Bernard Ducomet

Raphaël Danchin

Rapha Ël Danchin

EXISTENCE OF STRONG SOLUTIONS WITH CRITICAL REGULARITY TO A POLYTROPIC MODEL FOR RADIATING FLOWS

Keywords: Radiation hydrodynamics, Navier-Stokes system, diffusive limit, critical regularity, P 1-approximation

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Introduction i

We consider the barotropic version of a model of radiation hydrodynamics. Our main goal is to provide the rigorous justification of asymptotics that have been investigated formally and numerically by Lowrie, Morel and Hittinger [START_REF] Lowrie | The coupling of radiation and hydrodynamics[END_REF], and mathematically by the second author and Š. Nečasová in [10,11,12] in the finite energy weak solutions framework.

The fluid is described by standard classical fluid mechanics for the mass density ̺ and the velocity field u as functions of the time t ∈ R + and of the (Eulerian) spatial coordinate x ∈ Ω where Ω is either the whole space R n or some periodic box T n with n ≥ 2.

Radiation acts through some radiative momentum source S F which is given by

S F = 1 c ∞ 0 S n-1 ωS d ω dν,
where c is the light speed.

The radiative source S = S(t, x, ω, ν) depends on the direction vector ω ∈ S n-1 (where S n-1 denotes the unit sphere of R n ), and on the frequency ν ≥ 0 of the photons, and is given by

S = σ a B(ν, ̺) -I + σ s Ĩ -I where Ĩ := 1 |S n-1 | S n-1 I d ω.
The radiative intensity I obeys the transfer equation

i4 i4 (1.1) 1 c ∂ t I + ω • ∇ x I = S in (0, T ) × Ω × S n-1 × (0, ∞).
Date: March 23, 2016.

In the present paper, as in [7, 8], we make the following simplifying assumptions (1) Isotropy : the transport coefficients σ a and σ s are independent of ω ;

(2) 'Gray' hypothesis : σ a and σ s are independent of ν ;

(3) 'P 1 hypothesis' : the averaged radiative intensity I := ∞ 0 I dν is given by the ansatz Idec Idec (1.2)

I = I 0 + ω • I 1 ,
where I 0 and I 1 are independent of ω and ν. Plugging (1.2) in (1.1), and computing the 0th and 1st order momentum with respect to ω, we find out the following evolution equations for I 0 and I 1 (keeping the same notation B for the distribution function averaged in ν )

J4bis J4bis (1.3) 1 c ∂ t I 0 + 1 n div x I 1 = σ a (̺)(B(̺) -I 0 ), J4ter J4ter (1.4) 1 c ∂ t I 1 + ∇ x I 0 = (σ a (̺) + σ s (̺)) I 1 .
Besides, the radiative force is now given by eq:SF eq:SF (1.5)

S F = σ a (̺) + σ s (̺) n I 1 .
In order to identify the most relevant asymptotic regimes, we rewrite the equations in dimensionless form. To this end, introduce some reference hydrodynamical quantities (length, time, velocity, density, pressure): L, T , Ū , ̺, p, and reference radiative quantities (radiative intensity, absorption and scattering coefficients and equilibrium function): Ī, σa , σs and B.

Let Sr := L/ T Ū , M a := Ū / √ ̺p and Re := Ū ̺ L/μ be the Strouhal, Mach and Reynolds numbers corresponding to hydrodynamics. Let also define C := c/ Ū , L := Lσ a , L s := σs /σ a , various dimensionless numbers corresponding to radiation. In all that follows, we assume our flow to be strongly under-relativistic so that C is large.

Choosing B = Ī, we discover that the evolution of the dimensionless unknowns (still denoted in the same way) is governed by the following system of equations

               Sr ∂ t ̺ + div (̺ u) = 0, Sr ∂ t (̺ u) + div (̺ u ⊗ u) + 1 M a 2 ∇p -1 Re div (µ∇ u+ t ∇ u)+∇(λdiv u) = L σa+Lsσs n I 1 , Sr C ∂ t I 0 + 1 n div I 1 = Lσ a (B -I 0 ) , Sr C ∂ t I 1 + ∇I 0 = -L (σ a + L s σ s ) I 1
, where ̺ = ̺(t, x) ∈ R + and u = u(t, x) stand for the density and pressure, respectively, p = P (ρ) is the pressure, λ = λ(ρ) and µ = µ(ρ) are the viscosity coefficients. The given functions P, λ and µ are supposed sufficiently smooth, and we make the following strict ellipticity assumption ν := λ + 2µ > 0 and µ > 0. In our recent work [8], we gave a mathematical justification of the low Mach number asymptotics. In the present paper, we investigate another type of physically relevant asymptotic regimes, which are of diffusive type. They correspond to the case where C is large and all the other dimensionless numbers, but L and L s , are of order 1. To make it more concrete, take

M a = Sr = Re = 1, C = ε -1 , ̺ = P ′ (̺) = B ′ (̺) = σ a (̺) = σ s (̺) = 1,
where ε is a small positive number, bound to tend to 0.

Because we shall focus on small perturbations of the reference density ̺ = 1, it is convenient to introduce the new unknown b := B(̺) -B [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]. In this context, all the functions of ̺ may be written in terms of b. Setting j 0 := I 0 -B(1) and j 1 := I 1 , and using exponents to emphasize the dependency with respect to ε, we eventually get the following system eq:NSrad eq:NSrad (1.6)

                 ∂ t b ε + u ε • ∇b ε + (1 + k 1 (b ε ))div u ε = 0, ∂ t u ε + u ε • ∇ u ε -(1 + k 2 (b ε ))A u ε + (1 + k 3 (b ε ))∇b ε = L(1+Ls) n (1 + k 4 (b ε )) j ε 1 , ε∂ t j ε 0 + 1 n div j ε 1 = L(b ε -j ε 0 ), ε∂ t j ε 1 + ∇j ε 0 = -L(1 + L s ) j ε 1 ,
with A := µ∆ + (λ + µ)∇div and where k 1 , k 2 , k 3 , k 4 are smooth functions vanishing at 0.

Formal asymptotics s:formal

Let us first present some formal computations so as to exhibit the limit equations we can get from (1.6) in different types of diffusive asymptotic regimes. We restrict to the case where the following necessary and sufficient linear stability condition (derived in [7]) is fulfilled eq:stabcond eq:stabcond (2.1)

nνL > ε 2 + L s 1 + L s •
Note that (2.1) implies that lim inf Lε -1 > 0 for ε going to 0.

In all that follows, it is assumed that (b ε , u ε , j ε 0 , j ε 1 ) converges to (b, u, j 0 , j 1 ) in some suitable space with enough regularity to pass to the limit in the nonlinear terms.

2.1. Case L ≈ ε and L s → +∞. Denoting by P the L 2 orthogonal projector on divergence free vector fields, we get eq:Pj1 eq:Pj1 (2.2) P j ε 1 (t) = e -L ε (1+Ls)t P j ε 1 (0).

Hence P j ε 1 tends to 0 for ε → 0.

2.1.1. Subcase L 2 L s → 0. Setting Q := Id -P, we see that the equation for j ε 0 entails that Q j ε 1 = O(ε). Next, the equation for Q j ε 1 implies that ∇j ε 0 goes to 0, too, because ε 2 L s → 0. Assuming that j 0 decays to 0 at infinity, this yields j 0 = 0.

From the equation for j ε 1 , we also get eq:rad eq:rad (2.3) -L(1 + L s ) j ε 1 = ∇j ε 0 + O(ε).

Hence ε(1 + L s ) j ε 1 goes to 0 and (b, u) thus satisfies the barotropic Navier-Stokes equations. In other words, the radiative effect becomes negligible in the asymptotic L ≈ ε and ε 2 L s → 0 with L s → +∞ 2.1.2. Subcase lim ε→0 L 2 L s ∈ (0, +∞). This is the so-called nonequilibrium diffusion regime. The analysis of the previous paragraph shows that j ε 1 = O(ε) (hence j 1 = 0) and that (2.3) holds true. The new fact is that the equation for j ε 0 combined with (2.3) implies that eq:radb eq:radb (2.4)

∂ t j ε 0 + L ε j ε 0 -b ε - 1 n L ε 1 L 2 L s ∆j ε 0 = O(ε).
Now, if we assume that L ε → κ nν and L 2 L s → m ν 2 , for some m ∈ (0, +∞) and κ > 1 (see (2.1)), then (b, u) satisfies the following compressible Navier-Stokes equations coupled with a parabolic equation eq:noneq1 eq:noneq1 (2.5)

       ∂ t b + u • ∇b + (1 + k 1 (b))div u = 0, ∂ t u + u • ∇ u -(1 + k 2 (b))A u + (1 + k 3 (b))∇b + 1 n (1 + k 4 (b))∇j 0 = 0, ∂ t j 0 + κ nν j 0 -b -ν 2 nm ∆j 0 = 0.
2.1.3. Subcase L 2 L s → +∞. We still have j ε 1 = O(ε), (2.3) and thus (2.4) holds true. Now, as L 2 L s → +∞ and L ≈ ε, the r.h.s. of (2.4) tends to 0. Therefore, if we assume as before that L/ε → κ/(nν) then we find out that (b, u, j 0 ) satisfies the following degenerate nonequilibrium diffusion system eq:noneq2 eq:noneq2 (2.6)

     ∂ t b + u • ∇b + (1 + k 1 (b))div u = 0, ∂ t u + u • ∇ u -(1 + k 2 (b))A u + (1 + k 3 (b))∇b + 1 n (1 + k 4 (b))∇j 0 = 0, ∂ t j 0 + κ nν (j 0 -b) = 0. 2.2. Case ε ≪ L ≪ 1.
Recall that we have (2.3) while the equation for j ε 0 implies that eq:radter eq:radter (2.7) div j ε 1 = nL(b ε -j ε 0 ) + O(ε). Hence Q j 1 = 0 (as L → 0), and eq:radquater eq:radquater (2.8) ∆j ε 0 + nL 2 (1 + L s )(b ε -j ε 0 ) = O(ε) + O(εL(1 + L s )). Subcase L 2 L s → 0. Then (2.8) implies that ∆j 0 = 0 and thus j 0 ≡ 0 (if one assumes that j 0 → 0 at ∞). Consequently, (2.3) implies that the radiative force in the velocity equation tends to 0 when ε goes to 0. Therefore (b, u) just satisfies the classical compressible Navier-Stokes equation.

Subcase ν 2 L 2 L s → m ∈ (0, +∞). We have j 1 = 0, and Relations (2.3), (2.8) imply that (b, u, j 0 ) fulfills the following Navier-Stokes-Poisson system eq:eq1 eq:eq1 (2.9)

     ∂ t b + u • ∇b + (1 + k 1 (b))div u = 0, ∂ t u + u • ∇ u -(1 + k 2 (b))A u + (1 + k 3 (b))∇b + 1 n (1 + k 4 (b))∇j 0 = 0, -ν 2 ∆j 0 + mn(j 0 -b) = 0.
Subcase L 2 L s → +∞. Then (2.8) implies that j 0 = b. Combining with (2.3), we thus find out that (b, u) fulfills the following compressible Navier-Stokes equation with modified pressure law eq:eq2 eq:eq2 (2.10)

∂ t b + u • ∇b + (1 + k 1 (b))div u = 0, ∂ t u + u • ∇ u -(1 + k 2 (b))A u + 1 + 1 n + k 3 (b) + 1 n k 4 (b) ∇b = 0.
2.3. Case νL → ℓ ∈ (0, +∞).

Subcase ν 2 L 2 L s → m ∈ [0, +∞). Passing to the limit in (2.8) gives eq:eq1bis eq:eq1bis (2.11) -ν 2 ∆j 0 + n(ℓ 2 + m)(j 0 -b) = 0.

So we get System (2.9) for (b, j 0 , u) with the last equation replaced by (2.11).

Subcase L s → +∞. Exactly as in the case L → 0, we get j 0 = b, j 1 = 0, and (b, u) satisfies (2.10).

2.4. Case L → +∞. Relation (2.3) implies that j 1 = 0, and thus, according to (2.7), we have j 0 = b. Therefore (2.3) implies that L(1 + L s ) j ε 1 → ∇b, and (b, u) thus satisfies (2.10).

To make a long story short, the above formal computations pointed out five types of asymptotic regimes. They are governed by [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF] The ordinary compressible Navier-Stokes equations with null radiation (if L → 0 and The rest of the paper is devoted to justifying rigorously the last four asymptotics globally in time in the framework of small solutions with critical regularity. In the next section, we introduce a few notations that will be needed to define our functional framework, and give an overview of the strategy. Section 4 is devoted to a fine analysis of the linearized equations (1.6) about (0, 0, 0, 0), which turns out to be essentially the key to proving global results and justifying the diffusive asymptotics we have in mind. The next three sections are devoted to the rigorous justification of the nonequilibrium diffusion regime L ≈ ε and L s L 2 1, the equilibrium diffusion regime L → +∞ and of the Poisson type diffusion regime (ε ≪ L 1 and ν 2 L 2 L s → m ∈ (0, +∞)). In all of those sections, we establish a global-in-time existence result for the expected limit system, and for (1.6) supplemented with uniform estimates (for coefficients L and L s satisfying the assumptions of the studied regime), and eventually show the convergence of the solutions of (1.6) to those of the expected limit system. Some estimates, of independent interest, for the solutions to a class of linear ODEs corresponding to the linearized equations of (1.6) in the Fourier space are postponed in the appendix.

L 2 L s → 0); (

Functional framework and overview of the method s:results

The functional framework we shall work in is modeled on the linearized equations corresponding to (1.6), and is thus the same as in our first paper [7] devoted to the global well-posedness issue in critical regularity spaces for small perturbations of a stable constant state. The key to proving asymptotic results however, is to prescribe norms depending on the parameters ε, L and L s , so as to get optimal uniform estimates, enabling our justifying rigorously the different diffusive asymptotics exhibited above.

Let us first very briefly recall the definition of homogeneous Besov spaces Ḃs 2,1 (the reader is referred to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2 for more details). For simplicity, we focus on the R n case (adapting the construction to the torus being quite straightforward). Fix some smooth radial bump function χ : R n → [0, 1] with χ ≡ 1 on B(0, 1/2) and χ ≡ 0 outside B(0, 1), nonincreasing with respect to the radial variable. Let ϕ(ξ) := χ(ξ/2) -χ(ξ). The elementary spectral cut-off operator entering in the Littlewood-Paley decomposition is defined by ∆j

u := ϕ(2 -j D)u = F -1 (ϕ(2 -j D)Fu), j ∈ Z
where we denote by F the standard Fourier transform in R n .

For any s ∈ R, the homogeneous Besov space Ḃs 2,1 is the set of tempered distributions u so that u Ḃs

2,1 := j∈Z 2 js ∆j u L 2 < ∞,
and eq:LFconv eq:LFconv (3.1)

lim λ→+∞ χ(λD)u = 0 in L ∞ .
As pointed out in [7], scaling considerations that neglect low order terms of System (1.6)

suggest that critical regularity is Ḃ n 2 -1 2,1
for u 0 , j 0,0 and j 1,0 , and Ḃ n 2 2,1 for b 0 . However, to handle lower order terms, one has to make additional assumptions for the low frequencies. To this end, it is convenient to introduce the following notation (where η stands for a positive parameter)

u ℓ,η Ḃs 2,1 := 2 k ≤2η 2 ks ∆k u L 2 and u h,η Ḃs 2,1 := 2 k ≥η/2 2 ks ∆k u L 2 ,
and also u ℓ,η := 2 k ≤η ∆k u and u h,η :=

2 k >η ∆k u. Note that u ℓ,η Ḃs 2,1 ≤ C u ℓ,η Ḃs 2,1 and u h,η Ḃs 2,1 ≤ C u h,η Ḃs 2,1
. Because the Littlewood-Paley decomposition is not quite orthogonal, it is important to allow for a small overlap in the above definition of norms.

In some places, we will have to specify also the behavior for the middle frequencies, by considering for given 0

< η < η ′ , u m,η,η ′ Ḃs 2,1 := η≤2 k ≤η ′ 2 ks ∆k u L 2 .
Broadly speaking, our strategy to justify the different types of diffusive limits is as follows:

• Step 1: We prove 'uniform estimates' for the global solutions to (1.6), uniform meaning that we want a bound independent of ε, but the norm itself may depend 'in a nice way' of the parameters ε, L and L s . • Step 2: We show that the limit system is globally well-posed in the small data case.

• Step 3: We take advantage of estimates of Step 1 to exhibit weak compactness properties. Combining with the uniqueness result of Step 2, this allows to conclude to the convergence of the whole family of solutions of (1.6) to those of the limit system.

The most technical part is step 1, as it requires a fine analysis of the linearized equations of (1.6) about 0 that keeps track of the coefficients L, L s and ε. Schematically, in the Fourier space, one has to resort to different types of estimates for low, medium and high frequencies. The low frequency analysis is carried out by considering approximate eigenmodes of the system, that are constructed by a perturbative method from the (explicit) eigenmodes corresponding to null frequency. A part of the difficulty is that the 'fluid modes' are of parabolic type, hence the corresponding eigenvalues tend quadratically to 0 when the frequency size tends to 0 while the radiative modes are expected to be exponentially damped. The high frequency analysis is inspired by the corresponding one for the barotropic Navier-Stokes equations, after noticing that coupling between radiative and fluid unknowns occurs only through 0 order terms, and thus tend to be negligible for very high frequencies. Last but not least, medium frequency regime has to be looked at with the greatest care, as the low and high frequency regimes need not overlap. There is no general strategy for handling them, apart from guessing approximate eigenmodes of the system.

Uniform estimates for the linearized equations s:linear

In order to reduce the study to the case where the total viscosity ν := λ + 2µ is 1, and to get a symmetric first order system for the radiative unknowns, let us set eq:nu eq:nu (4.1) b, u, j 0 , j

1 (t, x) := (b ε , u ε , √ n j ε 0 , j ε 1 )(νt, νx).
Then (b ε , u ε , j ε 0 , j ε 1 ) satisfies (1.6) if and only if (b, u, j 0 , j 1 ) satisfies eq:NSradbis eq:NSradbis (4.2)

                 ∂ t b + u • ∇b + (1 + k 1 (b))div u = 0, ∂ t u + u • ∇ u -(1 + k 2 (b)) A u + (1 + k 3 (b))∇b = LM n (1 + k 4 (b)) j 1 , ε∂ t j 0 + 1 √ n div j 1 = L(b - √ n j 0 ), ε∂ t j 1 + 1 √ n ∇j 0 = -LM j 1 ,
with eq:notcoeff eq:notcoeff (4.3) M := 1 + L s , L := νL and A := ν -1 A.

The corresponding linearized system reads eq:diff eq:diff (4.4)

             ∂ t b + div u = f, ∂ t u -A u + ∇b -LM n j 1 = g, ε∂ t j 0 + 1 √ n div j 1 + L(j 0 - √ n b) = 0, ε∂ t j 1 + 1 √ n ∇j 0 + LM j 1 = 0.
On one hand, the coupling between the incompressible part of u and j 1 that is P u and P j 1 where P stands for the projector on divergence-free vector-fields is obvious as eq:Pu eq:Pu (4.5)

∂ t P u - µ ν ∆P u = LM n P j 1 , and
P j 1 (t) = e -LMt ε P j 1 (0),
hence in any functional space X we have eq:Pj1b eq:Pj1b (4.6)

LM P j 1 L 1 (X) ≤ ε P j 1 (0) X .
On the other hand, the coupling between b, d := Λ -1 div u, j 0 and j 1 := Λ -1 div j 1 (where Λ s := (-∆) s/2 ) is quite complicated: in Fourier variables, we have eq:diff0 eq:diff0 (4.7)

d dt      b d j 0 j 1      +       0 ρ 0 0 -ρ ρ 2 0 -LM n - √ n L ε 0 L ε ρ ε √ n 0 0 -ρ ε √ n LM ε            b d j 0 j 1      =     0 0 0 0     •
The analysis that has been performed in [7] pointed out the following necessary and sufficient stability condition eq:stabdiffu eq:stabdiffu (4.8)

L > ε n (1 + M -1 ).
So we shall make this assumption in all that follows. Of course one also has to keep in mind that M > 1, a consequence of M := 1+L s . For notational simplicity, we shall simply denote L by L in the following computations.

