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The low match number limit for a barotropic model of
radiative flow
Raphaël Danchin, Bernard Ducomet
To cite this version:
Raphaël Danchin, Bernard Ducomet. The low match number limit for a barotropic model of radiative flow. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2016, 48 (2), pp.1025-1053. �hal-01126759�
RADIATIVE FLOW
RAPHA ¨EL DANCHIN∗, BERNARD DUCOMET∗∗
∗Universit´e Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, Institut Universitaire de France,
61 avenue du G´en´eral de Gaulle, 94010 Cr´eteil Cedex 10
∗∗CEA, DAM, DIF, D´epartement de Physique Th´eorique et Appliqu´ee
F-91297 Arpajon, France
Abstract. We aim at justifying rigorously the low Mach number asymptotics for a model of compressible fluid coupled to the radiation. For simplicity, we restrict ourselves to the barotropic situation, and adopt the so-called P 1 -approximation to model the effects of radiation. We focus on small perturbations of stable constant equilibria. In the critical regularity framework, we establish estimates independent of the Mach number, and convergence results to the incompressible Navier-Stokes equation, exactly as in the non radiative case. Our results hold true in the whole space Rn as well as in a periodic box Tn with n ≥ 2.
Keywords: Radiation hydrodynamics, Navier-Stokes system, low Mach number, criti-cal regularity, P 1 -approximation.
1. Introduction
We consider the barotropic version of a model of radiation hydrodynamics. Our main goal is to provide the rigorous justification of the Low Mach number limit that has been recently investigated formally and numerically by Sea¨ıd et al. [18] [9] [17] in order to simulate fire propagation models in open vehicle tunnels.
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 that belongs to the set Ω which is either the whole space Rn or some periodic box Tn with n ≥ 2. Denoting by ~SF the radiative momentum source acting on the fluid, we thus
have:
(1.1) ∂t% + divx(%~u) = 0 in (0, T ) × Ω,
(1.2) ∂t(%~u)+divx(%~u⊗~u)+∇xp = divx µ(∇xu+~ t∇x~u) +∇x(λdivx~u)− ~SF in (0, T )×Ω,
where p stands for the pressure, which is given by p = P (%) (barotropic assumption) for some smooth enough function P. The viscosity coefficients µ and λ are smooth functions of % satisfying
µ > 0 and λ + 2µ > 0.
To model radiative effects, we follow the approach of [4]: we introduce a global distribution function, the radiative intensity I = I(t, x, ~ω, ν) , depending on the direction vector ~ω ∈ Sn−1, where Sn−1 denotes the unit sphere of Rn, and on the frequency ν ≥ 0 . The action
of radiation is then expressed in terms of integral means (with respect to the variables
~
ω and ν ) of quantities depending on I . The radiative intensity I evolves through the following radiative transfer equation:
(1.3) 1
c ∂tI + ~ω · ∇xI = S in (0, T ) × Ω × S
n−1× (0, ∞),
where c is the speed of light.
The radiative source S := Sa+ Ss is the sum of an emission-absorption term:
Sa,e:= σa(B(ν, %) − I)
and of a scattering contribution: Ss:= σs ˜I − I where I :=˜ 1 |Sn−1| Z Sn−1 I d~ω.
The transport coefficients σa(%, ~ω, ν) and σs(%, ~ω, ν) are (given) nonnegative functions.
The distribution function B(ν, %) which appears in Sa,e, measuring the discrepancy from
equilibrium, is a barotropic equivalent of the Planck’s function, depending smoothly on %. Finally, the radiative momentum source ~SF in the r.h.s. of (1.2) is given by
~ SF = 1 c Z ∞ 0 Z Sn−1 ~ ωS d~ωdν.
System (1.1), (1.2), (1.3) can be viewed as a simplified model in radiation hydrodynamics [15], [16]. More realistic systems (as regards astrophysics and asymptotic regimes) have been proposed by Lowrie, Morel and Hittinger [14] and revisited recently by Buet and Despr´es [3].
