+1 , we have v 0 -v0 ∈ H m+1 and the solution to (2.11) satisfies ∇ρ ∈ C([0, T 0 ]; H m+1 ) and ∇u ∈ L 2 (0, T 0 ; H m+2 ). It follows that

W ε 1 (t) 2 m + ε z ε (t) 2 m + t 0 z ε (t ) 2 m dt ≤ c t 0 W ε 1 (t ) 2 m dt + c t 0 W ε 1 (t ) m e -t ε dt + cε 2 .
Let

y(t) = c t 0 W ε 1 (t ) 2 m dt + c t 0 W ε 1 (t ) m e -t ε dt + cε 2 1 2
.

Then W ε 1 (t) m ≤ y(t), and
y (t) ≤ cy(t) + ce -t ε , y(0) = cε. Noting that T ε ≤ T 0 and t 0 e -t ε dt ≤ ε,
by a Gronwall inequality, we obtain y(t) ≤ cε, which implies that

W ε 1 (t) 2 m + ε z ε (t) 2 m + t 0 z ε (t ) 2 m dt ≤ cε 2 , ∀ t ∈ [0, T ε ).
This estimate shows (3.2)-(3.3) and Theorem 3.1 by standard arguments (see [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF][START_REF] Brenier | Derivation of particle, string, and membrane motions from the Born-Infeld electromagnetism[END_REF]). It also justifies (3.8) by a bootstrap argument (see [START_REF] Tao | Nonlinear Dispersive Equations. Local and Global Analysis[END_REF]). 2

Uniform global existence and global convergence

In this section, we want to prove the following result.

Let ε = (ε 1 , ε 2 ). Let m > d 2 + 1 be an integer and (ρ ε 0 -1, u ε 0 , v I,ε 0 , v II,ε 0 ) ∈ H m .
There are two positive constants δ and c (independent of ε) such that if

(4.1) ρ ε 0 -1 m + u ε 0 m + √ ε 1 v I,ε 0 m + √ ε 2 v II,ε 0 m ≤ δ, then for all ε 1 , ε 2 ∈ (0, 1], the Cauchy problem (2.10) admits a unique global solution (ρ ε , u ε , v I,ε , v II,ε ) satisfying ρ ε (t) -1 2 m + u ε (t) 2 m + ε 1 v I,ε (t) 2 m + ε 2 v II,ε (t) 2 m + t 0 ∇ρ ε (t ) 2 m-1 + ∇u ε (t ) 2 m-1 + v ε (t ) 2 m dt (4.2) ≤ c ρ ε 0 -1 2 m + u ε 0 2 m + ε 1 v I,ε 0 2 m + ε 2 v II,ε 0 2 m , ∀ t ≥ 0. Moreover, there exist functions (ρ, u, v) with (ρ -1, u) ∈ L ∞ (R + ; H m ) and v ∈ L 2 (R + ; H m ),
such that, as ε → 0 and up to subsequences,

(4.3) (ρ ε , u ε ) -(ρ, u), weakly- * in L ∞ (R + ; H m ), (4.4) v ε -v, weakly in L 2 (R + ; H m ),
where

(4.5) v = - d j=1 N T j ∂ x j u,
and (ρ, u) is a unique solution to (2.11) for the compressible Navier-Stokes equations with initial value (ρ 0 , u 0 ) being the weak limit of (ρ ε 0 , u ε 0 ) in H m (up to subsequences).

4.1.

Energy estimates for relaxed Euler equations.

Recall that ε = (ε 1 , ε 2 ). According to [START_REF] Nishida | Nonlinear Hyperbolic Equations and Related Topics in Fluids Dynamics[END_REF], the global existence of smooth solutions follows from the local existence and uniform estimates of solutions with respect to t. The result will be uniform with respect to ε if the constants in the energy estimates are independent of ε. Since the local existence to (2.10) is known, it remains to establish uniform estimates with respect to t and ε. For simplicity, in this section the subscript ε in the expression of the solutions is dropped.

Let T > 0 be any time for which the smooth solution (ρ, u, v) to (2.10) is defined on time

interval [0, T ], (ρ -1, u, v I , v II ) ∈ C([0, T ]; H m ) ∩ C 1 [0, T ]; H m-1 .
In what follows, we denote

(4.6) B T = sup 0≤t≤T ρ(t) -1 m + u(t) m + √ ε 1 v I (t) m + √ ε 2 v II (t) m .
Since we consider smooth solutions near equilibrium state (1, 0, 0) for (ρ, u, v), we may suppose that B T is bounded by a sufficiently small constant independent of ε and T . Then

1 2 ≤ ρ ≤ 3 2 .
It follows that A 0 (ρ) is uniformly positive definite with respect to ε. The proof of Theorem 4.1 follows from an L 2 estimate, a higher order estimate and a dissipation estimate for ∇ρ and ∇u. Recall that in Lemmas 4.1-4.5 below, c > 0 and c i > 0 (i ∈ N) are generic constants independent of ε, T and any time. Let us start with the L 2 estimate. Lemma 4.1. (L 2 estimate) For all ε ∈ (0, 1] 2 and all t ∈ [0, T ], it holds

ρ(t) -1 2 + u(t) 2 + ε 1 v I (t) 2 + ε 2 v II (t) 2 + t 0 v(t ) 2 dt (4.7) ≤ c ρ 0 -1 2 + u 0 2 + ε 1 v I 0 2 + ε 2 v II 0 2 .
Proof. Let us recall the entropy equality in (2.9), which is

∂ t E(U ) + div F (U ) + |v| 2 = 0,
where E and F are defined in (2.8). By the Taylor formulae, there is a ρ * between 1 and ρ such that

H(ρ) = H(1) + h(1)(ρ -1) + h (ρ * )(ρ -1) 2 .
Using the density conservation in (2.10), we obtain

(4.8) ∂ t 1 2 ρ|u| 2 + h (ρ * )(ρ -1) 2 + ε 1 2 |v I | 2 + ε 2 2 |v II | 2 + div F (U ) -h(1)ρu + |v| 2 = 0.
Since h (ρ) > 0 for ρ > 0, when B T is sufficiently small, we have

c 1 (ρ -1) 2 + |u| 2 + ε|v| 2 ≤ 1 2 ρ|u| 2 + h (ρ * )(ρ -1) 2 + ε 2 |v| 2 ≤ c (ρ -1) 2 + |u| 2 + ε|v| 2 .
Integrating (4.8) over [0, t] × R d with t ∈ [0, T ], it yields (4.7). 2

Now we consider higher order estimates of U .

Lemma 4.2. (Higher order estimate) For all ε ∈ (0, 1] 2 and all t ∈ [0, T ], it holds

ρ(t) -1 2 m + u(t) 2 m + ε 1 v I (t) 2 m + ε 2 v II (t) 2 m + t 0 v(t ) 2 m dt ≤ c ρ 0 -1 2 m + u 0 2 m + ε 1 v I 0 2 m + ε 2 v II 0 2 m (4.9) + c t 0 ∇ρ(t ) m-1 + ∇u(t ) m-1 ∇U (t ) 2 m-1 dt . Proof. Let 1 ≤ |α| ≤ m. Applying ∂ α x to (2.6), we get D 1 (ε)∂ t (∂ α x U ) + d j=1 A j (ρ, u)∂ x j (∂ α x U ) = S(∂ α x v) - d j=1 J α j ,
where

J α j = ∂ α x A j (ρ, u)∂ x j U -A j (ρ, u)∂ α x (∂ x j U ).
Similarly to the proof of Theorem 3.1, we obtain the classical energy equality :

d dt A 0 (ρ)D 1 (ε)∂ α x U, ∂ α x U = 2 A 0 (ρ)∂ α x U, S(∂ α x v) -2 d j=1 A 0 (ρ)∂ α x U, J α j + div A(ρ, u)∂ α x U, ∂ α x U , ∀ t ∈ [0, T ], (4.10)
where div A(ρ, u) is defined in (3.7).