4.1.

Estimates for small ρ. In order to prove estimates in the case 0 ≤ ρ ≤ C 1 (with

C 1 ≥ √ 1 + n -1
), we shall use that (4.7) enters in the class of ODEs that has been considered in the Appendix. Indeed, it corresponds to (A.3) with eq:coeffdiff eq:coeffdiff (4.9)

ς = LM n , η = √ n L ε , β = L ε , α = 1 ε √ n , γ = LM ε • 4.1.1.
The case L 1 and LεM 1. We shall follow the first approach proposed in Appendix A. It corresponds to the following matrices A 0 , A 1 , A 2 and B 1

A 0 =     0 0 0 0 0 0 0 0 0 0 L ε 0 0 0 0 LM ε     , A 1 =      0 1 0 0 -1 -1 n 0 0 0 0 0 0 1+ε 2 √ n ε 0 0 -1 √ n ε 0      , B 1 = -     0 0 0 ε n 0 0 1 n 3/2 0 0 √ n 0 0 1 ε 0 0 0     and A 2 =     0 0 0 0 0 1 0 -ε n 0 0 0 0 0 0 0 0     •
Therefore we set eq:Pdiffu eq:Pdiffu (4.10)

P :=      0 0 0 ε 2 nLM 0 0 ε n 3/2 L 0 0 -ε √ n L 0 0 -1 LM 0 0 0      ,
which corresponds to the change of unknowns eq:chgdiffu eq:chgdiffu (4.11)

     b d j 0 j 1      :=      1 0 0 ε 2 nLM ρ -ε nL ρ 1 ε n 3/2 L ρ ε n - √ n - √ n ε L ρ 1 -ε 2 √ n L ρ -1 LM ρ 0 0 1           b d j 0 j 1      •
According to (A.8), working with ( a, d, j 0 , j 1 ) or ( b, d, j 0 , j 1 ) is equivalent whenever mallrhodiffu1 mallrhodiffu1 (4.12) ρ L min(ε -1 , M).

Let us first compute the matrices P B 1 , [P, A 1 ] and A 3 appearing in (A.2)

P B 1 = ε nL     -M -1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 M -1     , [P, A 1 ] = 1 L       0 0 -ε(1+M -1 ) n 3/2 0 0 0 0 1+ε 2 (1+M -1 (n+1)) n 2 1+ε 2 (1+M(1+n)) ε √ n M 0 0 0 0 -1 -M -1 0 0       , A 3 = 1 n       0 ε 2 L 2 0 ε 3 nL 2 M 2 ε LM -(1+n -1 )ε 2 L 2 M 2 0 ε L √ n -ε 2 L 2 n 3/2 0 0 0 0 ε(1+ε 2 ) √ n(LM) 2 0 0 -ε L 2 √ n 0       • Because L 1,
d + ρ 0 1 -1 -n -1 0 b d + ρ 2 -ε nLM 0 0 1 -ε nL b d = ρ 2 ε(1+M -1 ) n 3/2 L 0 0 ε n -1+ε 2 (1+M -1 (n+1)) n 2 L j 0 j 1 •
In order to estimate ( b, d), we just follow the method of Appendix B, which requires Condition (4.8) and eq:rholf eq:rholf (4.13)

ρ ≤ √ 1 + n -1 1 -(1-M -1 )ε nL •
Keeping (4.12) in mind and noticing that

ν = 1 - ε nL (1 + M -1 ),
is of order 1 for small ε, we thus conclude that if

fuassumption1 fuassumption1 (4.14) ρ ≤ 1 + n -1 and ρ L min(ε -1 , M), then eq:diffu1 eq:diffu1 (4.15) |( b, d)(t)| + ρ 2 t 0 |( b, d)| dτ |( b, d)(0)| + ρ 2 t 0 ε L | j 0 | + ε + 1 L | j 1 | dτ + ερ 3 L t 0 |( b, d, j 0 , j 1 )| dτ.
Next, we see that the equations for ( j 0 , j 1 ) read (omitting the O(ερ 3 /L) term)

eq:j0j1 eq:j0j1 (4.16) d dt

j 0 j 1 + L ε + ερ 2 nL 0 0 LM ε + ερ 2 nLM j 0 j 1 + ρ √ n ε 0 1 + ε 2 -1 0 j 0 j 1 = ρ 2 -1+ε 2 (1+M(1+n)) ε √ n LM 0 0 1+M -1 L b d •
Therefore, computing eq:diffu0 eq:diffu0 (4.17)

d dt | j 0 | 2 + (1 + ε 2 )| j 1 | 2 ,
so as to eliminate the term in ρ, we end up with eq:diffu2 eq:diffu2 (4.18)

|( j 0 , j 1 )(t)| + L ε t 0 | j 0 , j 1 | dτ |( j 0 , j 1 )(0)| + ρ 2 t 0 1 εLM + ε L | b| + 1 L | d| dτ + ερ 3 L t 0 |( b, d, j 0 , j 1 )| dτ.
)(t)| + ρ 2 t 0 |( b, d)| dτ + L ε t 0 |( j 0 , j 1 )| dτ |( b, d, j 0 , j 1 )(0)|, whenever 0 ≤ ρ ≤ √ 1 + n -1 .
Now, resuming to the j 1 equation in (4. [START_REF] Mihalas | Foundations of radiation hydrodynamics[END_REF], and evaluating the first order term according to (4.20), we deduce that, in addition eq:diffulf1b eq:diffulf1b (4.21)

LM ε t 0 | j 1 | dτ |( b, d, j 0 , j 1 )(0)|. 4.1.2. The case ε ≪ L 1 with εL 2 L 2 s ≫ 1 and L 2 L s ≫ 1.
If L ≪ 1 then plugging (4.15) in (4.18) does not allow to get (4.20) any longer. In order to overcome this, we shall follow the second approach proposed in Appendix A with coefficients defined as in (4.9): we set

P =      0 0 0 ε 2 nLM 0 0 ε n 3/2 L 0 0 -ε √ n L 0 1+ε 2 √ n(1-M)L -1 LM 0 1 √ n (1-M)L 0      ,
and we thus have, remembering that M -1 = L s eq:chgdiffu2 eq:chgdiffu2 (4.22)

V =      b d j 0 j 1      :=      1 0 0 ε 2 nLM ρ -ε nL ρ 1 ε n 3/2 L ρ ε n - √ n - √ n ε L ρ 1 -1+ε 2 M √ n LLs ρ ρ LMLs 0 -ρ √ n LsL 1           b d j 0 j 1      •
The determinant of the above matrix is

1 + ε 2 nL 2 ρ 2 1 + ε 2 nL 2 M 2 ρ 2 - 1 + ε 2 nL 2 L 2 s ρ 2 ,
Hence working with ( b, d, j 0 , j 1 ) or ( b, d, j 0 , j 1 ) is equivalent whenever

mallrhodiffu2 mallrhodiffu2 (4.23) ρ L ε and ρ ≤ √ n LL s .
Then following the computations of Appendix A, second approach, and setting A 3 := (P A 0 -A 1 )P 2 + A 2 P leads to eq:Vdiffu eq:Vdiffu (4.24)

d dt V + ρ     0 1 0 0 -1 -1 n 0 0 0 0 0 0 0 0 0 0 0     V +      -ε nLM ρ 2 0 0 0 0 1-ε nL ρ 2 0 0 0 0 L ε + ε + 1+ε 2 εLs ρ 2 nL 0 0 0 0 LM ε + ε nLM -1+ε 2 nεLLs ρ 2      V = ρ 2       0 0 (1+M -1 )ε n 3/2 L 0 0 0 0 ε n -1+ε 2 n 2 L -ε 2 (1+n) n 2 LM -1+ε 2 ε √ n LsL -(n+1)ε √ n L 0 0 0 0 -1 LLsM 0 0       V + ρ 3 (I + ρP )A 3 (I + ρP ) -1 V.
q:diffuboundP q:diffuboundP (4.25) |P | 1 L max ε, 1 L s ,
does not provide an accurate enough bound for A 3 . Hence one has to go to further computations. Now, we get

P A 0 P 2 =        0 ε 2 nL 2 Ls 0 ε n 2 L 2 1+ε 2 L 2 s -ε 2 M 2 1+ε 2 n 2 L 2 MLs 0 1 n 5/2 L 2 1+ε 2 L 2 s -ε 2 0 0 -(1+ε 2 )M √ n L 2 L 2 s 0 (1+ε 2 )M n 3/2 εLsL 2 ε 2 M 2 -1+ε 2 L 2 s -ε(1+ε 2 ) nL 2 ML 2 s 0 ε n 3/2 LsL 2 ε 2 -1+ε 2 L 2 s 0        , A 2 P =     0 0 0 0 ε nLM 0 ε n 3/2 L 0 0 0 0 0 0 0 0 0     and A 1 P 2 =      0 -ε 2 nL 2 0 -ε(1+ε 2 ) nL 2 Ls (1+n -1 )ε 2 nL 2 M 2 0 (1+n -1 )ε 2 n 3/2 L 2 LsM 0 0 0 0 0 0 0 0      •
Hence, given that ε ≪ L 1 and that L s ≈ M in the regime that we are considering, one may conclude that eq:diffuA3 eq:diffuA3 (4.26)

|A 3 | max ε L , 1 L 2 L s , 1 εL 2 L 2 s •
Note that we still have ν → 1 for ε → 0. Hence applying the method of the appendix to handle ( b, d), we find out that if (4.8), (4.13) and (4.23) are fulfilled then

|( b, d)(t)| + ρ 2 t 0 |( b, d)| dτ |( b, d)(0)| + ρ 2 t 0 ε L | j 0 | + 1 L | j 1 | dτ +ρ 3 max ε L , 1 L 2 L s , 1 εL 2 L 2 s t 0 |V | dτ.
As regards the radiative modes, we have

| j 0 (t)| + L ε + ε + 1 + ε 2 εL s ρ 2 nL t 0 | j 0 | dτ ≤ | j 0 (0)| + Cρ 2 t 0 1 εLL s + ε L | b| dτ +Cρ 3 max ε L , 1 L 2 L s , 1 εL 2 L 2 s t 0 |V | dτ, | j 1 (t)| + LM ε + ε nLM - 1 + ε 2 nεLL s ρ 2 t 0 | j 1 | dτ ≤ | j 1 (0)| + C ρ 2 LL s M t 0 | d| dτ +Cρ 3 max ε L , 1 L 2 L s , 1 εL 2 L 2 s t 0 |V | dτ.
From the above three inequalities, we get for any A ∈ (0, 1]

|( b, d)(t)| + A| j 0 (t)| + | j 1 (t)| + ρ 2 t 0 |( b, d)| dτ + A L ε t 0 | j 0 | dτ + LL s ε t 0 | j 1 | dτ |( b, d)(0)| + A| j 0 (0)| + | j 1 (0)| + ρ 2 t 0 ε L | j 0 | + 1 L | j 1 | dτ + Aρ 2 t 0 1 εLL s + ε L | b| dτ + ρ 2 LL 2 s t 0 | d| dτ + ρ 3 max ε L , 1 L 2 L s , 1 εL 2 L 2 s t 0 |V | dτ.
Now, we notice that taking A = c 0 min(1, εLL s ) for a sufficiently small constant c 0 allows to absorb all the terms of the r.h.s. (but the data) by the l.h.s. provided we have 0

≤ ρ ≤ √ 1 + n -1 (4.27) ε ≪ L 1, εL 2 L 2 s ≫ 1 and L 2 L s ≫ 1. We thus conclude that for all 0 ≤ ρ ≤ √ 1 + n -1 , we have eq:diffulf2 eq:diffulf2 (4.28) |( b, d, min(1, εLL s ) j 0 , j 1 )(t)| + ρ 2 t 0 |( b, d)| dτ + L ε min(1, εLL s ) t 0 | j 0 | dτ + LM ε t 0 | j 1 | dτ ≤ C|( b, d, min(1, εLL s ) j 0 , j 1 )(0)|.
Let us point out that in the case where L 2 L s ≈ 1 (even if L ≈ ε in fact) then the same computation will lead to (4.28), but only for 0 ≤ ρ ≤ c, with c a small enough constant.

4.1.3. The case ε ≪ L ε 1/2 and L 2 L s ≈ 1. As the value √ 1 + n -1 will not play any particular role, we fix some C 1 > c in the following computations. We want to get (4.28) for ρ ∈ [c, C 1 ]. To this end, we introduce ζ 0 and ζ 1 such that

ζ 0 := j 0 - √ n 1 + ρ 2 nL 2 M b and ζ 1 := j 1 - ρ √ n LM j 0 .
Then we discover that ( b, d,

ζ 0 , ζ 1 ) fulfills                        ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ 1 + 1 n 1 + ρ 2 nL 2 M b = ρ n 3/2 ζ 0 + LM n ζ 1 , ∂ t ζ 0 + L ε ζ 0 = - ρ ε √ n ζ 1 - √ n ρ 1 + ρ 2 nL 2 M d, ∂ t ζ 1 + L ε M - ρ 2 nL 2 M ζ 1 = ρ √ n εM 1 + ρ 2 nL 2 M ζ 0 . Let ρ be in [c, C 1 ].
For the first two equations, performing the standard barotropic estimates (which rely on the use of U ρ defined in (4.43)) leads to eq:diffulf5 eq:diffulf5 (4.29)

|( b, d)(t)| + t 0 |( b, d)| dτ |( b, d)(0)| + t 0 | ζ 0 | dτ + LL s t 0 | ζ 1 | dτ.
For ζ 0 , it is obvious that eq:diffulf6 eq:diffulf6 (4.30)

| ζ 0 (t)| + L ε t 0 | ζ 0 | dτ ≤ | ζ 0 (0)| + ρ ε √ n t 0 | ζ 1 | dτ + ρ √ n t 0 | d| dτ,
and, as our conditions on L and L s guarantee that ρ ≤ n/2 LM for small enough ε, we also have eq:diffulf7 eq:diffulf7 (4.31)

| ζ 1 (t)| + LM 2ε t 0 | ζ 1 | dτ ≤ | ζ 1 (0)| + ρ √ n εM 1 + ρ 2 nL 2 M t 0 | d| dτ.
Putting together those three inequalities, we readily get for all A, B > 0, observing that

ρ 2 ≈ L 2 M ≈ 1 |( b, d)(t)| + A ε L | ζ 0 (t)| + Bε| ζ 1 (t)| + t 0 |( b, d)| dτ + A t 0 | ζ 0 | dτ + BLM t 0 | ζ 1 | dτ |( b, d)(0)| + A ε L | ζ 0 (0)| + Bε| ζ 1 (0)| + LM t 0 | ζ 1 | dτ + t 0 | ζ 0 | dτ + A L t 0 | ζ 1 | dτ + A ε L t 0 | d| dτ + B M t 0 | ζ 0 | dτ.
It is now clear that if one takes first A large enough (independently of ε) and B much larger, all the integrals of the r.h.s. may be absorbed by the left-hand side, as M -1 ≪ ρ, and we

thus get for all ρ ∈ [c, C 1 ] eq:diffulf3a eq:diffulf3a (4.32) |( b, d)(t)| + ε L | ζ 0 (t)| + ε| ζ 1 (t)| + t 0 |( b, d)| dτ + t 0 | ζ 0 | dτ + LM t 0 | ζ 1 | dτ |( b, d)(0)| + ε L | ζ 0 (0)| + ε| ζ 1 (0)|.
Plugging this new inequality in (4.31), we easily deduce that

| ζ 1 (t)| + LM ε t 0 | ζ 1 | dτ | ζ 1 (0)| + L| ζ 0 (0)| + 1 εM |( b, d)(0)|,
then inserting this information and (4.32) in (4.30), we discover that

| ζ 0 (0)| + L ε t 0 | ζ 0 | dτ | ζ 0 (0)| + L| ζ 1 (0)| + 1 + 1 εLM 2 |( b, d)(0)|.
Therefore, putting (4.32) and the above two inequalities together, using that |( b, d, ζ 0 , ζ 1 )| ≈ |( b, d, j 0 , j 1 )| and assuming in addition that L ε 1/2 , we conclude that eq:diffulf9 eq:diffulf9 (4.33)

|( b, d, ε L j 0 , j 1 )(t)| + t 0 |( b, d)| dτ + L ε t 0 | ζ 0 | dτ + LL s ε t 0 | ζ 1 | dτ |( b, d, ε L j 0 , j 1 )(0)| for all c ≤ ρ ≤ C 1 .
Note that due to the expression of ζ 1 , one may replace ζ 1 with j 1 of j 1 in the integral.

Still in the case L ε 1/2 , we claim that we have the following inequality eq:diffulf10 eq:diffulf10 (4.34)

|ρ j 0 (t)| + L ε t 0 |ρ ζ 0 | dτ |( b, d, ρ j 0 , ε L j 0 , j 1 )(0)| for all 0 ≤ ρ ≤ C 1 ,
which turns out to be crucial in the justification of the asymptotics toward (2.9).

Indeed, inequality (4.30) does not require any assumption on ρ and thus implies that

|ρ ζ 0 (t)| + L ε t 0 |ρ ζ 0 | dτ ≤ |ρ ζ 0 (0)| + ρ 2 ε √ n t 0 | ζ 1 | dτ + ρ 2 √ n t 0 | d| dτ. For 0 ≤ ρ ≤ c (resp. c ≤ ρ ≤ C 1 )
, the last term may be bounded according to (4.28) (resp. (4.33)). Regarding the term with ζ 1 , we notice that

ζ 1 = j 1 + ρ LL s M j 0 √ n + b ,
hence (4.28) and the fact that

L 2 L s ≈ 1 guarantee that for 0 ≤ ρ ≤ c ρ 2 ε √ n t 0 | ζ 1 | dτ L LM ε t 0 | j 1 | dτ + L 2 ε t 0 ρ 2 | j 0 + √ n b| dτ L(1 + L 2 ε )|( b, d, εLL s j 0 , j 1 )(0)|.
In the case L ε 1/2 , it is obvious that Inequality (4.33) implies that the above inequality is also true in the range c ≤ ρ ≤ C 1 , which completes the proof of (4.34).

4.1.4. The case L ≈ ε and L s ε 2 1. We saw that if (4.8) is fulfilled then (4.28) holds true on some small interval [0, c]. So we have to fill in the gap between c and √ 1 + n -1 . As the value √ 1 + n -1 does not play any particular role, we fix some C 1 > c and look for estimates if ρ ∈ [c, C 1 ]. For simplicity, take L = κε/n (with κ > 1 owing to (4.8)).