Obviously,

c 1 ∂ α x ρ 2 + ∂ α x u 2 + ε 1 ∂ α x v I 2 + ε 2 ∂ α x v II 2 ≤ A 0 (ρ)D 1 (ε)∂ α x U, ∂ α x U (4.11) ≤ c ∂ α x ρ 2 + ∂ α x u 2 + ε 1 ∂ α x v I 2 + ε 2 ∂ α x v II 2 , and (4.12) 2 A 0 (ρ)∂ α x U, S(∂ α x v) = -2 ∂ α x v 2 . Moreover, A 0 (ρ)∂ α x U, J α j = K α j + ρ∂ α x u, N j ∂ α x (ρ -1 ∂ x j v) -ρ -1 ∂ α x ∂ x j v
, where K α j are crossed terms from the Euler equations defined by

K α j = h (ρ)∂ α x ρ, ∂ α x (u j ∂ x j ρ) -u j ∂ α x ∂ x j ρ + h (ρ)∂ α x ρ, ∂ α x (ρ∂ x j u j ) -ρ∂ α x ∂ x j u j (4.13) + ρ∂ α x u j , ∂ α x (h (ρ)∂ x j ρ) -h (ρ)∂ α x ∂ x j ρ + ρ∂ α x u, ∂ α x (u j ∂ x j u) -u j ∂ α x ∂ x j u .
Applying the Moser-type calculus inequalities and the Young inequality, we have

(4.14) K α j ≤ c ∇u m-1 ∇ρ 2 m-1 + ∇u 2 m-1 , (4.15) A 0 (ρ)∂ α x U, J α j ≤ c ∇ρ m-1 + ∇u m-1 ∇U 2 m-1 . Let us denote U 1 = ρ u , Bj (U 1 ) = h (ρ)u j p (ρ)e T j p (ρ)e j ρu j I d , div B(U 1 ) = diag(h (ρ), I d )∂ t ρ + d j=1 ∂ x j Bj (U 1 ).
Since each N j is a constant matrix, from (3.7) and the definition of A 0 and Ãj , it is easy to see that

(4.16) div A(ρ, u)∂ α x U, ∂ α x U = div B(U 1 )∂ α x U 1 , ∂ α x U 1 ,
which is independent of v. Using ∂ t ρ = -div(ρu), we obtain

(4.17) div A(ρ, u)∂ α x U, ∂ α x U ≤ c ∇ρ m-1 + ∇u m-1 3 .
Adding (4.10) for all 1 ≤ |α| and integrating the resulting inequality over [0, t], from Lemma 4.1 and (4.11)-(4.17), we obtain (4.9). 2

Now we consider dissipation estimates of ∇ρ and ∇u. For this purpose, we introduce

ũ = d j=1 N T j ∂ x j u, ṽ = d j=1 N j ∂ x j v.
We first establish the following relations.

Lemma 4.3. For all β ∈ N d , it holds

∂ β x ũ 2 = ν ∂ β x (∇u) 2 + µ ∂ β x (div u) 2 , ∂ β x ṽ ≤ c ∂ β x (∇v) , ∂ β x v, ∂ β x ũ = -∂ β x u, ∂ β x ṽ , D 0 (ε)∂ t ∂ β x v, ∂ β x ũ = d dt D 0 (ε)∂ β x v, ∂ β x ũ + d j=1 ∂ t ∂ β x u, N j D 0 (ε)∂ x j ∂ β x v .
Proof. Obviously, it suffices to prove the result for β = 0. By the definition, N j = ( √ νM j , √ µe j ). Since e T j u = u j , we have

N T i ∂ x i u, N T j ∂ x j u = -ν M i M T j ∂ 2 x i x j u, u + µ ∂ x i u i , ∂ x j u j .
Hence, by (2.1),

ũ 2 = ν ∇u 2 + µ div u 2 .
This proves the first equality in the lemma. The other three relations can be proved in a similar way. 2

D 0 (ε)∂ β x v, ∂ β x ũ + c 2 ∂ β x u, ∂ β x (∇h(ρ)) + c 3 ∇ρ 2 m-1 + c 4 ∇u 2 m-1 ≤ c v 2 m + c u m ∇ρ 2 m-1 + ∇u 2 m-1 . (4.18) Proof. Let β ∈ N d with |β| ≤ m -1. Applying ∂ β
x to the system in (2.10), we have

(4.19)          ∂ t ∂ β x ρ + div ∂ β x (ρu) = 0, ∂ t ∂ β x u + ∂ β x (u • ∇)u + ∂ β x (∇h(ρ)) + ∂ β x 1 ρ ṽ = 0, D 0 (ε)∂ t ∂ β x v + ∂ β x ũ = -∂ β x v.
Taking the inner product of the third equation of (4.19) with ∂ β x ũ in L 2 (R d ), by Lemma 4.3 we obtain

ν ∂ β x (∇u) 2 + µ ∂ β x (div u) 2 = -∂ β x v, ∂ β x ũ -D 0 (ε)∂ t ∂ β x v, ∂ β x ũ = -∂ β x v, ∂ β x ũ - d dt D 0 (ε)∂ β x v, ∂ β x ũ - d j=1 ∂ t ∂ β x u, N j D 0 (ε)∂ x j ∂ β x v .
Since µ ≥ 0, by the Young inequality, the above equality implies that

(4.20) 2 d dt D 0 (ε)∂ β x v, ∂ β x ũ + ν ∂ β x (∇u) 2 ≤ c v 2 m -2 d j=1 ∂ t ∂ β x u, N j D 0 (ε)∂ x j ∂ β x v .
For the last term in (4.20), we use the second equation in (4. [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF]) to obtain

-∂ t ∂ β x u, N j D 0 (ε)∂ x j ∂ β x v = ∂ β x (u • ∇)u + ∇h(ρ) + ρ -1 ṽ , N j D 0 (ε)∂ x j ∂ β x v . Since ε ∈ (0, 1] 2 ,
by Lemma 4.3 and the Moser-type calculus inequalities, we have

- d j=1 ∂ t ∂ β x u, N j D 0 (ε)∂ x j ∂ β x v ≤ η ∂ β x (∇h(ρ)) 2 + c v 2 m + c u m ∇u m-1 v m ,
where η > 0 is a small constant to be chosen. This inequality together with (4.20) gives

(4.21) 2 d dt D 0 (ε)∂ β x v, ∂ β x ũ + ν ∂ β x (∇u) 2 ≤ η ∂ β x (∇h(ρ)) 2 + c v 2 m + c u m ∇u 2 m-1 .
Next, taking the inner product of the second equation of (4.19) with

∂ β x (∇h(ρ)) in L 2 (R d ) yields ∂ β x (∇h(ρ)) 2 = -∂ t ∂ β x u, ∂ β x (∇h(ρ)) -∂ β x (u • ∇)u + ρ -1 ṽ , ∂ β x (∇h(ρ))
. Hence, by the Young inequality,