Setting

ζ 1 := j 1 -1 √ n LM ρ j 0 = j 1 - √ n
κεM ρ j 0 as before, we observe that eq:diff-lf eq:diff-lf (4.35)

             ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ b = ρ n 3/2 j 0 + κεM n 2 ζ 1 , ∂ t j 0 + κ n + ρ 2 κε 2 M j 0 + ρ √ nε ζ 1 = κ √ n b, ∂ t ζ 1 + κM n -ρ 2 κε 2 M ζ 1 = ρ √ n εM 1 + ρ 2 n κ 2 ε 2 M j 0 -ρ εM b.
Let us focus on the subsystem corresponding to the first three equations, namely eq:sub eq:sub (4.36)

       ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ b -ρ n 3/2 j 0 = f , ∂ t j 0 + κ n + ρ 2 κε 2 M j 0 -κ √ n b = g. If we have the stronger condition ε 2 M → ∞ then we rewrite System (4.36) as follows eq:diff-mf2 eq:diff-mf2 (4.37) d dt   b d j 0   +   0 ρ 0 -ρ ρ 2 -ρ n 3/2 -κ √ n 0 κ n     b d j 0   =   0 f -ρ 2 κε 2 M j 0 + g   •
The eigenvalues of the matrix M ρ in the left-hand side are the roots of the polynomial

-X 3 + a 1 (ρ)X 2 -a 2 (ρ)X + a 3 (ρ) with a 1 (ρ) = κ n + ρ 2 , a 2 (ρ) = 1 + κ n ρ 2 , a 3 (ρ) = 1 + 1 n ρ 2 κ n •
According to Routh-Hurwitz criterion, those roots have positive real part if and only if

a 1 (ρ) > 0, a 1 (ρ) 1 a 3 (ρ) a 2 (ρ)
> 0 and

a 1 (ρ) 1 0 a 3 (ρ) a 2 (ρ) a 1 (ρ) 0 0 a 3 (ρ) > 0.
As a 1 (ρ) and a 3 (ρ) are positive, it suffices to check the second condition, that is

a 1 (ρ)a 2 (ρ) -a 3 (ρ) = 1 + κ n ρ 4 + ρ 2 κ n 2 (κ -1)
> 0, and this is indeed the case for all ρ > 0, as κ > 1.

In particular, all the eigenvalues of the matrix M ρ have positive real part if we assume ρ to belong to the compact set [c, C 1 ]. Therefore (see [7]) there exist two positive constants c 2 and C 2 depending only on c and C 1 , so that the matrix M ρ satisfies

e -tMρ ≤ C 2 e -c 2 t for all t ≥ 0 and ρ ∈ [c, C 1 ].
By taking advantage of Duhamel's formula, we thus deduce that

|( b, d, j 0 )(t)| + t 0 |( a, d, j 0 )| dτ |( b, d, j 0 )(0)| + t 0 |( f , g)| dτ + 1 ε 2 M t 0 | j 0 | dτ.
Of course, owing to the assumption ε 2 M → ∞, the last term of the r.h.s. may be absorbed by the l.h.s., for ε going to 0. So we get eq:diffulf3 eq:diffulf3 (4.38)

|( b, d, j 0 )(t)| + t 0 |( b, d, j 0 )| dτ |( b, d, j 0 )(0)| + t 0 |( f , g)| dτ.
In the case where ε 2 M does not go to ∞ then we have to proceed slightly differently. If we assume (for simplicity) that ε 2 M tends to some m > 0, then the matrix M ρ in (4.37) has to be changed in

N ρ =    0 ρ 0 -ρ ρ 2 -ρ n 3/2 -κ √ n 0 κ n + ρ 2 κm    •
The above analysis based on Routh-Hurwitz theorem still holds as the additional term has 'the good sign', and one may conclude, as before, that (4.38

) is satisfied for all ρ ∈ [c, C 1 ].
In every case ε 2 M 1, resuming to (4.35), Inequality (4.38) allows us to get for all

ρ ∈ [c, C 1 ] eq:diffulf4 eq:diffulf4 (4.39) |( b, d, j 0 )(t)| + t 0 |( b, d, j 0 )| dτ |( b, d, j 0 )(0)| + εM t 0 | ζ 1 | dτ.
Next, from the equation of ζ 1 , we readily get

| ζ 1 (t)| + M t 0 | ζ 1 | dτ | ζ 1 (0)| + (εM) -1 t 0 |( b, j 0 )| dτ.
Hence, adding up to Inequality (4.39), we conclude that (4.28) is also true for all ρ ∈ [c, C 1 ]. This completes the proof of estimates in the low frequency regime ρ ∈ [0, C 1 ].

4.2.

Estimates for middle frequencies.

The case lim inf M = +∞. As in the previous paragraph, introduce

ζ 1 := j 1 - ρ √ n LM j 0 .
The system fulfilled by ( b, d, j 0 , ζ 1 ) reads eq:diff-mf eq:diff-mf (4.40)

             ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ b = ρ n 3/2 j 0 + LM n ζ 1 , ∂ t j 0 + 1 ε L + ρ 2 nLM j 0 + ρ √ n ε ζ 1 = √ n L ε b, ∂ t ζ 1 + LM ε -ρ 2 nεLM ζ 1 = ρ √ nLMε L + ρ 2 nLM j 0 -ρ εM b.
The subsystem corresponding to the first three equations is eq:sub1 eq:sub1 (4.41)

       ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ b -ρ n 3/2 j 0 = f , ∂ t j 0 + L ε 1 + ρ 2 nL 2 M j 0 - √ n L ε b = g, with eq:diffumf00 eq:diffumf00 (4.42) f = LM n ζ 1 and g = - ρ √ n ε ζ 1 .
Assume that ( f , g) ≡ (0, 0) for a while and set

def:U def:U (4.43) U 2 ρ := 2|( b, d)| 2 -2ρRe ( b d) + |ρ b| 2 .
On one hand, we have eq:diff-mf0 eq:diff-mf0 (4.44) 1 2

d dt U 2 ρ + ρ 2 | b| 2 + ρ 2 | d| 2 = ρ n 3/2 Re (2 d -ρ b) j 0 ,
and on the other hand, 1 2

d dt | j 0 | 2 + L ε 1 + ρ 2 nL 2 M | j 0 | 2 = √ n L ε Re ( b j 0 ). Therefore 1 2 d dt U 2 ρ + ε n 2 L |ρ j 0 | 2 + ρ 2 |( b, d)| 2 + ρ 2 n 2 1 + ρ 2 nL 2 M | j 0 | 2 = 2 ρ n 3/2 Re ( d j 0 ). Now, by using the fact that 2 ρ n 3/2 Re ( d j 0 ) ≤ 1 An | d| 2 + Aρ 2 n 2 | j 0 | 2 ,
and by taking A = 3/4, we conclude that for ρ 2 ≥ 16n 3(4n 2 -1) , we have

d dt U 2 ρ + ε n 2 L |ρ j 0 | 2 + ρ 2 2n 2 |( b, d, j 0 )| 2 ≤ 0, whence, because 16n 3(4n 2 -1) ≤ 1 + 1 n ,
we get for some universal positive constants c 0 and C

|(ρ b, d)(t)| + ε L |ρ j 0 (t)| ≤ Ce -c 0 t |(ρ b, d)(0)| + ε L |ρ j 0 (0)| for ρ ≥ 1 + n -1 •
Resuming to the equation fulfilled by j 0 in (4.41), the above inequality implies (still assuming that f = g = 0) that

L ε 1 + ρ 2 nL 2 M t 0 |ρ j 0 | dτ ≤ |ρ j 0 (0)| + √ nL ε t 0 |ρb| dτ L ε |ρ j 0 (0)| + L ε |( d, ρ b)(0)|,
then plugging this inequality in the equation for d, we get in addition

ρ 2 t 0 | d| dτ | d(0)| + |ρ b(0)| + ε L |ρ j 0 (0)|.
Repeating the above computations in the case of general source terms f and g, we conclude that the solution (b, d, j 0 ) to (4.41) satisfies for all ρ ≥ √ 1 + n -1 , assuming only that L ε

|(ρ b, d)(t)| + ε L |ρ j 0 (t)| + t 0 |(ρ b, ρ 2 d)| dτ + L ε 1 + ρ 2 nL 2 M t 0 |ρ j 0 | dτ ≤ C |(ρ b, d)(0)| + ε L |ρ j 0 (0)| + t 0 | f | + ε L |ρ g| dτ •
Resuming to the value of f and g in (4.42), we thus get for ρ ≥ √ 1 + n -1 and L ε eq:diffumf eq:diffumf (4.45)

|(ρ b, d)(t)| + ε L |ρ j 0 (t)| + t 0 |(ρ b, ρ 2 d)| dτ + L ε 1 + ρ 2 nL 2 M t 0 |ρ j 0 | dτ ≤ C |(ρ b, d)(0)| + ε L |ρ j 0 (0)| + LM + ρ 2 √ Lε t 0 | ζ 1 | dτ • Next, it is clear that the equation for ζ 1 implies that whenever ρ ≤ n 2 LM eq:diff-mf5 eq:diff-mf5 (4.46) | ζ 1 (t)|+ LM 2ε t 0 | ζ 1 | dτ ≤ | ζ 1 (0)|+ 1 √ nεM 1+ ρ 2 nL 2 M t 0 |ρ j 0 | dτ + 1 εM t 0 |ρ b| dτ.
Hence, we get if M is large enough and ρ 2 ≪ L 3/2 M 2 ε 1/2 eq:diff-mf6 eq:diff-mf6 (4.47)

| ζ 1 (t)| + LM ε t 0 | ζ 1 | dτ ≤ C | ζ 1 (0)| + 1 εM |(ρ b, d)(0)| + ε L |ρ j 0 (0)| •
Then plugging that inequality in (4.45) implies that eq:diff-mf7 eq:diff-mf7 (4.48)

|(ρ b, d)(t)| + ε L |ρ j 0 (t)| + ρ 2 t 0 | d| dτ + t 0 |ρ b| dτ + L ε 1 + ρ 2 nL 2 M t 0 |ρ j 0 | dτ ≤ C |(ρ b, d)(0)| + ε L |ρ j 0 (0)| + ε + ε 1/2 ρ 2 ML 3/2 | ζ 1 (0)| , whenever 1 + 1/n ≤ ρ 2 ≪ L 3/2 M 2 ε 1/2 .
Here is another method that gives decay estimates in the range L √ M ≪ ρ ≪ LM if M is large enough. From the first two equations of (4.40), we have

eq:diff-mf1b eq:diff-mf1b (4.49) |(ρ b, d)(t)| + ρ 2 t 0 | d| dτ + t 0 |ρ b| dτ |(ρ b, d)(0)| + ρ n 3/2 t 0 | j 0 | dτ + LM n t 0 | ζ 1 | dτ.
The equations for j 0 and

ζ 1 give, if ρ ≤ n 2 LM eq:diff-mf2b eq:diff-mf2b (4.50) | j 0 (t)| + 1 ε L + ρ 2 nLM t 0 | j 0 | dτ ≤ | j 0 (0)| + ρ √ n ε t 0 | ζ 1 | dτ + L √ n ε t 0 | b| dτ, eq:diff-mf3b eq:diff-mf3b (4.51) | ζ 1 (t)|+ LM 2ε t 0 | ζ 1 | dτ ≤ | ζ 1 (0)| + ρ √ n LMε L+ ρ 2 nLM t 0 | j 0 | dτ + ρ εM t 0 | b| dτ.
Plugging (4.50) in (4.51), we discover if ρ ≪ LM that eq:diff-mf2c eq:diff-mf2c (4.52)

| ζ 1 (t)| + LM 4ε t 0 | ζ 1 | dτ ≤ | ζ 1 (0)| + ρ LM | j 0 (0)| + 1 εM t 0 ρ| b| dτ.
Inserting that inequality in (4.49), we conclude that the last term of (4.49) may be absorbed by the l.h.s. if M is large enough. Now, Inequality (4.50) guarantees that

ρ n 3/2 t 0 | j 0 | dτ ≤ 1 √ n ρεLM nL 2 M + ρ 2 | j 0 (0)| + ρ 2 LM n 2 L 2 M + ρ 2 n t 0 | ζ 1 | dτ + L 2 M nL 2 M + ρ 2 t 0 ρ| b| dτ.
Again, resuming to (4.49), we see that the second term in the r.h.s. may be absorbed by the l.h.s. This is also the case of the last one if

ρ 2 ≫ L 2 M. If all those conditions are fulfilled then we end up if L √ M ≪ ρ ≤ n 2 LM with eq:diff-mf4b eq:diff-mf4b (4.53) |(ρ b, d)(t)| + ρ 2 t 0 | d| dτ + ρ t 0 | b| dτ + ρ t 0 | j 0 | dτ + LM t 0 | ζ 1 | dτ |(ρ b, d)(0)| + εLM ρ | j 0 (0)| + ε| ζ 1 (0)|.
Resuming to (4.50) and (4.52), we thus easily deduce first that

| ζ 1 (t)| + LM ε t 0 | ζ 1 | dτ | ζ 1 (0)| + | j 0 (0)| + 1 εM |(ρ b, d)(0)|,
and next that

| j 0 (t)| + 1 ε L + ρ 2 LM t 0 | j 0 | dτ | j 0 (0)| + ρ LM | ζ 1 (0)| + L ερ |(ρ b, d)(0)|.
Our estimates and the definition of ζ 1 allow us to change ζ 1 to j 1 in (4.53). So finally, in the case εM 1 we get for some large enough C 1 and small enough c independent of ε eq:diff-hf7 eq:diff-hf7 (4.54)

|(ρ b, d, (1+ εLM ρ ) j 0 , j 1 )(t)| + t 0 |ρ b| dτ + ρ 2 t 0 | d| dτ + ρ εLM (1+ εLM ρ ) t 0 |ρ j 0 | dτ + LM ε t 0 | ζ 1 | dτ |(ρ b, d, (1+ εLM ρ ) j 0 , j 1 )(0)|, whenever C 1 L √ M ≤ ρ ≤ cLM.
The case

ε 2 M ≤ 1/2 and L ≫ ε. Let ζ 0 := j 0 - √ n b. We start from              ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ b = LM n j 1 , ∂ t ζ 0 + L ε ζ 0 + ρ √ n ε j 1 - √ n ρ d = 0, ∂ t j 1 + LM ε j 1 -ρ √ n ε ζ 0 -ρ ε b = 0.
In order to show the exponential decay, we set eq:Ur eq:Ur (4.55)

U 2 ρ := 2| b| 2 + 2| d| 2 + |ρ b| 2 -2ρRe ( b d) and J 2 ρ := | ζ 0 | 2 + | j 1 | 2 .
We easily get for all K ≥ 0, eq:diff-mf8 eq:diff-mf8 (4.56) 1 2

d dt U 2 ρ + KJ 2 ρ + ρ 2 |( b, d)| 2 + K L ε | ζ 0 | 2 + M| j 1 | 2 = 2 LM n Re ( j 1 d) + ρ K ε - LM n Re ( j 1 b) + K √ nRe (ρ d ζ 0 ).
It is thus natural to take K = ε n LM to cancel out the second term of the r.h.s. For the first and the last terms, we write that

K √ nRe ρ d ζ 0 ≤ 1 4 ρ 2 | d| 2 + nK 2 | j 0 - √ n b| 2 , 2LM n Re ( j 1 d) = 2K ε Re ( j 1 d) ≤ 2 3 ρ 2 | d| 2 + 3K 2 2ε 2 ρ 2 | j 1 | 2 .
Note that the last terms above may be absorbed by the l.h.s. of (4.56) if, say

nK ≤ L 2ε and 2K ε ≤ LMρ 2 .
Given the value of K, the first condition is equivalent to ε 2 M ≤ 1/2, whereas the second one means that ρ 2 n ≥ 2. Under this latter condition, we thus end up with eq:diff-hf1 eq:diff-hf1 (4.57)

d dt U 2 ρ + εLM n J 2 ρ + 1 6 ρ 2 |( b, d)| 2 + L 2 M n | ζ 0 | 2 + 1 6 M| j 1 | 2 ≤ 0,
which implies, according to (4.8), the following exponential decay estimate for some small enough κ > 0 eq:diff-hf2 eq:diff-hf2 (4.58)

U 2 ρ (t) + εLMJ 2 ρ (t) ≤ e -κt U 2 ρ (0) + εLM n J 2 ρ (0) if ρ ≥ 2/n.
To exhibit the parabolic decay for d, we introduce ζ 1 := j 1 -( √ nLM) -1 ρ j 0 , and get

             ∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρ 1 + 1 n b = LM n ζ 1 + ρ n 3/2 ζ 0 , ∂ t ζ 0 + 1 ε L + ρ 2 nLM ζ 0 + ρ √ n ε ζ 1 = √ n ρ d -ρ 2 √ n εLM b, ∂ t ζ 1 + LM ε -ρ 2 nεLM ζ 1 = ρ √ n LMε L + ρ 2 nLM ζ 0 + ρ 3 nεL 2 M 2 b.
We thus have for 2

/n ≤ ρ ≤ n/2 LM t 0 |ρ b| dτ + ρ 2 t 0 | d| dτ |(ρ b, d)(0)| + LM n t 0 | ζ 1 | dτ + ρ n t 0 | ζ 0 | dτ, 1 ε L + ρ 2 nLM t 0 | ζ 0 | dτ ≤ | ζ 0 (0)| + ρ √ n ε t 0 | ζ 1 | dτ + √ n ρ t 0 | d| dτ + ρ √ n εLM t 0 |ρ b| dτ, LM 2ε t 0 | ζ 1 | dτ ≤ | ζ 1 (0)| + ρ √ n εLM L + ρ 2 nLM t 0 | ζ 0 | dτ + ρ 2 nεL 2 M 2 t 0 |ρ b| dτ.
Combining the inequalities for ζ 0 and ζ 1 , we easily get if ρ ≤ cLM with c small enough

LM t 0 | ζ 1 | dτ ε| ζ 1 (0)| + ρ LM ε| ζ 0 (0)| + ρε t 0 | d| dτ + ρ LM t 0 |ρ b| dτ , L + ρ 2 LM t 0 | ζ 0 | dτ ε| ζ 0 (0)| + ρε LM | ζ 1 (0)| + ρε t 0 | d| dτ + ρ LM t 0 |ρ b| dτ.
Now, the exponential decay pointed out in (4.58) allows to bound the last terms above, and we get

LM t 0 | ζ 1 | dτ U ρ (0) + √ εLMJ ρ (0) + ε LM t 0 |ρ 2 d| dτ, eq:diff-mf10 (4.59) L+ ρ 2 LM t 0 | ζ 0 | dτ ε|( ζ 0 , ζ 1 )(0)| + ρ LM U ρ (0) + √ εLMJ ρ (0) + ρε t 0
| d| dτ, eq:diff-mf11 (4.60) whereas using U 2 ρ allows to get directly

ρ 2 t 0 | d| dτ |(ρ b, d)(0)| + LM t 0 | j 1 | dτ.
Using the definition of j 1 and, again, Inequality (4.58), we may replace j 1 with ζ 1 as follows eq:diff-mf12 eq:diff-mf12 (4.61)

ρ 2 t 0 | d| dτ U ρ (0) + √ εLMJ ρ (0) + LM t 0 | ζ 1 | dτ + t 0 |ρ ζ 0 | dτ.
Then plugging (4.59) and (4.60) in (4.61) and observing that ε ≪ LM, we get

ρ 2 t 0 | d| dτ U ρ (0) + √ εLMJ ρ (0) + ρLM L 2 M + ρ 2 ε|( ζ 0 , ζ 1 )(0)| + ρ LM U ρ (0) + √ εLMJ ρ (0) + ρε t 0 | d| dτ •
Because we assumed that L ≫ ε, the last term may be absorbed by the l.h.s. Using in addition that ε 2 M 1, we end up with

ρ 2 t 0 | d| dτ U ρ (0) + √ εLMJ ρ (0).
Then resuming to (4.59), (4.60), we obtain eq:diff-hf4 eq:diff-hf4 (4.62)

ρ t 0 | ζ 0 | dτ + LM t 0 | ζ 1 | dτ U ρ (0).
Obviously, this inequality implies that eq:diff-hf3 eq:diff-hf3 (4.63)

LM t 0 | j 1 | dτ U ρ (0) + √ εLM J ρ (0) + ε|( ζ 0 , ζ 1 )(0)|.
Of course, we get the same inequality if replacing ζ 0 and ζ 1 with j 0 and j 1 . So one can conclude that for 2/n ≤ ρ ≤ cLM, we have (4.58) and eq:diff-hf6 eq:diff-hf6 (4.64)

ρ 2 t 0 | d| dτ + LM t 0 | j 1 | dτ + ρ t 0 | j 0 | dτ |(ρ b, d)(0)| + √ εLM|( j 0 , j 1 )(0)|.
4.3. High frequencies. We eventually come to the proof of decay estimates for ρ ≥ cLM, where c is some given positive constant. We shall use that fact that the systems satisfied by (b, d) and by (j 0 , j 1 ), respectively, tend to be uncoupled for ρ → +∞. As regards (b, d), the classical approach for the barotropic Navier-Stokes equation, based on the study of

U 2 ρ := 2|( b, d)| 2 -2ρRe ( b d) + |ρ b| 2 , guarantees, if ρ ≥ c, that eq:hfdiff1 eq:hfdiff1 (4.65) |(ρ b, d)(t)| + ρ 2 t 0 | d| dτ + t 0 ρ| b| dτ |(ρ b, d)(0)| + LM n t 0 | j 1 | dτ.
Next, from the system fulfilled by ( j 0 , j 1 ), we get 1 2

d dt | j 0 | 2 + | j 1 | 2 + L ε | j 0 | 2 + LM ε | j 1 | 2 = √ n L ε Re ( b j 0 ), d dt Re ( j 0 j 1 ) + L ε (1 + M)Re ( j 0 j 1 ) + ρ ε √ n | j 1 | 2 - ρ ε √ n | j 0 | 2 = √ n L ε Re ( b j 1 ).
Therefore, for any κ > 0 1 2

d dt | j 0 | 2 + | j 1 | 2 - 2κLM ρ Re ( j 0 j 1 ) + 1 + κ √ n M L ε | j 0 | 2 + LM ε 1 - κ √ n | j 1 | 2 = κL 2 M ρε (1 + M)Re ( j 0 j 1 ) + √ n L ε Re b j 0 - κLM ρ j 1 •
For ρ ≥ cLM, it is clear that our choosing κ small enough implies that • the first term of the r.h.s. may be absorbed by the second and third ones of the l.h.s.,

• we have

| j 0 | 2 + | j 1 | 2 -2κLM ρ Re ( j 0 j 1 ) ≈ | j 0 | 2 + | j 1 | 2 , • we have 1 + κ √ n M L ε | j 0 | 2 + 1 -κ √ n LM ε | j 1 | 2 ≥ κ 2 LM √ n ε (| j 0 | 2 + | j 1 | 2 ).
Therefore, we end up with the following inequality eq:hfdiff2 eq:hfdiff2 (4.66) ε|( j 0 , j 1

)(t)| + LM t 0 |( j 0 , j 1 )| dτ ε|( j 0 , j 1 )(0)| + L t 0 | b| dτ.
Combining with (4.65), we conclude that if eq:hfdiff eq:hfdiff (4.67) ρ ≥ c max(1, LM) and ρ ≥ CL, for a large enough constant C then

|(ρ b, d)(t)| + ρ 2 t 0 | d| dτ + t 0 ρ| b| dτ |(ρ b, d, ε j 0 , ε j 1 )(0)|, eq:hfdiff3 (4.68) |( j 0 , j 1 )(t)| + LM ε t 0 |( j 0 , j 1 )| dτ L ρε |(ρ b, d)(0)| + |( j 0 , j 1 )(0)|, eq:hfdiff4 (4.69) whence eq:hfdiff3a eq:hfdiff3a (4.70) |(ρ b, d, j 0 , j 1 )(t)| + ρ 2 t 0 | d| dτ + t 0 ρ| b| dτ + LM ε t 0 |( j 0 , j 1 )| dτ |(ρ b, d, j 0 , j 1 )(0)|.
The only case where the condition ρ ≥ CL may be stronger than ρ ≥ cLM is when M is bounded. From our study for small ρ's, we must assume that LM ε -1 , and thus cLM is still much larger than 2/n. Therefore, one may take advantage of (4.58) to bound the r.h.s. of (4.66), and combining with (4.65), we get for ρ ≈ L eq:diff4 eq:diff4

The non-equilibrium diffusion regime

This section is devoted to the study of the so-called non-equilibrium diffusion asymptotics. Assuming that for some κ > 1 and m > 0, we have eq:diffasymp eq:diffasymp (5.1)

L ε → κ nν and L 2 L s → m ν 2 ,
we want to prove the convergence of the solutions of (1.6) to those of (2.5) or (2.6) if m < +∞ or m = +∞, respectively, when ε goes to 0.

The first subsection concerns the proof of global existence with 'uniform' estimates for the radiative Navier-Stokes equations (1.6) in the asymptotic (5.1), in the small data case with critical regularity. Next, still for small critical data, we establish the global existence for the limit systems (2.5) and (2.6). In the last part of the present section, we combine the uniform estimates with compactness arguments in order to justify the convergence of (1.6) to (2.5) or (2.6). 5.1. Global existence and uniform estimates for (1.6) with L ≈ ε and L 2 L s 1. In order to get a global-in-time existence statement for (1.6) in the non-equilibrium diffusion regime, we first put together the estimates that we obtained in the previous section, in the case L ≈ ε and L 2 L s 1. Even though localizing the linearized equations by means of Littlewood-Paley operators allows to get essentially optimal estimates for the linearized equations of (1.6), it is not enough for our purpose, owing to the convection term u • ∇b that may cause a loss of one derivative. The difficulty may be overcome by paralinearizing the whole system, as explained below. After that, it is easy to prove global in time estimates for the solutions to (1.6) just by combining the estimates for the paralinearized system, and standard product laws in Besov spaces to handle the other nonlinear terms. 5.1.1. Linear estimates. Performing the change of variables (4.1) reduces the study to that of the linear system (4.4).

Low frequencies estimates. Using (4.28), the comment that follows, (4.39) and the fact that |( b, d, j 0 , j 1 )| ≈ |( b, d, j 0 , j 1 )|, we get for the solution ( b, d, j 0 , j 1 ) to (4.7), eq:noneqlf eq:noneqlf (5.2) If L 2 L s → +∞ then (4.47) and (4.48) ensure that eq:noneqmf1 eq:noneqmf1 (5.4)

|( b, d, j 0 , j 1 )(t)| + ρ 2 t 0 |( b, d, j 0 , j 1 )| dτ + t 0 | j 0 | dτ + M t 0 | j 1 | dτ ≤ C|( b, d, j 0 , j 1 )(0)| for all 0 ≤ ρ ≤ C 1 , with 1 j 0 := j 0 - √ n b - √ n ε L ρ d and j 1 := j 1 -ρ √ n LM j 0 + ρ b LLsM • Middle frequencies estimates. If L 2 L s ≈
|(ρ b, d, ρ j 0 , j 1 )(t)| + ρ 2 t 0 | d| dτ + ρ t 0 |( b, j 0 )| dτ + M t 0 | j 1 | dτ ≤ C|(ρ b, d, ρ j 0 , j 1 )(0)| for all 1 + n -1 ≤ ρ ≤ εν √ M,
and, according to (4.54)

eq:noneqmf2 eq:noneqmf2 (5.5)

|(ρ b, d, (1+ ε 2 νM ρ ) j 0 , j 1 )(t)| + t 0 |ρ b| dτ + ρ 2 t 0 | d| dτ + (1+ ρ ε 2 νM ) t 0 |ρ j 0 | dτ + M t 0 | j 1 | dτ |(ρ b, d, (1+ ε 2 νM ρ ) j 0 , j 1 )(0)|, whenever C 1 εν √ M ≤ ρ ≤ cενM.
Hence eq:noneqmfbis eq:noneqmfbis (5.6)

|(ρ b, d, max(1, min(ρ, ρ -1 ε 2 νM)) j 0 , j 1 )(t)| + ρ t 0 |( b, min(1, ρ 2 ε 2 νM ) j 0 )| dτ + ρ 2 t 0 | d| dτ + M t 0 | j 1 | dτ ≤ C|(ρ b, d, max(1, min(ρ, ρ -1 ε 2 νM)) j 0 , j 1 )(0)|.
High frequencies estimates. Using (4.70), we have eq:noneqhf eq:noneqhf (5.7) |(ρ b, d, j 0 , j 1

)(t)| + ρ 2 t 0 | d| dτ + ρ t 0 | b| dτ + M t 0 |( j 0 , j 1 )| dτ ≤ C|(ρ b, d, j 0 , j 1 )(0)| for ρ ≥ c LM.
For notational simplicity, we shall slightly abusively change c and C 1 to 1 in all the following computations.

Optimal estimates in Besov spaces. If L 2 L s ≈ 1 then localizing (4.4) with nonzero source terms f and g according to Littlewood-Paley operator ∆k , using (4.5) and following the computations leading to (5.2), (5.3) and (5.7) (combined with Fourier-Plancherel theorem) we end up for all s ∈ R with 2 q:noneqestlin q:noneqestlin (5.8) ( u, j 0 , j 1 )(t) Ḃs

2,1 + b(t) ℓ,1 Ḃs 2,1 + b(t) h,1 Ḃs+1 2,1 + t 0 u Ḃs+2 2,1 dτ + t 0 (b, j 0 , j 1 ) ℓ,1 Ḃs+2 2,1 dτ + t 0 b h,1 Ḃs+1 2,1 dτ + t 0 j 0 ℓ,1 Ḃs 2,1 dτ + M t 0 j 1 ℓ, LM Ḃs 2,1 dτ + t 0 j 0 m,1, LM Ḃs+2 2,1 dτ + M t 0 (j 0 , j 1 ) h, LM Ḃs 2,1 dτ ( u, j 0 , j 1 )(0) Ḃs 2,1 + b(0) ℓ,1 Ḃs 2,1 + b(0) h,1 Ḃs+1 2,1 + t 0 f ℓ,1 Ḃs 2,1 + f h,1 Ḃs+1 2,1 + g Ḃs 2,1
dτ, with eq:J0J1 eq:J0J1 (5.9)

j 0 := j 0 - √ n b - √ n ε L div u and j 1 := j 1 + 1 √ n LM ∇j 0 - 1 LL s M ∇b.
According to our previous work in [7], the critical regularity framework corresponds to s = n/2 -1. Therefore, the following quantities will play an important role (b, u, j 0 , j 1 ) X := b ℓ,1

Ḃ n 2 -1 2,1 + b h,1 Ḃ n 2 2,1 + ( u, j 0 , j 1 ) Ḃ n 2 -1 2,1
, and (b, u, j 0 , j 1 )

Y := sup t≥0 (b, u, j 0 , j 1 )(t) X + R + b ℓ,1 Ḃ n 2 +1 2,1 + b h,1 Ḃ n 2 2,1 + u Ḃ n 2 +1 2,1 dτ + R + M j 1 ℓ,ενM Ḃ n 2 -1 2,1 + j 0 ℓ,1 Ḃ n 2 -1 2,1 + j 0 m,1,ενM Ḃ n 2 +1 2,1 + M (j 0 , j 1 ) h,ενM Ḃ n 2 -1 2,1
dτ.

We denote by X and Y the corresponding functional spaces (where time continuity is imposed rather than just boundedness) and agree that Y (t) stands for the restriction of Y to the interval [0, t].

2 Further explanations on the method will be supplied to the reader in the next paragraph.

In the case ε 2 L s → +∞, we have to change slightly the definition of the norms • X and • Y as the middle frequencies obey (5.6). Consequently, we change • X to • X∞ with (b, u, j 0 , j 1 ) X∞ := b ℓ,1

Ḃ n 2 -1 2,1 + b h,1 Ḃ n 2 2,1 + ( u, j 1 ) Ḃ n 2 -1 2,1 + j 0 ℓ,1 Ḃ n 2 -1 2,1 + j 0 h,ενM Ḃ n 2 -1 2,1 , + 1≤2 k ≤εM 2 k n 2 max(1, 2 k , min(2 -k ε 2 νM)) ∆k j 0 L 2 and • Y to (b, u, j 0 , j 1 ) Y∞ := sup t≥0 (b, u, j 0 , j 1 )(t) X∞ + R + b ℓ,1 Ḃ n 2 +1 2,1 + b h,1 Ḃ n 2 2,1 + u Ḃ n 2 +1 2,1 dτ + R + M j 1 ℓ,ενM Ḃ n 2 -1 2,1 + j 0 ℓ,1 Ḃ n 2 -1 2,1 + M (j 0 , j 1 ) h,ενM Ḃ n 2 -1 2,1 dτ + R + 1≤2 k ≤εM 2 k n 2 min(1, 2 2k ε -2 ν -1 M -1 ) ∆k j 0 L 2 dτ.
To prove global estimates for the nonlinear system (1.6), the natural next step would be to take advantage of (5.8) with s = n/2 -1 and all the nonlinear terms in the r.h.s. Unfortunately, this does not work for the convection term u • ∇b causes a loss of one derivative

(indeed, if b is in Ḃ n 2 2,1 then u • ∇b cannot be smoother than Ḃ n 2 -1
2,1 ). A nowadays standard way to overcome the difficulty is to paralinearize (1.6), that is to add to (4.4) the principal parts of the convection terms. This is the aim of the next paragraph.

ss:para 5.1.2. The paralinearized system. Before introducing the paralinearized system associated to (1.6), let us shortly recall the definition of the paraproduct, according to the pioneering paper [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] by J.-M. Bony. The (homogeneous) paraproduct between two tempered distributions U and V satisfying (3.1) is given by

T U V := k Ṡk-1 U ∆k V with Ṡk-1 := χ(2 k-1 D).
We also introduce

T ′ V U := k Ṡk+2 V ∆k U,
and, observe that, at least formally

U V = T U V + T ′ V U.
To some extent, if U is smooth enough then T U V may be seen as the principal part of the product U V. This motivates our considering the following system eq:par-diff eq:par-diff (5.10)

               ∂ t b + T v • ∇b + div u = F, ∂ t u + T v • ∇ u -A u + ∇b -LM n j 1 = G, ∂ t j 0 + 1 ε √ n div j 1 + L ε (j 0 - √ n b) = 0, ∂ t j 1 + 1 ε √ n ∇j 0 + LM ε j 1 = 0,
where A = µ∆+(λ+µ)∇div , M := 1+L s , v and G are given time dependent vector-fields, and F is a given real valued function.

p:para-diff Proposition 5.1. For any smooth solution (b, u, j 0 , j 1 ) we have the following a priori estimate if 0 < m < +∞ (b, u, j 0 , j 1 ) Y (t) ≤ C (b, u, j 0 , j 1 )(0

) X + t 0 ∇ v L ∞ (b, u, j 0 , j 1 ) X dτ + t 0 (∇F, G) h,1 Ḃ n 2 -1 2,1 dτ + t 0 (F -T v • ∇b, G -T v • ∇ u) ℓ,1 Ḃ n 2 -1 2,1 dτ •
A similar inequality holds if m = +∞, with X ∞ (t) and Y ∞ (t) instead of X(t) and Y (t).

Proof: Localizing System (5.10) by means of ∆k yields eq:par-diff-k eq:par-diff-k (5.11)

               ∂ t ∆k b + ∆k (T v • ∇b) + div ∆k u = ∆k F, ∂ t ∆k u + ∆k (T v • ∇ u) -A ∆k u + ∇ ∆k b -LM n ∆k j 1 = ∆k G, ∂ t ∆k j 0 + 1 ε √ n div ∆k j 1 + L ε ( ∆k j 0 - √ n ∆k b) = 0, ∂ t ∆k j 1 + 1 ε √ n ∇ ∆k j 0 + LM ε ∆k j 1 = 0.
The important point is that in order to obtain all the estimates corresponding to ρ ≥ C 1 , one only has to resort to combinations between fluid unknowns on one side, and radiative unknowns, on the other side. This will enable us to use exactly the same energy method for (5.10) as for (4.4), in the middle and high frequency regimes, without introducing unwanted parts of convection terms in the inequalities.

1. Low frequencies:

2 k ≤ C 1 .
Including the para-convection terms in the source terms of (5.11) and repeating the computations leading to (5.2), we get after taking L 2 norms and using Fourier-Plancherel theorem eq:lfnoneq1 eq:lfnoneq1 (5.12) ∆k (b, u, j 0 , j 1 )(t

) L 2 + 2 2k t 0 ∆k (b, u) L 2 dτ + t 0 ∆k j 0 L 2 dτ + νL s t 0 ∆k j 1 L 2 dτ ∆k (b, u, j 0 , j 1 )(0) L 2 + t 0 ∆k (F -T u • ∇b) L 2 dτ + t 0 ∆k ( G -T u • ∇ u) L 2 dτ.

Medium frequencies:

C 1 ≤ 2 k ≤ c LL s .
Keeping in mind the proof of (5.3), we see that it is suitable to introduce

ζ 1 := j 1 + 1 √ n LM ∇j 0 .
Now, because we have

∂ t ∆k b + div ∆k u = ∆k F, ∂ t ∆k u -A ∆k u + ∇ ∆k b = 1 n 3/2 ∇ ∆k j 0 + LM n ∆k ζ 1 + ∆k G,
we easily get by computing eq:lya1 eq:lya1 (5.13) 1 2

d dt 2 ( ∆k b, ∆k u) 2 L 2 + ∆k ∇b 2 L 2 + 2( ∆k ∇b| ∆k u) L 2 ,
and by using Lemma 4.1 in [7] to handle the para-convection terms, the following inequality for all 2 k ≥ C 1

( ∆k ∇b, ∆k u)(t) L 2 + 2 2k t 0 ∆k u L 2 dτ + t 0 ∆k ∇b L 2 dτ ( ∆k ∇b, ∆k u)(0) L 2 + t 0 ( ∆k ∇F, ∆k G) L 2 dτ + t 0 ∇ ∆k j 0 L 2 dτ + LM t 0 ∇ ∆k j 1 L 2 dτ + k ′ ∼k t 0 ∇ v L ∞ ( ∆k ′ ∇b, ∆k ′ u) L 2 dτ.
Then looking at the equations satisfied by ∆k j 0 and ∆k ζ 1 (in the spirit of (4.40)), we derive inequalities similar to (4.50) and (4.51) for ∆k j 0 L 2 and ∆k ζ 1 L 2 , and thus following the computations leading to (4.54), we end up in the case m < +∞ with eq:mfnoneq1 eq:mfnoneq1 (5.14) ∆k (∇b, u, ∇j 0 , j 1 )(t

) L 2 + 2 2k t 0 ( ∆k u, ∆k j 0 ) L 2 dτ + t 0 ∆k ∇b L 2 dτ + νL s t 0 ∆k ζ 1 L 2 dτ ∆k (∇b, u, ∇j 0 , j 1 )(0) L 2 + t 0 ∆k (∇F, G) L 2 dτ + k ′ ∼k t 0 ∇ v L ∞ ∆k ′ (∇b, u) L 2 dτ.
Comparing the definition of ζ 1 and j 1 , we see that one may replace ζ 1 with j 1 above, if

C 1 ≤ 2 k ≤ c LL s .
The obvious modifications to be done if m = +∞ are left to the reader.