(4.22) ∂ β x (∇h(ρ)) 2 ≤ c ∇u 2 m-1 + c v 2 m -2 ∂ t ∂ β x u, ∂ β x (∇h(ρ))
. For the last term in (4.22), we use the equation of the density conservation

∂ t ρ = -div(ρu).
It follows from an integration by parts that

-2 ∂ t ∂ β x u, ∂ β x (∇h(ρ)) = -2 d dt ∂ β x u, ∂ β x (∇h(ρ)) + 2 ∂ β x div u, ∂ β x (h (ρ) div(ρu)) .
Therefore, by the Young inequality and the Moser-type calculus inequalities,

-2 ∂ t ∂ β x u, ∂ β x (∇h(ρ)) ≤ -2 d dt ∂ β x u, ∂ β x (∇h(ρ)) + c ∇u 2 m-1 + c ∇ρ 2 m-1 u m .
This inequality together with (4.22) gives

(4.23) 2 d dt ∂ β x u, ∂ β x (∇h(ρ)) + ∂ β x (∇h(ρ)) 2 ≤ c ∇u 2 m-1 + c v 2 m + c ∇ρ 2 m-1 u m .
Multiplying (4.23) by 2η and adding with (4.21) yields

d dt 2 D 0 (ε)∂ β x v, ∂ β x ũ + 4η ∂ β x u, ∂ β x (∇h(ρ)) + η ∂ β x (∇h(ρ)) 2 + ν ∂ β x (∇u) 2 ≤ cη ∇u 2 m-1 + c v 2 m + c u m ∇ρ 2 m-1 + ∇u 2 m-1 .
Adding these inequalities for all β and taking η small enough, since

|β|≤m-1 ∂ β x (∇u) 2 = ∇u 2 m-1 , we have d dt |β|≤m-1 2 D 0 (ε)∂ β x v, ∂ β x ũ + 4η ∂ β x u, ∂ β x (∇h(ρ)) + η ∇h(ρ) 2 m-1 + ν 2 ∇u 2 m-1 ≤ c v 2 m + c u m ∇ρ 2 m-1 + ∇u 2 m-1 . Finally, recall that h is defined by h (ρ) = p (ρ)/ρ. Since p is sufficiently smooth, so is h (say h ∈ C m (R + * ))
. By the Moser-type calculus inequalities (see Proposition 2.1 (C) in [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF], p.43), we have ∇h(ρ) m-1 ≤ c ∇ρ m-1 . On the other hand, h (1) = p (1) > 0 implies that h is a C m -diffeomorphism at least in a neighborhood of ρ = 1. Thus, ∇h(ρ) m-1 is uniformly equivalent to ∇ρ m-1 . This proves (4.18). 2

Lemma 4.5. (Final estimate) For all ε ∈ (0, 1] 2 and all t ∈ [0, T ], it holds

ρ(t) -1 2 m + u(t) 2 m + ε 1 v I (t) 2 m + ε 2 v II (t) 2 m + t 0 ∇ρ(t ) 2 m-1 + ∇u(t ) 2 m-1 + v(t ) 2 m dt (4.24) ≤ c ρ 0 -1 2 m + u 0 2 m + ε 1 v I 0 2 m + ε 2 v II 0 2 m + cB T t 0 ∇U (t ) 2 m-1 dt ,
where B T is defined in (4.6).

Proof. Notice that, for all β ∈ N d with |β| ≤ m -1,

D 0 (ε)∂ β x v, ∂ β x ũ + c 2 ∂ β x u, ∂ β x (∇h(ρ)) ≤ c ∇ρ 2 m-1 + u 2 m + ε 1 v I 2 m + ε 2 v II 2
m . Integrating (4.18) over [0, t] and combining the result with (4.9), we obtain (4.24). 2

Proof of Theorem 4.1.

When the solution U is uniformly small in L ∞ (0, T ; H m ), the integral on the right hand-side can be controlled by that of the left-side in (4.24). This implies the uniform estimate (4.2) and the uniform global existence result. Now we prove the global convergence. The uniform estimate (4.2) implies that the sequences (ρ ε -1) ε>0 and (u ε ) ε>0 are bounded in L ∞ (R + ; H m ) and the sequence (v ε ) ε>0 is bounded in L 2 (R + ; H m ). This ensures the convergence (4.3)-(4.4) and

D 0 (ε)v ε -→ 0, strongly in L 2 (R + ; H m ). Since (ρ ε , u ε , v ε ) is a solution to (2.5), we have (4.25)                  ∂ t ρ ε + div(ρ ε u ε ) = 0, ∂ t (ρ ε u ε ) + div(ρ ε u ε ⊗ u ε ) + ∇p(ρ ε ) + d j=1 N j ∂ x j v ε = 0, D 0 (ε)∂ t v ε + d j=1 N T j ∂ x j u ε = -v ε .
For all T > 0, it is easy to see that both (∂ t ρ ε ) ε>0 and (∂ t u ε ) ε>0 are bounded in L 2 (0, T ; H m-1 ). Hence, (ρ, u) ∈ C([0, T ]; H m-1 ). Moreover, by a classical compactness theorem (see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]), for all m 1 ∈ (0, m), (ρ ε ) ε>0 and (u ε ) ε>0 are relatively compact in C([0, T ]; H m 1 loc ). As a consequence, as ε → 0 and up to subsequences, (ρ ε , u ε ) -→ (ρ, u), strongly in C([0, T ]; H m 1 loc ). Passing to the limit in the sense of distributions in (4.25), we obtain the Navier-Stokes equations (2.11) and also (4.5). Now we describe the initial condition for (ρ, u). The convergence above is uniform with respect to time. Hence,

(ρ ε 0 , u ε 0 ) = (ρ ε (0, •), u ε (0, •)) -→ (ρ(0, •), u(0, •)), strongly in H m 1 loc . Moreover, (ρ, u) ∈ C([0, T ]; H m-1 ) implies that (ρ(0, •), u(0, •)) ∈ H m-1 .
On the other hand, the boundedness of (ρ ε 0 -1, u ε 0 ) in H m implies that, up to a subsequence, (ρ ε 0 , u ε 0 ) admits a weak limit in H m , denoted by (ρ 0 , u 0 ) ∈ H m . It follows that, for all R > 0

(ρ(0, •), u(0, •)) = (ρ 0 , u 0 ), in B R ,
where B R is the ball of radius R and center 0 in R d . Now m -1 > d/2 and the embedding

H m-1 (R d ) → C(R d ) is continuous. Hence, both (ρ(0, •), u(0, •)) and (ρ 0 , u 0 ) are continuous functions in R d . Since R > 0 is arbitrary, we conclude that (ρ(0, •), u(0, •)) = (ρ 0 , u 0 ), in R d .
This ends the proof of Theorem 4.1. whose solution is given by

Incompressible relaxed Euler systems

p(t, x) = R d G(x -y)tr (∇u(t, y)) 2 dy,
where G is the fundamental solution of the Laplace equation :

G(x) =        - 1 2π ln |x|, if d = 2, C d |x| d-2 , if d ≥ 3, with C d > 0 being a constant.
From the discussion in the compressible case, we propose the relaxed incompressible Euler systems as follows (5.5) div u = 0, (

           ∂ t u + (u • ∇)u + ∇p + √ ν d j=1 M j ∂ x j v = 0, ε∂ t v + √ ν d j=1 M T j ∂ x j u = -v, in R + × R d , where v ∈ R r . 5.6)
Consider the Cauchy problem for (5.5)-(5.6) with initial condition (5.7) t = 0 : (u, v) = (u ε 0 , v ε 0 ). Suppose u ε 0 and v ε 0 are smooth and

(5.8) div u ε 0 = 0, div d j=1 M j ∂ x j v ε 0 = 0.
For incompressible Navier-Stokes equations, suppose (5.1) and div u(0, x) = 0 hold. It is known that (see [START_REF] Majda | Vorticity and Incompressible Flow[END_REF]) the incompressibility condition (5.5) is equivalent to (5.4) for all t > 0. Now we establish a similar result for the relaxed Euler systems. Proposition 5.1. Let T > 0 and (u ε 0 , v ε 0 ) ∈ H 2 satisfying (5.8). Let (u, p, v) be a solution of (5.6)-(5.7) with regularity

(u, v) ∈ L ∞ (0, T ; H 2 ) ∩ W 1,∞ (0, T ; H 1 ), ∇u ∈ L ∞ (0, T ) × R d , p ∈ L ∞ (0, T ; H 2 ).
Then (5.5) is equivalent to (5.4) in (0, T ) × R d . As a consequence, we also have

(5.9) div d j=1 M j ∂ x j v(t, •) = 0, ∀ t ∈ [0, T ].
Proof. As in Section 4, we denote ṽ = 2), we have

ε∂ t ṽ + √ ν∆u = -ṽ.
Applying div to the above equation and to the first equation in (5.6), by (5.3), we have

∂ t (div u) + u • ∇ div u + tr (∇u) 2 + ∆p + √ ν div ṽ = 0, ε∂ t (div ṽ) + √ ν∆(div u) + div ṽ = 0.
If (5.5) holds, then tr (∇u) 2 + ∆p + √ ν div ṽ = 0, ε∂ t (div ṽ) + div ṽ = 0.

Obviously, the second equation in the above system together with the second condition in (5.8) yields div ṽ = 0 which is (5.9). This implies (5.4). Conversely, if (5.4) holds, then (5.10)

∂ t u D + u • ∇u D + √ νv D = 0, ε∂ t v D + √ ν∆u D + v D = 0,
where (u D , v D ) = (div u, div ṽ). A standard energy estimate for v D yields ε 2

d dt v D 2 + v D 2 = √ ν ∇v D , ∇u D . Since √ νv D = -∂ t u D -u • ∇u D , we have √ ν ∇v D , ∇u D = - 1 2 d dt ∇u D 2 -∇(u • ∇u D ), ∇u D .
Hence, 1 2

d dt ε v D 2 + ∇u D 2 + v D 2 = -∇(u • ∇u D ), ∇u D .
A straightforward calculation shows that

∇(u • ∇u D ), ∇u D = d j=1 ∇u j • ∂ x j u D , ∇u D -u D , |∇u D | 2 .
Then, by denoting C T = ∇u L ∞ ((0,T )×R d ) , we have

∇(u • ∇u D ), ∇u D ≤ C T ∇u D 2 ,
which implies that 1 2

d dt ε v D 2 + ∇u D 2 + v D 2 ≤ C T ∇u D 2 .
By the Gronwall inequality together with (5.8), we obtain

v D (t, •) = 0, ∇u D (t, •) = 0, ∀ t ∈ (0, T ).
By (5.10), we further obtain ∂ t u D (t, •) = 0 for all t ∈ (0, T ), which implies (5.5) and (5.9). 2

Let m > d 2 + 1 be an integer. Suppose u ε 0 , v ε 0 ∈ H m .
Similarly to the energy estimates in Sections 4-5, it is easy to see that the solution of (5.5)-(5.7) satisfies

d dt u 2 m + ε v 2 m + v 2 m ≤ C u 3 m ,
where C > 0 is independent of (u, v). Thus, we may adapt the proof of existence of solutions for incompressible Euler equations [START_REF] Lichtenstein | Über einige existenz probleme der hydrodynamik, homogener, unzusammendrückbarer, reibungsloser flüssigkeiten und die Helmholtzschen wirbelsätze[END_REF][START_REF] Kato | Quasi-linear equations of evolution, with applications to partial differential equations[END_REF][START_REF] Majda | Vorticity and Incompressible Flow[END_REF]. There exist a maximal time T ε 0 > 0 and a unique smooth solution (u, p, v) to the Cauchy problem (5.5)-(5.7) such that

u, v ∈ C([0, T ε 0 ); H m ) ∩ C 1 [0, T ε 0 ); H m-1 , ∇p ∈ L 2 (0, T ε 0 ); H m-1 . Theorem 5.1. (Local convergence) Let m > d 2 + 1 be an integer. Let (u ε 0 , v ε 0 ) ∈ H m , u 0 ∈ H m+2 and v 0 ∈ H m+1 .
We assume (5.8) holds and

(5.11) u ε 0 -u 0 m + √ ε v ε 0 -v 0 m ≤ c 1 ε, where c 1 > 0 is a constant independent of ε.
Let (u, p) be the unique solution on [0, T 0 ] to (5.1)-(5.2) with initial data u 0 . Then there exists a constant ε 0 ∈ (0, 1] depending on T 0 such that for all ε ∈ (0, ε 0 ], the unique solution (u ε , p ε , v ε ) to (5.5)-(5.7) is defined on [0, T 0 ], and we have

(5.12) u ε (t) -u(t) 2 m + t 0 v ε (t ) -v(t ) -v ε (t ) 2 m dt ≤ cε 2 , ∀ t ∈ [0, T 0 ], (5.13) v ε (t) -v(t) -v ε (t) m ≤ c √ ε, ∀ t ∈ [0, T 0 ], (5.14)
t 0 ∇(p ε (t ) -p(t )) 2 m-1 dt ≤ cε, ∀ t ∈ [0, T 0 ],
where v and v ε are defined in (2.12)-(2.13), and c > 0 is a constant independent of ε.

Y.J.Peng

Proof. Let us introduce

w ε = u ε -u, q ε = p ε -p, z ε = v ε -(v + v ε ), w ε 0 = u ε 0 -u 0 , z ε 0 = v ε 0 -v 0 . From (5.1)-(5.4), we have (5.15)            ∂ t w ε + (u ε • ∇)w ε + ∇q ε + √ ν d j=1 M j ∂ x j z ε = -(w ε • ∇)u - √ ν d j=1 M j ∂ x j v ε , ε∂ t z ε + √ ν d j=1 M T j ∂ x j w ε = -z ε -ε∂ t v, (t, x) ∈ [0, T ε ) × R d ,
where

T ε = min(T ε 0 , T 0 ) ∈ (0, T 0 ]. For α ∈ N d with |α| ≤ m, applying ∂ α x to (5.15) yields            ∂ t ∂ α x w ε + (u ε • ∇)∂ α x w ε + ∇∂ α x q ε + √ ν d j=1 M j ∂ x j ∂ α x z ε = L ε α - √ ν d j=1 M j ∂ x j ∂ α x v ε , ε∂ t ∂ α x z ε + √ ν d j=1 M T j ∂ α x ∂ x j w ε = -∂ α x z ε -ε∂ α x ∂ t v, (t, x) ∈ [0, T ε ) × R d , where L ε α = (u ε • ∇)∂ α x w ε -∂ α x (u ε • ∇)w ε -∂ α x (w ε • ∇)u . Since div u ε = div w ε = 0, we have ∂ α x w ε , ∇∂ α x q ε = 0, ∂ α x w ε , (u ε • ∇)∂ α x w ε = 0. We also have ∂ α x w ε , M j ∂ α x j ∂ α x z ε = -∂ α x z ε , M T j ∂ α x j ∂ α x w ε . Hence, a classical energy estimate yields d dt ∂ α x w ε 2 + ε ∂ α x z ε 2 + 2 ∂ α x z ε 2 = 2 ∂ α x w ε , L ε α -2 √ ν d j=1 ∂ α x w ε , M j ∂ x j ∂ α x v ε -2ε ∂ α x z ε , ∂ α x ∂ t v . (5.16)
By the Moser-type calculus inequalities, we have