High frequencies:

2 k ≥ c LL s .
Again, we compute (5.13) to bound the fluid unknowns. In addition, to handle radiative unknowns, we compute for some small enough κ (see the proof of (4.66)) the following quantity 1 2

d dt ∆k j 0 2 L 2 + ∆k j 1 2 L 2 -κ LM 2 -2k ( ∆k j 0 | ∆k div j 1 ) L 2 .
Combining the computations leading to (5.7) with Fourier-Plancherel theorem and Lemma 4.1 in [7] eventually yields ∆k (∇b, u, j 0 , j 1 )(t)

L 2 +2 2k t 0 ∆k u L 2 dτ +2 k t 0 ∆k b L 2 dτ +νL s t 0 ( ∆k j 0 , ∆k j 1 ) L 2 dτ ∆k (∇b, u, j 0 , j 1 )(0) L 2 + t 0 ∆k (∇F, G) L 2 dτ + k ′ ∼k t 0 ∇ v L ∞ ∆k ′ (∇b, u) L 2 dτ.
Finally, multiplying (5.12), (5.14) and the above inequality by 2 k( n 2 -1) and summing up over k completes the proof of the proposition.

5.1.3.

A global existence result. According to the computations of the previous paragraph and to the change of variables (4.1), it is suitable to introduce the following norms for getting global solutions with uniform estimates in the case 3 m < +∞ (b, u, j 0 , j 1 )

X ν ε := b ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b h,ν -1 Ḃ n 2 2,1 + ( u, j 0 , j 1 ) Ḃ n 2 -1 2,1
, 3 Writing out the corresponding definition if m = +∞ is left to the reader.

and

(b, u, j 0 , j 1 ) Y ν ε := sup t≥0 (b, u, j 0 , j 1 )(t)

X ν + R + ν b ℓ,ν -1 Ḃ n 2 +1 2,1 + b h,ν -1 Ḃ n 2 2,1 + ν u Ḃ n 2 +1 2,1 dτ + R + ν -1 M j 1 ℓ,εM Ḃ n 2 -1 2,1 + ν -1 j 0 ℓ,ν -1 Ḃ n 2 -1 2,1 + ν j 0 m,ν -1 ,εM Ḃ n 2 +1 2,1 + ν -1 M (j 0 , j 1 ) h,εM Ḃ n 2 -1 2,1
dτ,

with j 0 := j 0 -b -ε L div u and j 1 := j 1 + 1 LM ∇j 0 -1 LLsM ∇b. Of course, if (b ′ , u ′ , j ′ 0 , j ′ 1 )
and (b, u, j 0 , j 1 ) are interrelated through (4.1) and νL is used for (b ′ , u ′ , j ′ 0 , j ′ 1 ) instead of L, then we have

(b ′ , u ′ , j ′ 0 , j ′ 1 ) X 1 ε = ν -1 (b, u, j 0 , j 1 ) X ν ε and (b ′ , u ′ , j ′ 0 , j ′ 1 ) Y 1 ε = ν -1 (b, u, j 0 , j 1 ) Y ν ε . th:diff1 Theorem 5.1. Assume that L ≈ 1, that lim inf ε -1 nνL > 1 and that L 2 L s ≈ 1.
There exists a positive constant η depending only on µ/ν, n and on the pressure law such that if ε is small enough and the data (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) satisfy eq:small-diff eq:small-diff (5.15)

(b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) X ν ε ≤ ην, then System (1.6) admits a unique global solution (b ε , u ε , j ε 0 , j ε 1 ) in Y ν ε .
In addition, we have eq:unif-diff eq:unif-diff (5.16)

(b ε , u ε , j ε 0 , j ε 1 ) Y ν ε ≤ C (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) X ν ε . A similar result holds true if L 2 L s → +∞.
Proof: Performing the change of variables proposed in (4.1) reduces the proof to the case ν = 1 (changing L into L := νL). Hence we consider a smooth enough solution to (4.2), and show that one may close the estimates globally 4 under Assumption (5.15).

Let us set U

0 := (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) X 1 ε and U (t) := (b ε , u ε , j ε 0 , j ε 1 ) Y 1 ε (t) .
In what follows, we drop exponents ε for notational simplicity. Finally, to shorten the presentation, we just treat the case where L 2 L s ≈ 1.

Now applying Proposition 5.1 with

v = u, F := -T ′ ∇b • u -k 1 (b)div u and G := -T ′ ∇ u • u + k 2 (b) A u -k 3 (b)∇b + LM n k 4 (b) j 1 ,
yields for all t ≥ 0 eq:U-diff eq:U-diff (5.17)

U (t) ≤ C U 0 + t 0 ∇ u L ∞ (b, u, j 0 , j 1 ) X 1 ε dτ + t 0 F Ḃ n 2 2,1 + F -T u • ∇b Ḃ n 2 -1 2,1 + G Ḃ n 2 -1 2,1 + T u • ∇ u Ḃ n 2 -1 2,1 dτ •
Using standard continuity results for the paraproduct and remainder, and composition estimates leads to

T ′ ∇b • u Ḃ n 2 2,1 ≤ C ∇b Ḃ n 2 -1 2,1 u Ḃ n 2 +1 2,1 , k 1 (b)div u Ḃ n 2 2,1 ≤ C b Ḃ n 2 2,1 div u Ḃ n 2 2,1
.

4 Existence follows from spectral truncation as in e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 10, and is thus omitted. As for uniqueness, we refer to [7].

Hence we have

eq:U-diff1 eq:U-diff1 (5.18)

t 0 F Ḃ n 2 2,1 dτ ≤ CU 2 (t).
We also have

T u • ∇ u Ḃ n 2 -1 2,1 ≤ C u Ḃ n 2 -1 2,1 ∇ u Ḃ n 2 2,1 , T ′ ∇ u • u Ḃ n 2 2,1 ≤ C ∇ u Ḃ n 2 2,1 u Ḃ n 2 -1 2,1 , k 2 (b) A u Ḃ n 2 -1 2,1 ≤ C b Ḃ n 2 2,1 ∇ 2 u Ḃ n 2 -1 2,1 , k 3 (b)∇b Ḃ n 2 -1 2,1 ≤ C b Ḃ n 2 2,1 ∇b Ḃ n 2 -1 2,1
.

Bounding LMk 4 (b) j 1 is slightly more involved as it is not true that the low frequencies of

j 1 are bounded in L 1 (R + ; Ḃ n 2 -1 2,1
). However, one may write that

LM j 1 = LM j h, LM 1 + LM j ℓ, LM 1 + L -1 s ∇b ℓ, LM -n -1/2 ∇j ℓ, LM 0 . Therefore LM k 4 (b) j 1 L 1 t ( Ḃ n 2 -1 2,1 ) LM j 1 h, LM L 1 t ( Ḃ n 2 -1 2,1 ) + j 1 ℓ, LM L 1 t ( Ḃ n 2 -1 2,1 ) b L ∞ t ( Ḃ n 2 2,1 ) + b L 2 t ( Ḃ n 2 2,1 ) L -1 s ∇b ℓ, LM L 2 t ( Ḃ n 2 -1 2,1 ) + ∇j 0 ℓ, LM L 2 t ( Ḃ n 2 -1 2,1 )
.

Hence eq:U-diff2 eq:U-diff2 (5.19)

t 0 T u • ∇ u Ḃ n 2 -1 2,1 + G Ḃ n 2 -1 2,1 dτ ≤ CU 2 (t). Finally u • ∇b Ḃ n 2 -1 2,1 ≤C u Ḃ n 2 2,1 ∇b Ḃ n 2 -1 2,1 , k 1 (b)div u Ḃ n 2 -1 2,1 ≤C b Ḃ n 2 2,1 div u Ḃ n 2 -1 2,1
.

Therefore, by Cauchy-Schwarz inequality eq:U-diff3 eq:U-diff3 (5.20)

t 0 F -T u • ∇b Ḃ n 2 -1 2,1 dτ ≤ CU 2 (t).
Inserting (5.18), (5.19), (5.20) in (5.17) and remembering that

Ḃ n 2 2,1 ֒→ L ∞ (to ensure that, say, |b| ≤ 1/2 if b Ḃ n 2 2,1
is small enough), we end up with

U (t) ≤ C(U 0 + U 2 (t)) for all t ≥ 0.
By a standard bootstrap argument, we easily deduce that U (t) ≤ 2CU 0 for all t ≥ 0, provided the data have been chosen so that 4C 2 U 0 ≤ 1. 5.2. Study of the limit system. In this paragraph, we prove the existence and uniqueness of strong (small) solutions with critical regularity for Systems (2.5) and (2.6). We shall give a common proof that works for both systems.

Before giving the global existence statement, let us introduce the solution space • If m ∈ (0, +∞) (that is for System (2.5)) then Initial data will be taken in the space X ν which is the set of triplets (b, u, j 0 ) satisfying

(b, u, j 0 ) X ν := b ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b h,ν -1 Ḃ n 2 2,1 + u Ḃ n 2 -1 2,1 + ν j 0 ℓ,ν -1 Ḃ n 2 2,1 + ν -1 j 0 h,ν -1 Ḃ n 2 -2 2,1 < ∞,
and the solution space Y ν will be the set of triplets (b, u, j 0 ) in

C b (R + ; X ν ) satisfying (b, u, j 0 ) Y ν := sup t≥0 (b, u, j 0 )(t) X ν + R + j 0 -b Ḃ n 2 2,1 + ν b ℓ,ν -1 Ḃ n 2 +1 2,1 + ν u Ḃ n 2 +1 2,1 + j 0 h,ν -1 B n 2 2,1
dτ.

• If m = +∞ (that is for System (2.6)), Initial data will be taken in the space X ν ∞ which is the set of triplets (b, u, j 0 ) satisfying

(b, u, j 0 ) X ν ∞ := b ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b h,ν -1 Ḃ n 2 2,1 + u Ḃ n 2 -1 2,1 + ν j 0 Ḃ n 2 2,1 < ∞,
and the solution space Y ν ∞ will be the set of triplets (b, u, j 0 ) in

C b (R + ; X ν ∞ ) satisfying (b, u, j 0 ) Y ν ∞ := sup t≥0 (b, u, j 0 )(t) X ν ∞ + R + j 0 -b Ḃ n 2 2,1 + ν b ℓ,ν -1 Ḃ n 2 +1 2,1 + ν u Ḃ n 2 +1 2,1 + j 0 h,ν -1 B n 2 2,1
dτ < ∞.

th:noneqlim1 Theorem 5.2. There exist two positive constants c and C so that if

(b 0 , u 0 , j 0,0 ) X ν ≤ cν (case m < +∞), q:smallnoneq2 (5.21)
or (b 0 , u 0 , j 0,0 ) X ν ∞ ≤ cν (case m = +∞), q:smallnoneq1 (5.22) then System (2.5) (resp. (2.6)) admits a unique solution in the space Y ν (resp. Y ν ∞ ) satisfying in addition,

(b, u, j 0 ) Y ν ≤ C (b 0 , u 0 , j 0,0 ) X ν if m < +∞, eq:estnoneq2 (5.23) (b, u, j 0 ) Y ν ∞ ≤ C (b 0 , u 0 , j 0,0 ) X ν ∞ if m = +∞• eq:estnoneq1 (5.24)
Proof: Set κ := κ/n and m := mn. As usual, it is enough to treat the case ν = 1 as performing the change of unknowns

(b, u, j 0 )(t, x) = ( b, u, j 0 )(ν -1 t, ν -1 x),
gives Systems (2.5) or (2.6) for ( b, u, j 0 ) with ν = 1 and A := A/ν and, obviously

(b, u, j 0 )(t) X ν = ν ( b, u, j 0 )(ν -1 t) X 1 and (b, u, j 0 ) Y ν = ν ( b, u, j 0 ) Y 1 .
Let us start with the study of the linearized equations with no source term, namely

(5.25)

     ∂ t b + div u = 0, ∂ t u -A u + ∇b + n -1 ∇j 0 = 0, ∂ t j 0 + κ(j 0 -m -1 ∆j 0 -b) = 0.
The divergence-free part P u of the velocity satisfies

∂ t P u -µ∆P u = 0,
while the coupling between b, d := Λ -1 div u and j 0 is described by eq:l-diff5 eq:l-diff5 (5.26)

     ∂ t b + Λd = 0, ∂ t d -∆d -Λb -n -1 Λj 0 = 0, ∂ t j 0 + κ(j 0 -m -1 ∆j 0 -b) = 0.
Note that the stability of a similar system has already been established in the previous section for κ > 1 (or, equivalently, κ > 1/n).

Linear estimates for low frequencies. We introduce

ζ 0 := j 0 -b -κ -1 Λd and notice that          ∂ t b + ρ d = 0, ∂ t d + ρ 2 1 -1 κn d -1 + 1 n ρ b = 1 n ρ ζ 0 , ∂ t ζ 0 + κ + ( κ m + 1 κn )ρ 2 ζ 0 = -1 + 1 n 1 κ + κ m ρ 2 b + 1 -1 κn 1 κ -1 m ρ 3 d.
On one hand, because κn > 1, the method described in the appendix (see in particular (B.7)) allows to write that, omitting the dependence with respect to κ

|( b, d)(t)| + ρ 2 t 0 |( b, d)| dτ |( b, d)(0)| + ρ t 0 | ζ 0 | dτ.
On the other hand, the last equation directly gives

| ζ 0 (t)| + κ + ( κ m + 1 κn )ρ 2 t 0 | ζ 0 | dτ ≤ | ζ 0 (0)| + C 1 + 1 m ρ 2 t 0 |( b, ρ d)| dτ.
Hence plugging the second inequality in the first one

|( b, d)(t)|+ρ 2 t 0 |( b, d)| dτ |( b, d)(0)|+ ρ 1+(1+m -1 )ρ 2 | ζ 0 (0)|+(1+m -1 )ρ 2 t 0 |( b, ρ d)| dτ •
It is clear that the last term may be absorbed by the integral of the l.h.s. if ρ ≪ m 1+m • Hence we eventually get for some small enough ρ ℓ > 0 eq:l-diff6 eq:l-diff6 (5.27)

|( b, d, ρ ζ 0 )(t)| + ρ 2 t 0 |( b, d)| dτ + t 0 |ρ ζ 0 | dτ |( b, d, ρ ζ 0 )(0)| if ρ ≤ m 1 + m ρ ℓ .
Linear estimates for high frequencies. We set δ := d -Λ -1 b and notice that

       ∂ t b + b = -ρ δ, ∂ t δ + (ρ 2 -1) δ = ρ -1 b + n -1 ρ j 0 , ∂ t j 0 + κ 1 + ρ 2 m j 0 = κ b. Therefore | δ(t)| + (ρ 2 -1) t 0 | δ| dτ ≤ | δ(0)| + ρ -1 t 0 | b| dτ + 1 n t 0 ρ| j 0 | dτ.
At the same time

| b(t)| + t 0 | b| dτ ≤ | b(0)| + ρ t 0 | δ| dτ, | j 0 (t)| + κ 1 + ρ 2 m t 0 | j 0 | dτ ≤ | j 0 (0)| + κ t 0 | b| dτ. Hence | δ(t)| + (ρ 2 -1) t 0 | δ| dτ ≤ | δ(0)| + 1 ρ + ρ n | b(0)| + n -1 κ -1 1+ m -1 ρ 2 ρ| j 0 (0)| + 1 + ρ 2 n(1 + m -1 ρ 2 ) t 0 | δ| dτ.
Therefore there exists a constant ρ h depending only on m and n (with n ≥ 2 if m = +∞) such that for ρ ≥ ρ h , we have eq:l-diff8 eq:l-diff8 (5.28)

|(ρ b, δ)(t)| + min(ρ, mρ -1 )| j 0 (t)| + ρ t 0 |( b, j 0 )| dτ + ρ 2 t 0 | δ| dτ |(ρ b, δ, ρ j 0 )(0)|.
Of course, one may replace δ with d in (5.28).

Linear estimates for medium frequencies. The stability argument used just below (4.37) allows to write that there exist two constants c and C depending continuously on 1/m, such that if ρ ∈ [ m m+1 ρ ℓ , ρ h ] then eq:l-diff9 eq:l-diff9 (5.29)

|( b, d, j 0 )(t)| ≤ Ce -ct |( b, d, j 0 )(0)|.
Estimates for the paralinearized system. The previous steps allow us to get handy estimates for the following paralinearized version of System (2.6) eq:l-diff10 eq:l-diff10 (5.30)

     ∂ t b + T v • ∇b + div u = F, ∂ t u + T v • ∇ u -A u + ∇b + n -1 ∇j 0 = G, ∂ t j 0 + κ(j 0 -m -1 ∆j 0 -b) = 0.
More precisely, following the steps leading to (5.27), (5.28) and (5.29), introducing ζ 0 := j 0 -b -κ -1 div u, and arguing as in Subsection 5.1.2 we end up with 5 eq:l-diff11 eq:l-diff11 (5.31) j 0 (t) ℓ,1

Ḃ n 2 2,1 + (b, u)(t) ℓ,1 Ḃ n 2 -1 2,1 + t 0 j 0 -b ℓ,1 Ḃ n 2 2,1 + (b, u) ℓ,1 Ḃ n 2 +1 2,1 dτ j 0 (0) ℓ,1 Ḃ n 2 2,1 + (b, u)(0) ℓ,1 Ḃ n 2 -1 2,1 + t 0 F -T v • ∇b ℓ,1 Ḃ n 2 -1 2,1 + G -T v • ∇ u ℓ,1 Ḃ n 2 -1 2,1
dτ.

For high frequencies, we get, in the case m = +∞ eq:l-diff12a eq:l-diff12a (5.32) (b, j 0 )(t) h,1

Ḃ n 2 2,1 + u(t) h,1 Ḃ n 2 -1 2,1 + t 0 (b, j 0 ) h,1 Ḃ n 2 2,1 + u h,1 Ḃ n 2 +1 2,1 dτ (b, j 0 )(0) h,1 Ḃ n 2 2,1 + u(0) h,1 Ḃ n 2 -1 2,1 + t 0 F h,1 Ḃ n 2 2,1 + G h,1 Ḃ n 2 -1 2,1 dτ + t 0 ∇ v L ∞ (b, j 0 ) Ḃ n 2 2,1 + u Ḃ n 2 -1 2,1
dτ, 5 Here we do not track the dependency with respect to m. and if 0 < m < +∞ eq:l-diff12b eq:l-diff12b (5.33) b(t) h,1

Ḃ n 2 2,1 + j 0 (t) h,1 Ḃ n 2 -2 2,1 + u(t) h,1 Ḃ n 2 -1 2,1 + t 0 (b, j 0 ) h,1 Ḃ n 2 2,1 + u h,1 Ḃ n 2 +1 2,1 dτ b(0) h,1 Ḃ n 2 2,1 + j 0 (0) h,1 Ḃ n 2 -2 2,1 + u(0) h,1 Ḃ n 2 -1 2,1 + t 0 F h,1 Ḃ n 2 2,1 + G h,1 Ḃ n 2 -1 2,1 dτ + t 0 ∇ v L ∞ b Ḃ n 2 2,1 + j 0 h Ḃ n 2 -2 2,1 + u Ḃ n 2 -1 2,1
dτ.

Proof of existence. We only establish global-in-time a priori bounds in the space Y 1 or Y 1

∞

for the solutions to (2.5) or (2.6) with data satisfying (5.21) or (5.22). Our proof is based on (5.31), (5.32) and (5.33

) with v = u F = -T ′ ∇b • v -k 1 (b)div u and G = -T ′ ∇ u • u + k 2 (b) A u -k 3 (b)∇b -n -1 k 4 (b)∇j 0 . Bounding F -T v • ∇b ℓ,1 Ḃ n 2 -1 2,1 and F h,1 Ḃ n 2 2,1
relies on (5.18) and (5.20). As regards G, the computations that we did in the proof of Theorem 5.1 ensure that the first three terms may be bounded as in (5.19). To handle the last term, k 4 (b)∇j 0 , in the case 6 m < +∞ we use the decomposition

k 4 (b)∇j 0 = k 4 (b)∇b + k 4 (b)∇(j 0 -b).
The first term may be bounded quadratically exactly as k 3 (b)∇b. As for the last term, we may write k 4 (b)∇(j 0 -b)

L 1 t ( Ḃ n 2 2,1 ) b L ∞ t ( Ḃ n 2 2,1 ) ∇(j 0 -b) L 1 t ( Ḃ n 2 -1 2,1 )
, hence it is also bounded by C (b, u, j 0 ) 2 Y 1 (t) . This enables to conclude that we do have for all

t ∈ R + (b, u, j 0 ) Y 1 (t) ≤ C (b, u, j 0 )(0) X 1 + (b, u, j 0 ) 2 Y 1 (t)
. This obviously yields (5.24) if (5.22) is fulfilled.

Proof of uniqueness. It works the same as for the standard barotropic Navier-Stokes equations: we look at the system satisfied by the difference (δb, δ u, δj 0 ) between two solutions (b 1 , u 1 , j 1 0 ) and (b 2 , u 2 , j 2 0 ) of (2.5), namely (denoting

K i = 1 + k i for i = 1, 2, 3, 4)          ∂ t δb + u 2 • ∇δb = -δ u • ∇b 1 + (K 1 (b 1 ) -K 1 (b 2 ))div u 2 -K 1 (b 1 )div δ u, ∂ t δ u + u 2 •∇δ u + δ u•∇ u 1 -(K 2 (b 2 )-K 2 (b 1 ))A u 2 -K 2 (b 1 )Aδ u + (K 3 (b 2 )-K 3 (b 1 ))∇b 2 +K 3 (b 1 )∇δb + n -1 (K 4 (b 2 )-K 4 (b 1 ))∇j 1 0 + n -1 K 4 (b 2 )∇δj 0 = 0, ∂ t δj 0 + κ δj 0 -δb -1
m ∆δj 0 = 0. Now, exactly as for the barotropic Navier-Stokes equations, it is possible to bound δb, δ u and δj 0 just resorting to basic estimates for the transport and heat equations. However, the hyperbolic nature of the first equation forces us to estimate (δb, δ u, δj 0 ) with one less derivative, namely in

L ∞ (0, T ; Ḃ n 2 -1 2,1 ) × L ∞ (0, T ; Ḃ n 2 -2 2,1 ) ∩ L 1 (0, T ; Ḃ n 2 2,1 ) n × L 1 (0, T ; Ḃ n 2 -1 2,1
). In dimension n = 3 combining estimates for the transport and the heat equation allows to get uniqueness on a small time interval, then on the whole R + by induction. In dimension 6 The case m = +∞ does not require that decomposition. n = 2, this is slightly more involved as some product laws do not work correctly if estimating (δb, δ u, δj 0 ) in the above space (some regularity exponents become too negative). Nevertheless this may be overcome by combining logarithmic interpolation and Osgood lemma (see e.g.

[6] for more details). This completes the proof of the theorem.

Remark 5.1. If 0 < m < +∞ then one may alternately assume that j 0 is in Ḃ n 2 2,1 . Taking advantage of the parabolic smoothing given by the equation for j 0 , it is not difficult to get a solution (b, u, j 0 ) with

m R + j 0 Ḃ n 2 +2 2,1 ≤ C b 0 Ḃ n 2 -1 2,1 + ν b 0 Ḃ n 2 2,1 + u 0 Ḃ n 2 -1 2,1 + j 0 Ḃ n 2 2,1 • 5.3. Weak convergence.
Here we justify the weak convergence of (1.6) to (2.5) or (2.6) under the assumption that lim inf ε 2 L s > 0 and that L tends to κε nν for some κ > 1.

th:diff-lim1 Theorem 5.3. Let the family of data (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) 0<ε<1 satisfy Condition (5.15). Assume in addition that eq:l-diff0 eq:l-diff0 (5.34)

L 2 L s ν 2 → m ∈ (0, +∞] and nνL ε → κ ∈ (1, +∞).
Then the global solution (b ε , u ε , j ε 0 , j ε 1 ) given by Theorem 5.1 satisfies

j ε 1 → 0 in L 1 (R + ; Ḃ n 2 -1 2,1 + Ḃ n 2 2,1 ), and, if (b ε 0 , u ε 0 , j ε 0,0 ) ⇀ (b 0 , u 0 , j 0,0 ) then (b ε , u ε , j ε 0 )
converges weakly to the unique solution (b, u, j 0 ) of (2.5) supplemented with initial data (b 0 , u 0 , j 0,0 ). Proof: From (5.16) we gather that

( j ε 1 ) ℓ,LM = O(M -1 ) and ( j ε 1 ) h,LM = O(M -1 ) in L 1 (R + ; Ḃ n 2 -1 2,1
). Therefore, taking advantage of the boundedness of the low frequencies of ∇b ε and ∇j ε 0 in

L 1 (R + ; Ḃ n 2 2,1
), and of the fact that

j ε 1 = j ε 1 - 1 LM ∇j ε 0 + 1 LL s M ∇b ε ,
we get eq:l-diff1 eq:l-diff1 (5.35)

j ε 1 = O(ε) in L 1 (R + ; Ḃ n 2 -1 2,1 + Ḃ n 2 2,1 ).
Next, we observe that (5.16) implies that (b ε ) and (

u ε ) are bounded in L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2 2,1 ) ∩ L 1 (R + ; Ḃ n 2 +1 2,1 + Ḃ n 2 2,1 ) and L ∞ (R + ; Ḃ n 2 -1 2,1 ) ∩ L 1 (R + ; Ḃ n 2 +1
2,1 ), respectively. Note that this implies that u ε is bounded in

L 2 (R + ; Ḃ n 2 2,1 ). Because ∂ t b ε = -u ε • ∇b ε -k 1 (b ε )div u ε ,
and the product maps

Ḃ n 2 -1 2,1 × Ḃ n 2 2,1 in Ḃ n 2 -1 2,1 , we thus get in addition that ∂ t b ε is bounded in L 2 (R + ; Ḃ n 2 -1 2,1 ), and thus (b ε ) is bounded in C 1 2 (R + ; Ḃ n 2 -1 2,1 ). Interpolating with the bound in C b (R + ; Ḃ n 2 2,1 ), we thus have (b ε ) bounded in C α 2 (R + ; Ḃ n 2 -α 2,1
) for all α ∈ [0, 1]. Then combining locally compact Besov embeddings and Ascoli theorem allows to conclude that there exists b

in L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2 2,1 ) ∩ L 1 (R + ; Ḃ n 2 +1 2,1 + Ḃ n 2 2,1
) and a sequence (ε k ) k∈N going to 0 so that, for all φ ∈ S and all α ∈ (0, 1] eq:l-diff2 eq:l-diff2 (5.36)

φ b ε k -→ φ b in L ∞ loc (R + ; Ḃ n 2 -α 2,1
). From (5.16), we readily get for some sequence (ε k ) k∈N tending to 0 eq:l-diff4 eq:l-diff4 (5.37)

u ε k ⇀ u in L ∞ (R + ; Ḃ n 2 -1 2,1 ) ∩ L 1 (R + ; Ḃ n 2 +1
2,1 ) weak *, which, combined with (5.36) is clearly enough to pass to the limit in the mass equation.

Next, we see that (5.16) implies that (j

ε 0 ) is bounded in L ∞ (R + ; Ḃ n 2 -1 2,1
). Hence there exists

j 0 ∈ L ∞ (R + ; Ḃ n 2 -1 2,1
) and a sequence (ε k ) k∈N going to 0 so that

j ε k 0 ⇀ j 0 in L ∞ (R + ; Ḃ n 2 -1
2,1 ) weak *. Because eq:l-diff3 eq:l-diff3 (5.38)

1 ε div j ε 1 = - L ε 1 L 2 (1 + L s ) ∆j ε 0 - 1 L(1 + L s ) ∂ t div j ε 1 ,
and (5.35) implies that ∂ t j ε 1 → 0 in the sense of distributions, we deduce that

1 ε div j ε 1 → - κν nm ∆j 0 in S ′ .
Note that the right-hand side is 0 if m = +∞. Therefore (b, j 0 ) satisfies the third line of (2.5) (case m < +∞) or (2.6) (case m = +∞).

Let us finally pass to the limit in the second equation of (1.6). The main difficulty is that, owing to the radiative term which is only bounded in a L 1 -in-time type space (namely

L 1 (R + ; Ḃ n 2 -1
2,1 ) or so), one cannot take advantage of some suitable bound of ∂ t u ε so as to glean some equicontinuity and then resort to Ascoli theorem. To overcome this, we use the fact that, owing to (5.38)

∂ t u ε + ε n (1 + k 4 (b ε )) j ε 1 = -u ε • ∇ u ε + (1 + k 2 (b ε ))A u ε -(1 + k 3 (b ε ))∇b ε - 1 n (1 + k 4 (b ε )) ∇j ε 0 + ε n k ′ 4 (b ε )∂ t b ε j ε 1 . Now, because ( u ε ) is bounded in the space L ∞ (R + ; Ḃ n 2 -1 2,1 ) ∩ L 2 (R + ; Ḃ n 2 2,1 ), (j ε 0 ) is bounded in L ∞ (R + ; Ḃ n 2 -1 2,1 ) and (b ε ) is bounded in (L 2 ∩ L ∞ )(R + ; Ḃ n 2 2,1
), product laws in Besov spaces ensure that the first four terms of the r.h.s. are bounded in

L 2 (R + ; Ḃ n 2 -2 2,1 ) (only in L 2 (R + ; Ḃ n 2 -2 2,∞ ) if n = 2)
. The same property holds true for the last term for

(∂ t b ε ) is bounded in L 2 (R + ; Ḃ n 2 -1 2,1 ) and ( j ε 1 ) is bounded in L ∞ (R + ; Ḃ n 2 -1 2,1
). Using locally compact Besov embedding and Ascoli theorem, one can now conclude that there exists some v in

L ∞ (R + ; Ḃ n 2 -1 2,1
) so that for all φ in S and α ∈ (0, 1), we have, up to extraction

φ u ε + ε n (1 + k 4 (b ε )) j ε 1 -→ φ v in L ∞ loc (R + ; Ḃ n 2 -1-α 2,1
).

Of course, combining with (5.35), we discover that v = u. Hence we also have

φ u ε -→ φ v in L ∞ loc (R + Ḃ n 2 -1-α 2,1
) for all φ ∈ S.

It is now easy to conclude that the second line of (2.5) is fulfilled by (b, u, j 0 ).

Finally, that the whole family (b ε , u ε , j ε 0 ) (and not only subsequences) converges to (b, u, j 0 ) stems from the fact that the solution to (2.5) or (2.6) is unique.

Remark 5.2. It is also possible to justify the strong convergence of the solutions of (1.6) to (2.5) or (2.6) using (5.35) and performing the difference between (b ε , u ε , j ε 0 ) and the solution (b, u, j 0 ) to the limit system. Again, taking advantage of the decay properties of j ε 1 is crucial. Note however that, exactly as in the proof of uniqueness, owing to the hyperbolic nature of the density equation, one cannot prove the strong convergence in the solution space. There is a loss of one derivative that may be partially compensated by combining with uniform estimates. As we do not think this approach to bring much compared to weak compactness, we leave the details to the reader.

The equilibrium diffusion regime

This section is devoted to the mathematical justification of the equilibrium diffusion regime given by (2.10). To avoid useless technicality, we focus on the case where eq:eqass eq:eqass (6.1)

L → +∞ and εLM ≈ 1.

6.1. Linear estimates. Let us gather the estimates we proved for (4.7) for the above asymptotics in Section 4.

Regarding low frequencies, one may combine (4.20) and (4.21) to get eq:eqlf eq:eqlf (6.2) |( b, d, j 0 , j 1 )(t

)| + ρ 2 t 0 |( b, d)| dτ + L ε t 0 |j 0 | dτ + LM ε t 0 |j 1 | dτ ≤ C|( b, d, j 0 , j 1 )(0)| for 0 ≤ ρ ≤ 1 + n -1 , with L := νL, j 0 := j 0 - √ n b - √ n ε L ρ d and j 1 = j 1 -ρ LM b.
For middle frequencies, we have according to (4.58) and (4.64)

eq:eqmf eq:eqmf (6.3)

|(ρ b, d, j 0 , j 1 )(t)| + t 0 |ρ b| dτ + ρ 2 t 0 | d| dτ + ρ t 0 | j 0 | dτ + LM t 0 | j 1 | dτ ≤ C|(ρ b, d, j 0 , j 1 )(0)| for 2/n ≤ ρ ≤ c LM,
and (4.70) gives, if M is large enough eq:eqhf eq:eqhf (6.4)

|(ρ b, d, j 0 , j 1 )(t)| + ρ 2 t 0 | d| dτ + ρ t 0 | b| dτ + LM ε t 0 |( j 0 , j 1 )| dτ ≤ C|(ρ b, d, j 0 , j 1 )(0)| for ρ ≥ c LM.
If M is bounded then we must assume that ρ ≥ C 1 LM for some C 1 > c. However, we have (4.71) and M bounded implies that ε L ≈ 1. Therefore (6.3) is satisfied up to ρ ≤ C 1 LM.

For the whole system (4.4) with nonzero source terms f and g , we thus obtain (taking slightly abusively c = C 1 = 1 for notational simplicity) eq:eqestlin eq:eqestlin (6.5) ( u, j 0 , j 1 )(t) Ḃs

2,1 + b(t) ℓ,1 Ḃs 2,1 + b(t) h,1 Ḃs+1 2,1 + t 0 u Ḃs+2 2,1 dτ + L ε t 0 j 0 ℓ,1 Ḃs 2,1 dτ + LM ε t 0 j 1 ℓ,1 Ḃs 2,1 dτ + t 0 (b, j 0 , j 1 ) ℓ,1 Ḃs+2 2,1 dτ + t 0 j 0 m,1, LM Ḃs+1 2,1 dτ + LM t 0 j 1 m,1, LM Ḃs 2,1 dτ + t 0 b h,1 Ḃs+1 2,1 dτ + LM ε t 0 (j 0 , j 1 ) h, LM Ḃs 2,1 dτ ( u, j 0 , j 1 )(0) Ḃs 2,1 + b(0) ℓ,1 Ḃs 2,1 + b(0) h,1 Ḃs+1 2,1 + t 0 f ℓ,1 Ḃs 2,1 + f h,1 Ḃs+1 2,1 + g Ḃs 2,1
dτ, with

j 0 := j 0 - √ nb - √ n ε L div u and j 1 = j 1 + 1 LM ∇b.
Back to the original variables, that linear analysis induces us to introduce the following norms (b, u, j 0 , j 1 )

X ν ε := b ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b h,ν -1 Ḃ n 2 2,1 + u Ḃ n 2 -1 2,1 + (j 0 , j 1 ) Ḃ n 2 -1 2,1 and (b, u, j 0 , j 1 ) Y ν ε := sup t≥0 (b, u, j 0 , j 1 )(t) X ν ε + ν R + (b, j 0 , j 1 ) ℓ,ν -1 Ḃ n 2 +1 2,1 + u Ḃ n 2 +1 2,1 dτ + R + b h,ν -1 Ḃ n 2 2,1 + LM ε j 1 ℓ,ν -1 Ḃ n 2 -1 2,1 + L ε j 0 ℓ,ν -1 Ḃ n 2 -1 2,1 dτ + R + j 0 m,ν -1 ,LM Ḃ n 2 2,1 + LM j 1 m,ν -1 ,LM Ḃ n 2 -1 2,1 + LM ε (j 0 , j 1 ) h,LM Ḃ n 2 -1 2,1
dτ,

with j 0 := j 0 -b -ε L div u and j 1 := j 1 + 1 LM ∇b.
We denote by X ν ε and Y ν ε the corresponding functional spaces (where time continuity is imposed rather than just boundedness). Of course, we still have

(b ′ , u ′ , j ′ 0 , j ′ 1 ) X 1 ε = ν -1 (b, u, j 0 , j 1 ) X ν ε and (b ′ , u ′ , j ′ 0 , j ′ 1 ) Y 1 ε = ν -1 (b, u, j 0 , j 1 ) Y ν ε ,
through the change of variables (4.1), if we replace L by L in the left-hand side.

6.2. The paralinearized equations. In the equilibrium diffusion limit case the estimates for the paralinearized system eq:par-diff2 eq:par-diff2 (6.6)

             ∂ t b + T v • ∇b + div u = F, ∂ t u + T v • ∇ u -A u + ∇b -L(1+Ls) n j 1 = G, ∂ t j 0 + div j 1 nε + L ε (j 0 -b) = 0, ∂ t j 1 + ∇j 0 ε + L(1+Ls) ε j 1 = 0.
recast as follows p:para-diff2 Proposition 6.1. For any smooth solution (b, u, j 0 , j 1 ) we have the following a priori estimate for (5.10)

(b, u, j 0 , j 1 ) Y ν ε (t) ≤ C (b, u, j 0 , j 1 )(0) X 1 ε + t 0 ∇ v L ∞ (b, u, j 0 , j 1 ) X ν ε dτ + t 0 (∇F, G) h,1 Ḃ n 2 -1 2,1 dτ + t 0 (F -T v •∇b, G-T v •∇ u) ℓ,1 Ḃ n 2 -1 2,1 dτ + t 0 T v •∇b-F m,1, LM Ḃ n 2 -1 2,1 dτ • Proof:
Except in the middle frequencies range, the proof goes along the lines of the corresponding result in the non-equilibrium case. Still assuming that ν = 1 and replacing L with L = νL then, working directly on the localized paralinearized system (6.6), and combining Inequalities (6.2) to (6.4) with estimates for the para-convection terms gives 1. Low frequencies:

2 k ≤ C 1 . ∆k (b, u, j 0 , j 1 )(t) L 2 + 2 2k t 0 ∆k (b, u, j 0 , j 1 ) L 2 dτ + LM ε t 0 ∆k j 1 L 2 dτ + L ε t 0 ∆k j 0 L 2 dτ ∆k (b, u, j 0 , j 1 )(0) L 2 + t 0 ∆k (F -T v • ∇b) L 2 dτ + t 0 ∆k ( G -T v • ∇ u) L 2 dτ. 2. Medium frequencies: C 1 ≤ 2 k ≤ c LM.
One has to keep in mind that in order to derive (6.3) from (4.58) and (4.64), one has to consider the system that is fulfilled by (b, u, ζ 0 , j 1 ) with ζ 0 := j 0 -√ n b. In particular, a part of the the paraconvection term of b enters in the equation for ζ 0 as we have

∂ t ζ 0 + L ε ζ 0 + 1 ε √ n div j 1 - √ ndiv u = √ n(T v • ∇b -F ).
Therefore, following the computations leading to (4.58) and (4.64), and using Lemma 4.1 in [7] to bound the convection terms coming from the equations for b and u, we end up with ∆k (∇b, u, j 0 , j 1 )(t

) L 2 + 2 2k t 0 ∆k u L 2 dτ + 2 k t 0 ( ∆k b, ∆k j 0 ) L 2 dτ +L(1 + L s ) t 0 ∆k j 1 L 2 dτ ∆k (∇b, u, j 0 , j 1 )(0) L 2 + t 0 ∆k (∇F, G) L 2 dτ + t 0 ∆k (T v • ∇b -F ) L 2 dτ + k ′ ∼k t 0 ∇ v L ∞ ∆k ′ (∇b, u) L 2 dτ.
3. High frequencies:

2 k ≥ c LM. We get ∆k (∇b, u, j 0 , j 1 )(t) L 2 + t 0 2 2k ∆k u L 2 + 2 k ∆k b L 2 dτ + LMε t 0 ∆k (j 0 , j 1 ) L 2 dτ ∆k (∇b, u, j 0 , j 1 )(0) L 2 + t 0 ∆k (∇F, G) L 2 dτ + k ′ ∼k t 0 ∇ v L ∞ ∆k ′ (∇b, u) L 2 dτ.
Putting together all those inequalities completes the proof.