2 ∂ α x w ε , L ε α ≤ c w ε 2 m .
In a similar way to (3.11) and (3.12), we estimate the last two terms in (5.16) and obtain

2 √ ν ∂ α x w ε , M j ∂ x j ∂ α x v ε ≤ c w ε m e -t ε , 2ε ∂ α x z ε , ∂ α x ∂ t v ≤ ∂ α x z ε 2 + cε 2 .
Taking into account these estimates, we add (5.16) for all α to get

d dt w ε 2 m + ε z ε 2 m + z ε 2 m ≤ c w ε 2 m + c w ε m e -t ε + cε 2 .
Thus, a Gronwall inequality together with condition (5.11) implies that

w ε (t) 2 m + ε z ε (t) 2 m + t 0 z ε (t ) 2 m ds ≤ cε 2 .
This proves (5.12)-(5.13).

Finally, noting z ε + v ε = v ε -v, taking the divergence of the first equation in (5.15), we have

-∆q ε = div (u ε • ∇)w ε + (w ε • ∇)u + √ ν d j=1 M j ∂ x j (v ε -v) .
By (5.12), this implies (5.14). The proof of Theorem 5.1 is finished. 2

As a variant of Theorem 4.1, we also have a result on the uniform global existence and global convergence for relaxed incompressible Euler equations. The proof is omitted here. (5.17)

u ε 0 m + √ ε v ε 0 m ≤ δ,
then for all ε ∈ (0, 1], the Cauchy problem (5.5)-(5.7) admits a unique global solution (u ε , p ε , v ε ) satisfying

u ε (t) 2 m + ε v ε (t) 2 m + t 0 ∇u ε (t ) 2 m-1 + ∇p ε (t ) 2 m-1 + v ε (t ) 2 m dt (5.18) ≤ c u ε 0 2 m + ε v ε 0 2 m , ∀ t ≥ 0.
Moreover, there exist functions u ∈ L ∞ (R + ; H m ) and p, v ∈ L 2 (R + ; H m ), such that, as ε → 0 and up to subsequences,

(5.19) u ε -u, weakly- * in L ∞ (R + ; H m ).
(5.20)

∇p ε -∇p, v ε -v, weakly in L 2 (R + ; H m ),
where

(5.21) v = - √ ν d j=1 M T j ∂ x j u,
and (u, p) is a unique solution to the Cauchy problem for incompressible Navier-Stokes equations (5.1)-(5.2) with initial value u 0 being the weak limit of u ε 0 in H m (up to subsequences).

6. Relaxed Euler systems with tensor variables 6.1. The system with Maxwell's constitutive relation.

We consider system (1.5), namely, (6.1)

     ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + div π = 0, ε∂ t π + νσ(u) + λ(div u)I d = -π, in R + × R d ,
with initial condition (6.2) t = 0 : (ρ, u, π) = (ρ ε 0 , u ε 0 , π ε 0 ), where π is a symmetric matrix variable of order d and ε > 0 is a small parameter.

Recall the inner product of two matrices as follows :

π : τ = d i,j=1 π ij τ ij , for π = (π ij ) 1≤i,j≤d , τ = (τ ij ) 1≤i,j≤d .
We denote

|π| 2 = π : π = d i,j=1 π 2 ij , and π, τ = R d π : τ dx, π 2 = π, π , π 2 m = |α|≤m ∂ α x π 2 .
Obviously, div(π T u) = u • div π + π : ∇u, π : I d = tr(π). Now we consider the pair of entropy-entropy flux. Let (E 0 , F 0 ) be the pair of entropy-entropy flux for the Euler equations, defined by (2.7). By (6.1), we have

∂ t E 0 (ρ, u) + div F 0 (ρ, u) + u • div π = 0.
By the symmetry of π and the definition of σ(u), we have

1 2 π : σ(u) = π : ∇u - 1 d (div u)tr(π).
Hence, the third equation in (6.1) yields an energy equality ε 4ν

∂ t |π| 2 + 1 2ν |π| 2 + π : ∇u + λ 2ν - 1 d (div u)tr(π) = 0.
In order to eliminate the term containing tr(π), we take the trace of the third equation in (6.1). Since tr(σ(u)) = 0, it yields ε∂ t tr(π) + dλ div u = -tr(π), which implies that ε 2 ∂ t (tr(π)) 2 + (tr(π)) 2 + dλ(div u)tr(π) = 0.

It follows from these two energy equalities that ε∂ t P 1 (π) + 2P 1 (π) + π : ∇u = 0. where (6.3)

P 1 (π) = 1 4ν |π| 2 + ω(tr(π)) 2 , with ω = 2ν d 2 λ - 1 d .
Let V 1 = (ρ, u, π). We define functions E 1 and F 1 by (6.4)

E 1 (V 1 ) = E 0 (ρ, u) + εP 1 (π), F 1 (V 1 ) = F 0 (ρ, u) + πu.
It is easy to check that a smooth solution V 1 of (6.1) satisfies the energy equality (6.5)

∂ t E 1 (V 1 ) + div F 1 (V 1 ) + 2P 1 (π) = 0.
Therefore, (E, F ) is a pair of entropy-entropy flux of system (6.1). Moreover, if ω ≥ 0, then

P 1 (π) ≥ 1 4ν |π| 2 ,
and if ω < 0, by using (tr(π)) 2 ≤ d|π| 2 , we have

P 1 (π) ≥ 1 4ν (1 + ωd)|π| 2 = 1 2dλ |π| 2 .
Hence, (6.6)

P 1 (π) ≥ min 1 4ν , 1 2dλ |π| 2 .
Since π -→ P 1 (π) is quadratic, E 1 is a strictly convex function with respect to the conservative variable (ρ, ρu, π) for ρ > 0. As a consequence, E 1 is a strictly convex entropy. If we use the columns of π as variables instead of π, system (6.1) can be written in a standard form. We conclude that system (6.1) is symmetrizable hyperbolic (see [START_REF] Godunov | An interesting class of quasi-linear systems (Russian)[END_REF][START_REF] Friedrichs | Systems of conservation equations with a convex extension[END_REF][START_REF] Boillat | Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques[END_REF]) and its Cauchy problem admits a unique smooth solution defined in a finite time interval (see [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF][START_REF] Kato | The Cauchy problem for quasilinear symmetric hyperbolic systems[END_REF][START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF]).

The local convergence of system (6.1) can be proved in a similar way to the proof of Theorem 3.1. Now we establish the global convergence of the system near a constant state (1, 0, 0) for V 1 . The result is stated as follows.

Theorem 6.1. Let m > d 2 + 1 be an integer. Assume (ρ ε 0 -1, u ε 0 , π ε 0 ) ∈ H m with π ε 0 being symmetric. There are two positive constants δ and c (independent of ε) such that if

ρ ε 0 -1 m + u ε 0 m + √ ε π ε 0 m ≤ δ,
then for all ε ∈ (0, 1], the Cauchy problem (6.1)-(6.2) admits a unique global solution (ρ ε , u ε , π ε ) satisfying

ρ ε (t) -1 2 m + u ε (t) 2 m + ε π ε (t) 2 m + t 0 ∇ρ ε (t ) 2 m-1 + ∇u ε (t ) 2 m-1 + π ε (t ) 2 m dt (6.7) ≤ c ρ ε 0 -1 2 m + u ε 0 2 m + ε π ε 0 2 m , ∀ t ≥ 0. Moreover, there exist functions (ρ, u, π) with (ρ -1, u) ∈ L ∞ (R + ; H m ) and π ∈ L 2 (R + ; H m ),
such that, as ε → 0 and up to subsequences,

(ρ ε , u ε ) -(ρ, u), weakly- * in L ∞ (R + ; H m ), π ε -π, weakly in L 2 (R + ; H m ),
where π = -νσ(u) -λ(div u)I d , and (ρ, u) is a unique solution to (2.11) for the compressible Navier-Stokes equations with initial value (ρ 0 , u 0 ) being the weak limit of (ρ ε 0 , u ε 0 ) in H m (up to subsequences). Proof. Let T > 0. We follow the same steps in the proof of Theorem 4.1 by considering energy estimates for V 1 defined on [0, T ]. Integrating (6.5) over R d , we have (6.8)

d dt R d E 1 (V 1 )dx + 2 R d P 1 (π)dx = 0,
which provides an L 2 estimate for V 1 with dissipation for π.

Next, for α ∈ N d with 1 ≤ |α| ≤ m, applying ∂ α x to (6.1) and taking the inner product in L 2 with 2h (ρ)∂ α x ρ, 2ρ∂ α x u, ν -1 ∂ α x π , we obtain an energy equality for V 1 . Then combining this equality with the energy for tr(∂ α x π), similarly to (4.10) and (6.5), we have

d dt R d h (ρ)|∂ α x ρ| 2 + ρ|∂ α x u| 2 + 2εP 1 (∂ α x π) dx + 4 R d P 1 (∂ α x π)dx = div B(U 1 )∂ α x U 1 , ∂ α x U 1 -2 d j=1 K α j -2 ∂ α x (ρ -1 div π) -ρ -1 ∂ α x div π, ρ∂ α x u ,
where the first two terms on the right-hand side are estimated in (4.14) and (4.16)-(4.17) with U 1 = (ρ, u). For the last term, since |α| ≥ 1, the Moser-type calculus inequalities yield

∂ α x (ρ -1 div π -ρ -1 div ∂ α x π, ρ∂ α x u ≤ c ∇ρ m-1 ∇u m-1 ∇π m-1 .
It follows that

d dt R d h (ρ)|∂ α x ρ| 2 + ρ|∂ α x u| 2 + 2εP 1 (∂ α x π) dx + 4 R d P 1 (∂ α x π)dx (6.9) ≤ c ∇ρ m-1 + ∇u m-1 ∇V 1 2 m-1 .
Adding (6.9) for all α with 1 ≤ |α| ≤ m, together with (6.8), we obtain (6.10)

E 1 (t) + 4D 1 (t) ≤ c ∇ρ m-1 + ∇u m-1 ∇V 1 2 m-1 .
where E 1 (t) is the total energy defined by

E 1 (t) = 2 R d E 0 (ρ, u)dx + 1≤|α≤m R d h (ρ)|∂ α x ρ| 2 + ρ|∂ α x u| 2 dx + 2εD 1 (t), with D 1 (t) = |α|≤m R d P 1 (∂ α x π)dx.
From (6.3)-(6.4) and (6.6), it is easy to see that D 1 (t) is uniformly equivalent to π 2 m , and

E 1 (t) is uniformly equivalent to ρ -1 2 m + u 2 m + ε π 2 m .
This provides an estimate in H m for V 1 with dissipation for π.

Finally, we consider a dissipation estimate for ∇ρ and ∇u. Let β ∈ N d with |β| ≤ m -1. Applying div ∂ β

x to the third equation in (6.1), we have

ε∂ t (∂ β x div π) + ν∆∂ β x u + µ∇(div ∂ β x u) = -∂ β x div π, where µ = ν + λ -2ν d .
Taking the inner product with ∂ β x u in L 2 , we have

ν ∇(∂ β x u) 2 + µ div(∂ β x u) 2 = ε ∂ t (∂ β x div π), ∂ β x u -∂ β x π, ∇∂ β
x u , which is a key relation to obtain the dissipation for ∇ρ and ∇u. Let 2κ = min(ν, λ) > 0. We know that ν

∇(∂ β x u) 2 + µ div(∂ β x u) 2 ≥ 2κ ∇(∂ β x u) 2 . Clearly, ∇∂ β x u, ∂ β x π ≤ κ ∇(∂ β x u) 2 + c ∂ β x π 2 , Therefore, κ ∂ β x (∇u) 2 ≤ c π 2 m + ε ∂ t (∂ β x div π), ∂ β x u .
The rest of the proof is similar to that of (4.18). By using analogous techniques, we obtain

∂ t (∂ β x div π), ∂ β x u = d dt ∂ β x div π, ∂ β x u + ∂ β x ((u • ∇)u + ∇h(ρ) + ρ -1 div π), ∂ β x div π , ∂ β x ∇h(ρ) 2 = - d dt ∂ β x ∇h(ρ), ∂ β x u + ∂ β x (h (ρ) div(ρu)), ∂ β x div u -∂ β x ((u • ∇)u + ρ -1 div π), ∂ β x ∇h(ρ) . Combining the last three relations yields d dt (2η 1 ∂ β x ∇h(ρ) -ε∂ β x div π), ∂ β x u + κ ∂ β x (∇u) 2 + η 1 ∂ β x ∇h(ρ) 2 ≤ c π 2 m + c ∇ρ m-1 + u m ∇V 1 2 m-1 ,
where η 1 > 0 is a sufficiently small constant. Adding this inequality for all |β| ≤ m-1, together with (6.10), we have

d dt   E 1 (t) + η 2 |β|≤m-1 (2η 1 ∂ β x ∇h(ρ) -ε∂ β x div π), ∂ β x u   + 4D 1 (t) + η 2 κ ∇u 2 m-1 + η 1 ∇h(ρ) 2 m-1 ≤ c π 2 m + c ∇ρ m-1 + u m ∇V 1 2 m-1
, where η 2 > 0 is a small constant to be chosen. Integration this inequality over [0, t] with t > 0 and taking η 2 > 0 sufficiently small, as in the proof of Lemma 4.5, we obtain

ρ(t) -1 2 m + u(t) 2 m + ε π(t) 2 m + t 0 ∇ρ(t ) 2 m-1 + ∇u(t ) 2 m-1 + π(t ) 2 m dt ≤ c ρ 0 -1 2 m + u 0 2 m + ε π 0 2 m , ∀ t ∈ [0, T ]
, which implies (6.7) and the uniform global existence of solutions. The global convergence of system (6.1) can be performed in a similar way to the proof of Theorem 4.1. 2 6.2. The system with revised Maxwell's constitutive relations.

We consider system (1.7) with the revised Maxwell's constitutive relations, namely, (

           ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + div π 1 + ∇π 2 = 0, ε 1 ∂ t π 1 + νσ(u) = -π 1 , ε 2 ∂ t π 2 + λ div u = -π 2 , in R + × R d . 6.11)
with initial condition (6.12) t = 0 : (ρ, u, π 1 , π 2 ) = (ρ ε 0 , u ε 0 , π ε 10 , π ε 20 ), where π 1 is a square matrix variable of order d, π 2 is a scalar variable and ε = (ε 1 , ε 2 ). Similarly to π in (6.1), we may assume that π 1 is symmetric and we know that this assumption is not a restriction condition.