6.3.

A global existence result. Our global existence result with uniform estimates reads th:diff2 Theorem 6.1. There exists a positive constant η depending only on µ/ν, n and on the pressure law such that if ε ∈ (0, 1) and the data (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) satisfy q:small-diff2 q:small-diff2 (6.7)

(b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) X ν ε ≤ ην,
then System (1.6) admits a unique global solution (b ε , u ε , j ε 0 , j ε 1 ) in Y ν ε . In addition, we have eq:unif-diff2 eq:unif-diff2 (6.8)

(b ε , u ε , j ε 0 , j ε 1 ) Y ν ε ≤ C (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) X ν ε .
The proof relies on Proposition 6.1. Note in particular that the 'new' last term in the estimate of (6.1) does not entail a loss of derivative as we simply have

T v • ∇b -F m,1, LM L 1 t ( Ḃ n 2 -1 2,1 ) ≤ v • ∇b L 1 t ( Ḃ n 2 -1 2,1 ) v L 2 t ( Ḃ n 2 2,1 ) b L 2 t ( Ḃ n 2 2,1 )
.

The rest of the proof works exactly the same as in the non-equilibrium case.

6.4. Weak convergence. Here we justify weak convergence to (2.10) when assumption (6.1) is fulfilled.

th:diff-lim2 Theorem 6.2. Let the family of data (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) 0<ε<1 satisfy (6.7). Then the global solution (b ε , u ε , j ε 0 , j ε 1 ) in Y ν ε given by Theorem 6.1 satisfies

j ε 1 → 0 in L 1 (R + ; Ḃ n 2 -1 2,1 + Ḃ n 2 2,1 ), and if (b ε 0 , u ε 0 ) ⇀ (b 0 , u 0 ) then (b ε , u ε , j ε 0 ) converges weakly to (b, u, b)
where (b, u) stands for the unique solution of eq:eqlim eq:eqlim (6.9)

∂ t b + u • ∇b + k 1 (b)div u = 0, ∂ t u + u • ∇ u -k 2 (b)A u + k 3 (b) + n -1 k 4 (b) ∇b = 0.
supplemented with initial data (b 0 , u 0 ).

Proof: Let j ε 1 = j ε 1 + ∇b ε LM • From (6.8), we have LM ε j ε 1 ℓ,ν -1 L 1 (R + ; Ḃ n 2 -1 2,1 ) + ν ∇b ε ℓ,ν -1 L 1 (R + ; Ḃ n 2 2,1 ) + LM j ε 1 h,ν -1 L 1 (R + ; Ḃ n 2 -1 2,1 )
≤ Cην.

Hence, given (6.1), we deduce that eq:l-diff4b eq:l-diff4b (6.10)

j ε 1 = O(ε) in L 1 (R + ; Ḃ n 2 -1 2,1 + Ḃ n 2 2,1
). Using the equation of j ε 1 , this gives eq:l-diff4a eq:l-diff4a (6.11) ∇j ε 0 + L(1 + L s ) j ε 1 → 0 in the sense of distributions. As in the non-equilibrium case, (6.8) implies that the families (b ε ) and ( u ε ) are bounded in

L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2 2,1 )∩L 1 (R + ; Ḃ n 2 +1 2,1 + Ḃ n 2 2,1 ) and L ∞ (R + ; Ḃ n 2 -1 2,1 )∩L 1 (R + ; Ḃ n 2 +1 2,1 ), respectively. Hence (∂ t b ε ) is bounded in L 2 (R + ; Ḃ n 2 -1 2,1
) and we can thus deduce that there exists b in

L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2 2,1 ) ∩ L 1 (R + ; Ḃ n 2 +1 2,1 + Ḃ n 2 2,1
) and a sequence (ε k ) k∈N going to 0 so that, for all φ ∈ S and all α ∈ (0, 1] eq:l-diff5b eq:l-diff5b (6.12)

φ b ε k -→ φ b in L ∞ (R + ; Ḃ n 2 -α 2,1
). For ( u ε ), we still have the weak convergence result given by (5.37), which suffices to pass to the limit in the mass equation.

Next, we observe that (6.8) implies that (j ε 0 ) is bounded in

L 1 (R + ; Ḃ n 2 -1 2,1 + Ḃ n 2 +1
2,1 ). Hence, there exists a sequence (ε k ) k∈N going to 0 so that j ε k 0 ⇀ j 0 in the sense of distributions. Moreover, we have

j ε 0 -b ε = -L -1 n -1 div j ε 1 -εL -1 ∂ t j ε 0 .
Remembering (6.10) and (6.1), we see that the first term of the r.h.s. is O(ε) in a suitable space. The second one also tends to 0 in the sense of distributions as εL -1 → 0. Hence eq:diff6 eq:diff6 (6.13) j 0 = b.

In order to pass to the limit in the velocity equation, we proceed as in the non-equilibrium case. First we use (5.37) and next, the fact that

∂ t u ε + ε n k 4 (b ε ) j ε 1 = -u ε • ∇ u ε + k 2 (b ε )A u ε -k 3 (b ε )∇b ε + ε n k ′ 4 (b ε )∂ t b ε j ε 1 - 1 n k 4 (b ε ) ∇j ε 0 .
Taking advantage of (6.8) and of product laws in Besov spaces, we readily obtain that the four first terms of the r.h.s. are bounded in

L 2 (R + ; Ḃ n 2 -2 2,1 ) (or only in L 2 (R + ; Ḃ n 2 -2 2,∞ ) if n = 2).
To handle the last term, we observe that according to (6.8),

∇j ε 0 is bounded in L ∞ (R + ; Ḃ n 2 -2 2,1 ) hence the term k 4 (b ε )∇j ε 0 is bounded in L ∞ (R + ; Ḃ n 2 -2 2,1 ) (or L ∞ (R + ; Ḃ n 2 -2 2,∞ ) if n = 2).
As in the non-equilibrium case, it is now easy to conclude that there exists some v in

L ∞ (R + ; Ḃ n 2 -1 2,1
) so that for all φ in S and α ∈ (0, 1), we have

φ u ε k + ε k n j ε k 1 -→ φ v in L ∞ loc (R + ; Ḃ n 2 -1-α 2,1
).

Of course, combining with (6.10), this implies that v = u, and (b, u) thus satisfies the second line of (6.9).

Finally, that the whole family (b ε , u ε , j ε 0 , j ε 1 ) converges to (b, u, b, 0) stems from the fact that the solution to (6.9) is unique (note that it is just the standard barotropic Navier-Stokes equations with a modified but still stable pressure law).

The Poisson diffusion regime

This section is devoted to the study of the asymptotics regime where eq:poissonsas eq:poissonsas (7.1)

ε ≪ L ε 1/2 and L 2 L s ≈ 1.
According to the formal computations of Section 2, we expect the solutions of (1.6) to tend to those of the Navier-Stokes-Poisson sytem (2.9). The general scheme of the proof that we here propose is the same as in the study of the other asymptotics: we first perform a fine analysis of the linearized equations so as to check the long-time stability and exhibit the quantities that are likely to be bounded uniformly when ε → 0, then tackle the proof of the global existence. We rapidly justify that the limit system is globally well-posed in a functional framework that is consistent with the analysis we used for (1.6), and eventually take advantage of compactness arguments so as to prove the expected convergence result. As in the other regimes, the fact that the limit system has a unique solution will guarantee that the whole family of solutions to (1.6) converges to the solution to (2.9). 7.1. Linear analysis of (1.6) in the Poisson regime (7.1). We here gather the estimates for (4.7) that have been obtained in Section 4. Recall that L := νL. Small frequencies. Using (4.28), (4.33), (4.34) and the fact that |( b, d, j 0 , j 1 )| ≈ |( b, d, j 0 , j 1 )| and that the last term in the original definition of j 0 in (4.22) has a negligible contribution with respect to j 1 , we get eq:noneqPlf eq:noneqPlf (7.2) |( b, d, ε

L j 0 , ρ j 0 , j 1 )(t)| + ρ 2 t 0 |( b, d, j 0 , j 1 )| dτ + t 0 | j 0 | dτ + L ε t 0 |ρ ζ 0 | + LM ε t 0 | j 1 | dτ ≤ C|( b, d, ε L j 0 , ρ j 0 , j 1 )(0)| for all 0 ≤ ρ ≤ C 1 , with j 0 := j 0 - √ n b - √ n ε L ρ d, ζ 0 := j 0 - √ n 1+ ρ 2 n L 2 M b and j 1 := j 1 -ρ √ n LM j 0 + ρ b LLsM •
Middle frequencies. Combining (4.54) and the definition of ζ 1 versus that of j 1 , we get eq:noneqPmf eq:noneqPmf (7.3) |(ρ b, d, j 0 , j 1 )(t)| + ρ

t 0 | b| dτ + ρ 2 t 0 | d| dτ + ρ 2 L ε t 0 | j 0 | dτ + LM ε t 0 | j 1 | dτ |(ρ b, d, j 0 , j 1 )(0)| for C 1 ≤ ρ ≤ c LM.
Large frequencies. Finally, using (4.70), we have eq:noneqPhf eq:noneqPhf (7.4)

|(ρ b, d, j 0 , j 1 )(t)| + ρ 2 t 0 | d| dτ + ρ t 0 | b| dτ + LM ε t 0 |( j 0 , j 1 )| dτ ≤ C|(ρ b, d, j 0 , j 1 )(0)| for ρ ≥ c LM.
Therefore, localizing (4.4) (with nonzero source terms f and g ) according to Littlewood-Paley operator ∆k , using (4.5), following the computations leading to the above three inequalities and using Fourier-Plancherel theorem, we end up with the following inequality for all s ∈ R :noneqPestlin :noneqPestlin (7.5) ( u, j 1 )(t) Ḃs

2,1 + b(t) ℓ,1 Ḃs 2,1 + b(t) h,1 Ḃs+1 2,1 + ε L j 0 (t) ℓ,1 Ḃs 2,1 + j 0 (t) ℓ,1 Ḃs+1 2,1 + j 0 (t) h,1 Ḃs 2,1 + t 0 u Ḃs+2 2,1 dτ + t 0 (b, j 0 , j 1 ) ℓ,1 Ḃs+2 2,1 dτ + L ε t 0 ζ 0 ℓ,1 Ḃs+1
2,1 dτ + t 0 j 0 ℓ,1 Ḃs 2,1 dτ + LM ε t 0 j 1 ℓ, LM Ḃs 2,1 dτ + L ε t 0 j 0 m,1, LM Ḃs+2 2,1 + t 0 b h,1 Ḃs+1 2,1 dτ + LM ε t 0 (j 0 , j 1 ) h, LM Ḃs 2,1 dτ ( u, j 1 )(0) Ḃs 2,1 + b(0) ℓ,1 Ḃs 2,1 + b(0) h,1 Ḃs+1 2,1 + ε L j 0 (0) ℓ,1 Ḃs 2,1 + j 0 (0) ℓ,1 Ḃs+1 2,1 + j 0 (0) h,1 Ḃs 2,1 + t 0 f ℓ,1 Ḃs 2,1 + f h,1 Ḃs+1 2,1 + g Ḃs 2,1
dτ,

with j 0 := j 0 - √ n b - √ n ε L div u, ζ 0 := j 0 - √ n Id - 1 n L 2 M ∆ -1 b and j 1 := j 1 + 1 √ n LM ∇j 0 - 1 LL s M ∇b.
As in the previous sections, owing to the convection term in the equation for b, the above inequality does not allow to prove the global existence for (1.6), and one has to consider the paralinearized system (5.10). Adapting the proof of Proposition 5.1, we get the following -diff-poisson Proposition 7.1. If the coefficients L, L s and ε fulfill (7.1) then for any smooth solution (b, u, j 0 , j 1 ) to (5.10), one has the following inequality

(b, u, j 0 , j 1 )(t) Y ν ε (t) ≤ C (b, u, j 0 , j 1 )(0) Xν ε + t 0 ∇ v L ∞ (b, u, j 0 , j 1 ) Xν ε dτ + t 0 (∇F, G) h,1 Ḃ n 2 -1 2,1 dτ + t 0 (F -T v • ∇b, G -T v • ∇ u) ℓ,1 Ḃ n 2 -1 2,1 dτ , with (b, u, j 0 , j 1 ) Xν ε := ( u, j 1 ) Ḃ n 2 -1 2,1 + b ℓ,ν -1 Ḃ n 2 -1 2,1 +ν b h,ν -1 Ḃ n 2 2,1 + ε Lν j 0 ℓ,ν -1 Ḃ n 2 -1 2,1 +ν j 0 ℓ,ν -1 Ḃ n 2 2,1 + j 0 h,ν -1 Ḃ n 2 -1 2,1 , and
(b, u, j 0 , j 1 )(t) Y ν ε (t) := sup 0≤τ ≤t (b, u, j 0 , j 1 )(τ ) Xν ε +ν t 0 u Ḃ n 2 +1 2,1 + (b, j 1 ) ℓ,ν -1 Ḃ n 2 +1 2,1 + Lν ε j 0 m,ν -1 ,LM Ḃ n 2 +1 2,1 dτ + L ε t 0 ζ 0 ℓ,ν -1 Ḃ n 2 2,1 dτ +ν -1 t 0 j 0 ℓ,ν -1 Ḃ n 2 -1 2,1 dτ + LM ε t 0 j 1 ℓ,LM Ḃ n 2 -1 2,1 dτ + t 0 b h,1 Ḃ n 2 2,1 dτ + LM ε t 0 (j 0 , j 1 ) h,LM Ḃ n 2 -1 2,1
dτ.

Above, we set

j 0 := j 0 -b- ε L div u, ζ 0 := j 0 -Id- 1 nL 2 M ∆ -1 b and j 1 := j 1 + 1 LM ∇j 0 - 1 LL s M ∇b.
7.2. Uniform global well-posedness in the Poisson regime. In this paragraph, we sketch the proof of the following global existence result.

th:diff3 Theorem 7.1. There exists a positive constant η depending only on µ/ν, n and on the pressure law such that if ε ∈ (0, 1) and if the coefficients L and L s fulfill (7.1) then any data (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) satisfying q:small-diff3 q:small-diff3 (7.6)

(b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) Xν ε ≤ ην, generates a unique global solution (b ε , u ε , j ε 0 , j ε 1 ) in Y ν ε to System (1.6). Furthermore, we have eq:unif-diff3 eq:unif-diff3 (7.7) (b ε , u ε , j ε 0 , j ε 1 ) Y ν ε ≤ C (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) Xν ε . Proof.
Assuming with no loss of generality that ν = 1, the proof relies on Proposition 7.1 with, dropping the indices ε for better readability, v = u

F := -T ′ ∇b • u -k 1 (b)div u and G := -T ′ ∇ u • u + k 2 (b) A u -k 3 (b)∇b + LM n k 4 (b) j 1 .
Let us just explain how to handle the last term, as it cannot be bounded exactly as in the proof of Theorems 5.1 or 6.1 due to the difference between the spaces Y ν ε and Y ν ε . We use the fact that LM j 1 = LM j 1 + L -1 s ∇b -∇j 0 , and thus

LMk 4 (b) j 1 = LMk 4 (b)( j h,LM 1 + j ℓ,LM 1 ) + L -1 s k 4 (b)∇b ℓ,LM -k 4 (b)∇j ℓ,LM 0 . It is clear that k 4 (b) j h,LM 1 L 1 ( Ḃ n 2 -1 2,1 ) b L ∞ ( Ḃ n 2 2,1 ) j h,LM 1 L 1 ( Ḃ n 2 -1 2,1 ) (LM) -1 (b, u, j 0 , j 1 ) Y 1 ε ,
that the second term in the r.h.s. may be bounded in the same way, and that the third one can be bounded as the pressure term k 3 (b)∇b. For the last term, one just has to observe that the definition of • Y 1 ε guarantees that eq:poisson3 eq:poisson3 (7.8)

∇j 0 ℓ,LM L 1 ( Ḃ n 2 2,1 ) (b, u, j 0 , j 1 ) Y 1 ε , and that k 4 (b) L ∞ ( Ḃ n 2 2,1 ) b L ∞ ( Ḃ n 2 2,1 )
.

The rest of the proof is standard, and thus left to the reader.