As mentioned in the introduction, tr(π 1 ) satisfies a linear equation (6.13) ε 1 ∂ t tr(π 1 ) + tr(π 1 ) = 0, and tr(π 1 (t, •)) = 0 for all t > 0 if and only if tr(π ε 10 ) = 0. Condition tr(π ε 10 ) = 0 is used in the study of (6.11)-(6.12) in [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF]. However, it is a real restriction on the initial data. In this subsection, we consider the Cauchy problem (6.11)-(6.12) without this condition. The local convergence of system (6.11) can be proved in a similar way to the proof of Theorem 3.1 as ε 1 = ε 2 → 0. Now we establish the global convergence of the system near a constant state (1, 0, 0, 0) for V 2 = (ρ, u, π 1 , π 2 ).

We start with the definition of a pair of entropy-entropy flux of the system. From (6.1) and (6.11), we have successively

∂ t E 0 (ρ, u) + div F 0 (ρ, u) + u • (div π 1 + ∇π 2 ) = 0, ε 1 4ν ∂ t |π 1 | 2 + 1 2ν |π 1 | 2 + 1 2 π 1 : σ(u) = 0, ε 2 2λ ∂ t |π 2 | 2 + 1 λ |π 2 | 2 + π 2 div u = 0. Since 1 2 π 1 : σ(u) = π 1 : ∇u - 1 d (div u)tr(π 1 ), we obtain ∂ t E 0 (ρ, u) + ε 1 4ν |π 1 | 2 + ε 2 2λ |π 2 | 2 + div F 0 (ρ, u) + (π 1 + π 2 I d )u + 1 2ν |π 1 | 2 + 1 λ |π 2 | 2 - 1 d (div u)tr(π 1 ) = 0.
In order to eliminate the last term on the right hand-side of the above equality, we multiply the last equation in (6.11) This provides a pair of entropy-entropy flux. However, the strict convexity of the entropy and the negativity of the entropy production are not guaranteed. To remedy this, we use again (6.13) to obtain (6.15) 2a(ε)ε 1 ∂ t |tr(π 1 )| 2 + 4a(ε)|tr(π 1 )| 2 = 0, where a(ε) > 0 is a constant defined by

a(ε) = 1 4d 2 λ ε 2 ε 1 + 1 2 .
Finally, adding (6.14) and (6.15), it yields (6.16) ∂ t E 2 (V 2 ) + div F 2 (V 2 ) + P 2 (π 1 , π 2 ) = 0, where V 2 = (ρ, u, π 1 , π 2 ), (6.17) Therefore, E 2 is a strictly convex entropy with respect to (ρ, ρu, π 1 , π 2 ) for ρ > 0 and the entropy production -P 2 is negative.

         E 2 (V 2 ) = E 0 (ρ, u) + Ẽ2 (π 1 , π 2 ), F 2 (V 2 ) = F 0 (ρ, u) + (π 1 + π 2 I d )
The main result of this subsection is stated as follows.

Theorem 6.2. Let ε = (ε 1 , ε 2 ) and m > d 2 +1 be an integer. Assume (ρ ε 0 -1, u ε 0 , π ε 10 , π ε 20 ) ∈ H m with π ε 10 being symmetric. There are two positive constants δ and c (independent of ε) such that if

ρ ε 0 -1 m + u ε 0 m + √ ε 1 π ε 10 m + √ ε 2 π ε 20 m + ε 2 √ ε 1 tr(π ε 10 ) m ≤ δ,
then for all ε 1 , ε 2 ∈ (0, 1], the Cauchy problem (6.11)-(6.12) admits a unique global solution

(ρ ε , u ε , π ε 1 , π ε 2 ) satisfying ρ ε (t) -1 2 m + u ε (t) 2 m + ε 1 π ε 1 (t) 2 m + ε 2 π ε 2 (t) 2 m + ε 2 2 ε 1 tr(π ε 1 (t)) 2 m + t 0 ∇ρ ε (t ) 2 m-1 + ∇u ε (t ) 2 m-1 + π ε 1 (t ) 2 m + π ε 2 (t ) 2 m dt (6.20) ≤ c ρ ε 0 -1 2 m + u ε 0 2 m + ε 1 π ε 10 2 m + +ε 2 π ε 20 2 m + ε 2 2 ε 1 tr(π ε 10 ) 2 m , ∀ t ≥ 0.
Moreover, there exist functions (ρ-1, u) ∈ L ∞ (R + ; H m ) and π 1 , π 2 ∈ L 2 (R + ; H m ), such that, as (ε 1 , ε 2 ) → 0 and up to subsequences, (ρ ε , u ε ) -(ρ, u), weakly- * in L ∞ (R + ; H m ), (π ε 1 , π ε 2 ) -(π 1 , π 2 ), weakly in L 2 (R + ; H m ), where π 1 = -νσ(u), π 2 = -λ div u, and (ρ, u) is a unique solution to (2.11) for the compressible Navier-Stokes equations with initial value (ρ 0 , u 0 ) being the weak limit of (ρ ε 0 , u ε 0 ) in H m (up to subsequences). Proof. The proof is similar to that of Theorem 6.1. It follows from energy and dissipation estimates that we give below. The detail of the proof is omitted here. Let T > 0 and V 2 be a smooth solution defined on [0, T ].

Firstly, the entropy equality (6. From (6.19), it is easy to see that

R d P 2 (∂ α x π 1 , ∂ α x π 2 )dx = div B(U 1 )∂ α x U 1 , ∂ α x U 1 -2 d j=1 K α j -2 ∂ α x (ρ -1 div π 1 ) -ρ -1 ∂ α x div π 1 , ρ∂ α x u -2 ∂ α x (ρ -1 ∇π 2 ) -ρ -1 ∂ α x ∇π 2 ,
D 2 (t) ≥ 1 2ν π 1 | 2 m + 1 2λ π 2 | 2 m .
Thus, (6.22) together with (6.17)-( 6.18) provides an estimate in H m for V 2 with dissipation for π 1 and π 2 .

For the dissipation estimate of ∇ρ and ∇u, let β ∈ N d with |β| ≤ m -1. From the equations in (6.11), we obtain successively

∂ β x ∇h(ρ) 2 = - d dt ∂ β x ∇h(ρ), ∂ β x u + ∂ β x (h (ρ) div(ρu)), ∂ β x div u -∂ β x ((u • ∇)u + ρ -1 (div π 1 + ∇π 2 )), ∂ β x ∇h(ρ) , ν ∇(∂ β x u) 2 + ν 1 - 2 d div(∂ β x u) 2 = ε 1 ∂ t (∂ β x div π 1 ), ∂ β x u -∂ β x π 1 , ∇∂ β x u , λ div(∂ β x u) 2 = ε 2 ∂ t (∂ β x ∇π 2 ), ∂ β x u -∂ β x π 2 , div ∂ β x u , ∂ t ∂ β x (ε 1 div π 1 + ε 2 ∇π 2 ), ∂ β x u = d dt ∂ β x (ε 1 div π 1 + ε 2 ∇π 2 ), ∂ β x u + ∂ β x (ε 1 div π 1 + ε 2 ∇π 2 ),

(1. 1 )

 1 ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + div π = 0, in R + × R d ,1 with the constitutive law for a Newtonian fluid (1.2) π = -νσ(u) -λ(div u)I d , where (1.3) σ(u) = ∇u + (∇u) T -2 d (div u)I d .

Theorem 4 . 1 .

 41 (Uniform global existence and global convergence for the relaxed Euler system)

Lemma 4 . 4 .

 44 (Dissipation estimates of ∇ρ and ∇u) There are positive constants c 2 , c 3 and c 4 such that for all ε ∈ (0, 1] 2 and all t ∈ [0, T ], it holds d dt |β|≤m-1

Let d ≥ 2

 2 and ν > 0. Consider incompressible Navier-Stokes equations (5.1) ∂ t u + (u • ∇)u + ∇p = ν∆u, (5.2) div u = 0, in R + × R d , where p(t, x) vanishes sufficiently fast as |x| → +∞. Using (5.3) div (u • ∇)u = (u • ∇) div u + tr (∇u) 2 , tr (∇u) 2 = d i,j=1 ∂u i ∂x j ∂u j ∂x i , it is known that p satisfies a Poisson equation (5.4) -∆p = tr (∇u) 2 ,

M

  j ∂ x j v. Multiplying the second equation in (5.6) on the left by M i and summing up for all i = 1, • • • , d, by (2.

Theorem 5 . 2 . 2 + 1

 5221 (Uniform global existence and global convergence) Let m > d be an integer. We assume (u ε 0 , v ε 0 ) ∈ H m and (5.8) holds. There are two positive constants δ and c (independent of ε) such that if

2 m- 1 ≤ c π 1 2 m + π 2 2 m + c ∇ρ m- 1 + u m ∇V 2 2 m- 1 . 2 6. 3 .

 212212123 ∂ β x ((u • ∇)u + ρ -1 (div π 1 + ∇π 2 )) + ∂ β x (ε 1 div π 1 + ε 2 ∇π 2 ), ∂ β x (∇h(ρ)) . It follows that d dt   E 2 (t) + η 2 |β|≤m-1 ∂ β x (2η 1 ∇h(ρ) -ε 1 div π 1 -ε 2 ∇π 2 ), ∂ β x u   + 2D 2 (t) + η 2 κ ∇u 2 m-1 + η 1 ∇h(ρ)This estimate implies (6.20) and then the result in Theorem 6.2 follows. Incompressible case. Let d ≥ 2. In the incompressible case, both systems (6.1) and (6.11) lead to the following system (6.23)∂ t u + (u • ∇)u + ∇p + div π = 0, ε∂ t π + ν ∇u + (∇u) T = -π, in R + × R dand (6.24) div u = 0, with initial condition (6.25) t = 0 : (u, π) = (u ε 0 , π ε 0 ),

  Case 1 : r ≥ d 2 . These matrices can be constructed explicitly as follows.Let O i (1 ≤ i ≤ d) be any orthogonal matrices of order d. We take

	The first condition in (2.1) implies that M i is a full-rank matrix for all 1 ≤ i ≤ d. If the entries
	of M i are regarded as unknown variables, (2.1) represents at most (d + 1)d 3 /2 independent
	equations with d 2 r unknown variables. We observe that for a fixed pair (d, r), if (2.1) admits a
	solution M i (1 ≤ i ≤ d), then for all r > r, the matrices Mi (1 ≤ i ≤ d) (of order d × r) defined
	by Mi = (M i , 0 d×(r-r) ) still satisfy (2.1), where 0 d×s is the zero matrix of order d × s for some
	integer s ∈ N. The solution for (2.1) with d = 1 is obvious (take r = 1). For d ≥ 2, we consider
	the following two cases where Case 1 shows that the solution of (2.1) exists for all r ≥ d 2 and
	Case 2 shows that the solution with square matrices of (2.1) exists only when d = 1, 2, 4, 8.

  by tr(π 1 )/dλ and (6.13) by ε 2 π 2 /ε 1 dλ to yield tr(π 1 ) + div F 0 (ρ, u)+ (π 1 + π 2 I d )u

			ε 2 dλ	tr(π 1 )∂ t π 2 +	1 d	(div u)tr(π 1 ) +	1 dλ	π 2 tr(π 1 ) = 0,
				ε 2 dλ	π 2 ∂ t tr(π 1 ) +	ε 2 dλε 1	π 2 tr(π 1 ) = 0.
	This implies that
		ε 2 dλ	∂ t π 2 tr(π 1 ) +	1 d	(div u)tr(π 1 ) +	1 dλ	ε 2 ε 1	+ 1 π 2 tr(π 1 ) = 0.
	Thus,
	(6.14)	∂ t E 0 (ρ, u) + π 2 + ε 1 4ν |π 1 | 2 + ε 2 2λ |π 2 | 2 + ε 2 dλ 1 2ν |π 1 | 2 + 1 λ |π 2 | 2 + 1 dλ ε 2 ε 1 + 1 π 2 tr(π 1 ) = 0.

  P 2 (π 1 , π 2 )dx = 0, which provides an L 2 estimate for V 2 with dissipation for π 1 and π2 . Next, let α ∈ N d with 1 ≤ |α| ≤ m. Applying ∂ αx to (6.11) and taking the inner product inL 2 with 2h (ρ)∂ α x ρ, 2ρ∂ α x u, ν -1 ∂ α x π 1 , 2λ -1 ∂ α x π 2, together with the energy equality for tr(∂ α x π 1 ), we obtain an energy equality for V 2 :d dt R d h (ρ)|∂ α x ρ| 2 + ρ|∂ α x u| 2 + 2 Ẽ2 (∂ α x π 1 , ∂ α x π 2 ) dx + 2

			16) yields
	(6.21)	d dt R d	E 2 (V 2 )dx +	R d

  ρ∂ α x u , where the first two terms on the right-hand side are estimated in (4.14) and (4.16)-(4.17). Together with (6.21), we further obtain(6.22) E 2 (t) + 2D 2 (t) ≤ c ∇ρ m-1 + ∇u m-1 ∇V 2 2 m-1 ,where E 2 (t) is the total energy defined byE 2 (t) = 2 R d E 0 (ρ, u)dx + 1≤|α≤m R d h (ρ)|∂ α x ρ| 2 + ρ|∂ α x u| 2 dx + 2

			|α|≤m R d	Ẽ2 (∂ α x π 1 , ∂ α x π 2 )dx,
	and
	D 2 (t) =	|α|≤m R d	P 2 (∂ α x π 1 , ∂ α x π 2 )dx.

where π is a square matrix variable of order d and ε > 0 is a small parameter. Obviously, (6.23) is an approximation of the incompressible Navier-Stokes equations (5.1)-(5.2). Applying div div to the second equation in (6.23), we have ε∂ t (div div π) + div div π = 0. Suppose now (6.26) div u ε 0 = 0, div div π ε 0 = 0. We obtain div div π(t, •) = 0 for all time t > 0. In this case, we still have (6.27) -∆p = tr (∇u) 2 .

Similarly to Proposition 5.1, we have the following result of which the proof is omitted.

Proposition 6.1. Let T > 0 and (u ε 0 , π ε 0 ) ∈ H 2 satisfying (6.26). Let (u, p, π) be a solution of (6.23)-(6.25) with regularity

Then the incompressibility condition (6.24) is equivalent to (6.27) in (0, T ) × R d . As a consequence, we also have div div π = 0 in (0, T ) × R d .

Similarly to Theorems 5.1-5.2, we can prove the local and global convergences of system (6.23)-(6.24) to incompressible Navier-Stokes equations (5.1)-(5.2). The detail is omitted here.


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The project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 857484