7.3. Study of the limit system. We introduce the following norms

(b, u, j 0 ) X ν := b ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b h,ν -1 Ḃ n 2 2,1 + u Ḃ n 2 -1 2,1 + j 0 ℓ,ν -1 Ḃ n 2 -1 2,1 + ν 3 j 0 h,ν -1 Ḃ n 2 +2 2,1 , and (b, u
, j 0 ) Yν := sup t≥0 (b, u, j 0 )(t) X ν +ν R + (b, j 0 ) ℓ,ν -1 Ḃ n 2 +1 2,1 + u Ḃ n 2 +1 2,1 dt + R + b h,ν -1 Ḃ n 2 2,1 + ν 2 j 0 h,ν -1 Ḃ n 2 +2 2,1
dt.

th:poisson Theorem 7.2. Let the data (b 0 , u 0 , j 0,0 ) satisfy for a small enough constant c > 0 eq:poisson1 eq:poisson1 (7.9) b 0

ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b 0 h,ν -1 Ḃ n 2 2,1 + u 0 Ḃ n 2 -1 2,1 ≤ cν,
and the compatibility condition

j 0,0 - ν 2 nm ∆j 0,0 = b 0 .
Then System (2.9) admits a unique global solution (b, u, j 0 ) in the space Yν , satisfying in addition for a large enough constant C independent of ν eq:poisson2 eq:poisson2 (7.10)

(b, u, j 0 ) Yν ≤ C b 0 ℓ,ν -1 Ḃ n 2 -1 2,1 + ν b 0 h,ν -1 Ḃ n 2 2,1 + u 0 Ḃ n 2 -1 2,1 .
Proof. We just sketch the proof as it is very similar to the standard one for the barotropic Navier-Stokes equations. As usual, it suffices to treat the case ν = 1. The first step is to analyse the linearized system eq:poissonl eq:poissonl (7.11)

     ∂ t b + div u = f, ∂ t u -A u + ∇b + 1 n ∇j 0 = g, Id -1 nm ∆ j 0 = b.
To this end, we set d = (-∆) -1/2 div u and observe that in the Fourier space, ( b, d) fulfills the following ODE if f = g = 0

∂ t b + ρ d = 0, ∂ t d + ρ 2 d -ρa ρ b = 0
with ρ := |ξ| and a ρ := 1 + m ρ 2 +nm • Of course, j 0 may be computed from b by the relation

j 0 = nm ρ 2 + nm b.
Introducing the following Lyapunov and diffusion functionals

L 2 ρ = 2a ρ | b| 2 + 2| d| 2 + |ρ b| 2 -2Re (ρ b d) and H 2 ρ = ρ 2 (a ρ | b| 2 + | d| 2 ), we see that 1 2 d dt L 2 ρ + H 2 ρ = 0.
Because we have a ±1 ρ ≤ c 0 for some c 0 independent of ρ, one can thus conclude exactly as in the standard barotropic case that for all t ≥ 0 and ρ ≥ 0

|( b, ρ b, d)(t)| + min(1, ρ) t 0 |ρ b| dτ + ρ 2 t 0 | d| dτ |( b, ρ b, d)(0)|.
Back to (7.11), one may combine Fourier-Plancherel theorem and Duhamel formula to get the following estimate for all s ∈ R

(b, ∇b, u)(t) Ḃs 2,1 + j 0 (t) Ḃs 2,1 ∩ Ḃs+3 2,1 + t 0 u Ḃs+2 2,1 + (b, j 0 ) ℓ,1 Ḃs+2 2,1 dτ + t 0 b h,1 Ḃs+1 2,1 + j 0 h,1 Ḃs+3 2,1 dτ ≤ C (b 0 , ∇b 0 , u 0 ) Ḃs 2,1 + t 0 (f, ∇f, g) Ḃs 2,1 dτ .
However, because of the convection term in the equation for b, this does not allow to prove estimates for the nonlinear system (2.9). Therefore, mimicking the standard approach for the compressible Navier-Stokes equation we 'paralinearize' the system and get the following result Proposition 7.2. The solutions to the following paralinearized system

     ∂ t b + T v • ∇ u + div u = f, ∂ t u + T v • ∇ u -A u + ∇b + 1 n ∇j 0 = g, Id -1 nm ∆ j 0 = b. fulfill the following a priori estimate (b, ∇b, u)(t) Ḃs 2,1 + j 0 (t) Ḃs 2,1 ∩ Ḃs+3 2,1 + t 0 u Ḃs+2 2,1 + (b, j 0 ) ℓ,1 Ḃs+2 2,1 dτ + t 0 b h,1 Ḃs+1 2,1 + j 0 h,1 Ḃs+3 2,1 dτ ≤ C (b 0 , ∇b 0 , u 0 ) Ḃs 2,1 + t 0 (f, ∇f, g) Ḃs 2,1 dτ + t 0 ∇ v L ∞ (b, ∇b, u) Ḃs 2,1
dτ . Now, taking the divergence of the equation of j ε 1 , then using the equation of j ε 0 gives eq:poisson5 eq:poisson5 (7.16) ∆j ε 0 = -L(1 + L s ) nL(b ε -j ε 0 ) -εn∂ t j ε 0 -ε∂ t div j ε 1 . Given (7.13), one can assert that the last term tends to 0 in the sense of distributions. We also know that, up to an omitted extraction, j ε 0 → j 0 in the sense of distributions, hence given that L(1 + L s )ε → 0, the term with ∂ t j ε 0 also tends to 0. Finally, exactly as in the cases treated before, (b

ε ) is bounded in L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2 2,1 ) hence weakly converges to some b ∈ L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2
2,1 ). As (7.12) has been assumed, passing to the limit in (7.16) gives ν 2 ∆j 0 = -nm b -j 0 ). Passing to the limit in the equation of b goes along the lines of the non-equilibrium case we notice that (

∂ t b ε ) is bounded in L 2 (R + ; Ḃ n 2 -1 2,1
) and we thus have, up to an omitted extraction eq:poisson6 eq:poisson6 (7.17)

φ b ε -→ φ b in L ∞ (R + ; Ḃ n 2 -α 2,1
) for all α ∈ (0, 1).

As (7.7) also implies that (

u ε ) is bounded in L ∞ (R + ; Ḃ n 2 -1 2,1 )∩L 1 (R + ; Ḃ n 2 +1
2,1 ), we have u ε ⇀ u weakly * in that space, which is enough to justify the first equation of (2.9).

In order to pass to the limit in the velocity equation, we use again the fact that

∂ t u ε + ε n k 4 (b ε ) j ε 1 = -u ε • ∇ u ε + k 2 (b ε )A u ε -k 3 (b ε )∇b ε + ε n k ′ 4 (b ε )∂ t b ε j ε 1 - 1 n k 4 (b ε ) ∇j ε 0 .
As in the other asymptotic regimes, the first four terms of the r.h.s. are bounded in

L 2 (R + ; Ḃ n 2 -2 2,1 ) (or in L 2 (R + ; Ḃ n 2 -2 2,∞ ) if n = 2).
To handle the last term, we observe that according to (7.7) and (7.15), (∇j

ε 0 ) is bounded in L 2 (R + ; Ḃ n 2 -2 2,1 + Ḃ n 2 -1 2,1 ). Because (b ε ) is bounded in L ∞ (R + ; Ḃ n 2 -1 2,1 ∩ Ḃ n 2 2,1 ), this implies that k 4 (b ε )∇j ε 0 is bounded in L 2 (R + ; Ḃ n 2 -2 2,1 ) (or L 2 (R + ; Ḃ n 2 -2
2,∞ ) if n = 2), and thus ∂ t u ε + ε n k 4 (b ε ) j ε 1 is bounded in the same space.

As in the already studied cases, we conclude that there exists some u in L ∞ (R + ; Ḃ n 2 -1 2,1 ) so that for all φ in S and α ∈ (0, 1), we have

φ u ε + ε n j ε 1 -→ φ u in L ∞ loc (R + ; Ḃ n 2 -1-α 2,1
).

Finally, in the case where (7.14) is fulfilled, the limit system (2.9) supplemented with initial data (b 0 , u 0 , j 0,0 ) possesses a unique solution (b, u, j 0 ) given by Theorem 7.2, and the whole family (b ε , u ε , j ε 0 ) thus converges to (b, u, j 0 ).

Appendix A. Estimates for a toy linear differential equation s:A

The appendix is devoted to the proof of decay estimates for the solutions to systems of ODEs of the form (E)

∂ t U + A 0 U + ρ (A 1 + B 1 ) U + ρ 2 A 2 U = 0,
where ρ is a nonnegative parameter, and A 0 , A 1 , B 1 and A 2 are given N × N matrices.

We have in mind System (4.7) in which case, after suitable change of unknowns (see (A.4)), A 0 is a degenerate nonnegative diagonal matrix, A 2 has nonnegative eigenvalues and A 1 is skewsymmetric up to some positive diagonal symmetrizer.

A.1. A general approach. The basic idea is to set V := (I + ρP )U where P is a suitable matrix, so as to eliminate the bad first order term ρB 1 U. Now, whenever (I+ρP ) is invertible, the equation for V reads ∂ t V + A 0 V + ρ A 1 + B 1 + [P, A 0 ] V + ρ 2 [A 0 , P ]P + [P, A 1 ] + [P, B 1 ] + A 2 V +ρ 3 (I + ρP ) (A 1 + B 1 )P 2 -A 0 P 3 -A 2 P (I + ρP ) -1 V = 0.

Therefore, if one can find some matrix P so that eq:Pcom eq:Pcom (A.1) [A 0 , P ] = B 1 , then we have eq:V eq:V (A.2)

∂ t V + A 0 V + ρA 1 V + ρ 2 (A 2 + P B 1 + [P, A 1 ]) V = ρ 3 (I + ρP ) A 3 (I + ρP ) -1 V,
where A 3 := (P A 0 -A 1 )P 2 + A 2 P.

The gain is clear as the matrix B 1 now appears at order 2 instead of order 1. Hence the system for V is more likely to be tractable for small enough ρ as we shall see below.

A.2. Application to the linearized system for barotropic radiative flows. The system we are interested in reads eq:class eq:class (A.3)

d dt     a d j 0 j 1     +     0 ρ 0 0 -ρ ρ 2 0 -ς -η 0 β αρ 0 0 -αρ γ         a d j 0 j 1     =     0 0 0 0     ,
where all the coefficients of the matrix are positive.

To bound the solutions of (A.3) for small enough ρ (under some stability condition that we will discover below), we propose two different approaches, the first one being appropriate to handle the case where β and γ are of the same order of magnitude, and the second one, to the case where β/γ ≪ 1 or γ/β ≪ 1 (of course only γ ≥ β is relevant as far as (4.7) is concerned). and is thus of order 1 whenever ρ satisfies the smallness condition eq:smallrho1 eq:smallrho1 (A.8) ρ 2 βγ αςη min(β 2 , γ 2 ).

In order to go further in the estimates of V, we compute

P B 1 = -B 1 1 ∆ -1 B 2 1 0 0 ∆ -1 B 2 1 B 1 1 =      -αςη βγ 2 0 0 0 0 -αςη β 2 γ 0 0 0 0 αςη β 2 γ 0 0 0 0 αςη βγ 2      and [P, A 1 ] = 0 -B 1 1 ∆ -1 A 2 1 + A 1 1 B 1 1 ∆ -1 ∆ -1 B 2 1 A 1 1 -A 2 1 ∆ -1 B 2 1 0 =      0 0 -ας γ ( 1 β + 1 γ ) 0 0 0 0 α ας βγ + ς γ 2 (1+ αςη βγ ) α αη βγ + η β 2 (1+ αςη βγ ) 0 0 0 0 -αη β ( 1 β + 1 γ ) 0 0      •
The modified radiative modes j 0 and j 1 fulfill eq:radialf eq:radialf (A.15) d dt

j 0 j 1 + ρ 0 α -α 0 j 0 j 1 + β + αςη β 2 γ ρ 2 0 0 γ + αςη βγ 2 ρ 2 j 0 j 1 = ρ 2 -α αη βγ -η β 2 (1 + αςη βγ ) 0 0 αη β ( 1 β + 1 γ ) b d + O(ρ 3 ).
Therefore we easily get 1 2

d dt | j 0 | 2 + α α | j 1 | 2 + β + αςη β 2 γ ρ 2 | j 0 | 2 + α α γ + αςη βγ 2 ρ 2 | j 1 | 2 ≤ C ρ 2 |( b, d)| + ρ 3 |( b, d, j 0 , j 1 )| .
Then, integrating and assuming that ρ ≪ 1 yields eq:J eq:J (A. Combining with (A.14), we can conclude that there exists some positive constants ρ 0 and C depending only on (α, β, γ, ς, η) so that for all eq:condlf eq:condlf (A.17 In this paragraph, we propose another approach to handle (A.3) in the case β = γ, still based on rewriting the system in the form (A.2), but with a different definition of A 1 and B 1 (A 0 and A 2 being unchanged). More precisely, we now set it is clear that working with ( a, d, j 0 , j 1 ) or ( b, d, j 0 , j 1 ) is equivalent whenever

ρ ≤ C|γ -β|,
for some positive constant C depending continuously on the coefficients of the system.

Putting together the previous computations, we see that V fulfills Next, arguing exactly as to handle (A.10), we discover that under the stability condition (A.11) and for ρ satisfying (A.13) (and of course also ρ ≤ c|β -γ| 3 ), we have

2 )

 2 The compressible Navier-Stokes equation with an extra pressure term see (2.10) (equilibrium diffusion regime corresponding to ε ≪ L and L 2 L s → +∞, or L → +∞); (3) The Navier-Stokes-Poisson equations (2.9) (or (2.11)) (case ε ≪ L 1 and ν 2 L 2 L s → m ∈ (0, +∞)); (4) The compressible Navier-Stokes equations coupled with a parabolic equation (2.5) (nonequilibrium diffusion regime L ≈ ε and ν 2 L s L 2 → m ∈ (0, +∞)); (5) The compressible Navier-Stokes equations coupled with a damped equation (2.6) (degenerate nonequilibrium diffusion regime L ≈ ε and L s L 2 → +∞).

  we thus have |A 3 | ε L • Hence, up to a O(ερ 3 /L) term, the equations for ( b, d) read d dt b

  (4.71) |(ρ b, d, √ εL j 0 , √ εL j 1 )(t)| + ρ 2 0 , j 1 )| dτ |(ρ b, d, √ εL j 0 , √ εL j 1 )(0)|.

2 t 0 | 0 | b| dτ + M t 0 | j 1

 20001 1 then using (4.54), and the definition of ζ 1 versus that of j 1 , we get eq:noneqmf eq:noneqmf (5.3) |(ρ b, d, j 0 , j 1 )(t)| + ρ ( d, j 0 )| dτ + ρ t | dτ ≤ C|(ρ b, d, j 0 , j 1 )(0)| for all C 1 ≤ ρ ≤ c LM.

A. 2 . 1 . 1 .

 211 First approach. Making the change of unknown eq:U eq:U (A.4) := α + ςη βγ , we see that U satisfies a system of type (E) with A Note that the above matrices may be written in block form as follows P 11 P 12 P 21 P 22 • Computing the commutator eq:A0P eq:A0P (A.5) [A 0 , P ] = 0 -P 12 ∆ ∆P 21 [∆, P 22 ] ,we see (A.1) is satisfied ifP 11 := 0, P 22 := 0, P 12 := -B 1 1 ∆ -1 , P 21 := ∆ -1 B 2In other words eq:P eq:P (A.6) (A.4), corresponds to the following change of unknowns eq:Vc eq:Vc (A.7)

  16) (|( j 0 , j 1 )(t)| + min(β, γ) t 0 |( j 0 , j 1 )| dτ ≤ C |( j 0 , j 1 )(0)| + ρ 2 t 0 |( b, d)| dτ •

0 | 0 |A. 2 . 2 .

 0022 ) 0 ≤ ρ ≤ min(1, ν) ρ 0 ,we have eq:lf eq:lf(A.18) |( b, d)(t)| + ν|( j 0 , j 1 )(t)| + νρ 2 t ( b, d)| dτ + ν min(β, γ) t ( j 0 , j 1 )| dτ ≤ C |( b, d)(0)| + ν|( j 0 , j 1 )(0)| . Second approach.In the case where β and γ are not of the same order of magnitude, Inequality (A.18) is not fully satisfactory, first because we would like to have a control on β t 0 | j 0 | dτ and γ t 0 | j 1 | dτ rather than just on min(β, γ) t 0 |( j 0 , j 1 )| dτ and, second, because the range for which (A.18) holds true tends to shrink to 0 if β ≪ γ or γ ≪ β.

1 ,

 1 Then writing the matrices coming into play in block form, we see according to (A.5), that a possible choice for P isP 11 := 0, P 12 := -B 1 1 ∆ -1 , P 21 := ∆ -1 B 2

+ρ 3 (

 3 I + ρP )A 3 (I + ρP ) -1 V, with A 3 := (P A 0 -A 1 )P 2 + A 2 P satisfying |A 3 | ≤ C 1 + |γ -β| -3 .

0 | 2 t 0 | 0 |c|A| 2 +

 02002 eq:radialf0 eq:radialf0 (A.20)|( b, d)(t)| + νρ 2 t ( b, d)| dτ ≤ C |( b, d)(0)| + ρ ( j 0 , j 1 )| dτ + ρ 3 1 + 1 |γ -β| 3 t ( b, d, j 0 , j 1 )| dτ • which leads if A ≡ B ≡0 to eq:ODE4 eq:ODE4 (B.5) eq:ODE5 (B.6) L ρ (t) ≤ e -( d-b 6 )ρ 2 t L ρ (0). Combining with (B.4) and Duhamel's formula, we deduce that for general source terms A and B we have eq:ODE7 eq:ODE7 (B.7) c|X(t)| 2 + a|Y (t)| 2 ≤ √ 3 e -( d-b 6 )ρ 2 t c|X(0)| 2 + a|Y (0)| 2 a|B| 2 dτ •

  Let us boundA 3 in order to determinate for which values of ρ the last term in (4.24) is indeed negligible. Just writing that |A 3 | ≤ |P | 2 (|P | |A 0 | + |A 1 |) + |A 2 | |P | using the explicit values of A 0 , A 1 and A 2 and

Note that the last term of j0 given by (4.22) is negligible for ρ 1 and may thus be omitted.

Now, in order to estimate the solutions of the nonlinear system (2.9), it suffices to apply the above proposition with v = u f = -T ′ u • ∇b -k 1 (b)div u and g = -T ′ u • ∇ u + k 2 (b) A u -k 3 (b)∇b -n -1 k 4 (b)∇j 0 . All the terms but the last one of g are already present in the barotropic Navier-Stokes equations, and may be bounded quadratically in terms of (b, u, j 0 ) Y1 . Now, we have

, and one can thus conclude that whenever the solution (b, u, j 0 ) exists we have

which allows to get (7.10) if (7.9) is fulfilled with a small enough c.

Weak convergence.

Here we justify weak convergence to (2.9) when assumption (7.1) is fulfilled and, in addition q:poissonsas1 q:poissonsas1 (7.12)

th:diff-lim3 Theorem 7.3. Let the family of data (b ε 0 , u ε 0 , j ε 0,0 , j ε 1,0 ) 0<ε<1 satisfy (7.6). Then the global solution (b ε , u ε , j ε 0 , j ε 1 ) in Y ν ε given by Theorem 7.1 satisfies eq:poisson4 eq:poisson4 (7.13)

), and, up to extraction, (b ε , u ε , j ε 0 ) converges weakly to some solution (b, u, j 0 ) in Yν of System (2.9) when ε goes to 0.

If in addition

eq:poisson7 eq:poisson7 (7.14) (b ε 0 , u ε 0 , j ε 0,0 ) ⇀ (b 0 , u 0 , j 0,0 ) with -ν 2 ∆j 0,0 + nm(j 0,0 -b 0 ) = 0, then the whole family (b ε , u ε , j ε 0 ) converges to the unique solution (b, u, j 0 ) corresponding to the initial data (b 0 , u 0 , j 0,0 ), given by Theorem 7.2.

Proof: Let us first prove (7.13). From (7.7), we already know that ( j ε 1 ) ℓ,LM and (

). Now, we have

It is easy to see that the last term is

2,1 ), and that, according to (7.8) and

), which completes the proof of (7.13).

Next, let us turn our attention to the convergence of j ε 0 . First, (7.7) and the definition of

). Next, using the bound for the middle frequencies of j 0 and for the low frequencies of ζ 0 , we discover that (

). Hence, up to an omitted extraction eq:poisson4a eq:poisson4a (7.15)

Finally, A 3 := (P A 0 -A 1 )P 2 + A 2 P reads eq:A_3 eq:A_3 (A.9)

Therefore, resuming to (A.2), we conclude that

Of course, the remainder term O(ρ 3 ) strongly depends on the coefficients of the system. We shall see below that the structure of [A 1 , P ] will enable us to treat ρ 2 [A 1 , P ] and the nondiagonal term of A 2 as small error terms as well.

Let us focus on the system satisfied by ( b, d) for a while. We have eq:hydrolf eq:hydrolf (A.10)

For small enough ρ, optimal estimates may be proved by taking advantage of the results of Appendix B. Indeed, denoting by F ρ the r.h.s. of (A.10), we see from (B.6) that if we set

then, under the following necessary and sufficient stability condition eq:stabcond1 eq:stabcond1 (A.11) ν := 1 -αςη βγ

we have (see (B.4) and (B.5))

whenever eq:hydrolf1a eq:hydrolf1a (A.13) ρ ≤

So finally, we get for some appropriate constant C = C(α, β, γ, ς, η)

which, if ρ ≪ ν, may be simplified into eq:hydrolf2 eq:hydrolf2 (A.14)

Now, in contrast with the first method, we can bound j 0 and j 1 independently from one another from the equation satisfied by j 0 , we readily get eq:radialf1 eq:radialf1 (A.21)

Putting inequalities (A.20), (A.21) and (A.22) together, it is now easy to conclude that eq:lf2 eq:lf2

if, for some small enough constant c depending continuously on α, β, γ and ς, we have

Optimal decay estimates for a toy system s:B

For the reader convenience, we here recall some results that have been obtained in our recent work [8] for the following linear system of ordinary differential equations eq:ODE eq:ODE (B.1)