In what follows, we assume that σa and σs do not depend on ~ω (isotropy), and make the
so-called “gray hypothesis”, that is, σa and σs do not depend on the frequency ν. After
integrating with respect to ν, and considering the ‘integrated’ quantities (still keeping the same notation), this enables us to omit the dependency with respect to ν in (1.1), (1.2), (1.3). Our second simplification is that we consider the so-called “P1-Approximation” of (1.3), that is, we postulate the following decomposition of I ≡R0∞I dν :
(1.4) I = I0+ ~ω~I1,
where I0 and ~I1 do not depend on ~ω and ν anymore.
This simplification amounts to assume a slight departure of radiation from isotropy 1. For more complete informations about P 1 moments method and closure relations, see [3] [2].
Plugging (1.4) into (1.3), taking the first two moments with respect to ~ω and integrating on Sn−1× R+ yields: (1.5) 1 c ∂tI0+ 1 n divxI~1 = σa(%)(B(%) − I0) in (0, T ) × Ω, (1.6) 1 c ∂tI~1+ ∇xI0 = (σa(%) + σs(%))~I1 in (0, T ) × Ω, while the radiative source ~SF rewrites
(1.7) S~F = σa(ρ) + σs(ρ) n ~ I1.
1One observes that a P0 approximation, retaining only the isotropic part I
0, would produce a complete
In order to identify the appropriate limit regime we perform a general scaling, denoting by ¯L, ¯T , ¯U , ¯%, ¯p, the reference hydrodynamical quantities (length, time, velocity, density, pressure) and by ¯I, ¯σa, ¯σs, B, the reference radiative quantities (radiative intensity,¯
absorption and scattering coefficients and equilibrium function).
Let Sr := ¯L/ ¯T ¯U , M a := ¯U /√%¯¯p and Re := ¯U ¯% ¯L/¯µ be the Strouhal, Mach, Reynolds (dimensionless) numbers corresponding to hydrodynamics. Let also define C := c/ ¯U , L :=
¯
L¯σa, Ls:= ¯σs/¯σa, various dimensionless numbers corresponding to radiation.
Denoting by ˆt and ˆx the renormalized time and space variables, setting σa= ¯σaσˆa, and
so on, we perform the change of unknowns:
(%, ~u, j0,~j1)(t, x) = ¯% ˆ%, ¯U ˆ~u, ¯I ˆj0, ¯I~jˆ1( ¯T ˆt, ¯Lˆx).
Choosing ¯B = ¯I, omitting the carets and the dependence with respect to x in the differen-tial operators ∇ and div leads to the following scaled continuity and momentum equations (keeping in mind (1.7)): (1.8) Sr ∂t% + div (%~u) = 0, (1.9) Sr ∂t(%~u) + div (%~u ⊗ ~u) + 1 M a2 ∇p(%) − 1 Re div µ(∇~u +
t∇~u) + ∇(λdiv ~u)
= L σa(%) + Lsσs(%) n
~ I1,
while the rescaled radiative unknowns I0 and ~I1 satisfy:
(1.10) Sr C ∂tI0+ 1 n div~I1 = Lσa(%) (B(%) − I0) , (1.11) Sr C ∂tI~1+ ∇I0 = − (Lσa(%) + LLsσs(%)) ~I1.
In all that follows, we suppose that a moderate amount of radiation is present ( L = O(1) ) in our strongly under-relativistic flow ( C−1 = o(1) ) and assume that ¯σa and ¯σs are
comparable (i.e. Ls≈ 1). To clarify the presentation, we shall focus on the case where
M a = ε, Sr = Re = 1, C = ε−1C,e L = 1 and Ls= 1,
where ε is a small positive number and2 C is bounded from below when ε → 0. Thereforee the rescaled unknowns (%ε, ~uε, I0ε, ~I1ε) satisfy
(1.12) ∂t%ε+ div (%ε~uε) = 0, (1.13) ∂t(%ε~uε)+div (%ε~uε⊗~uε)+ ∇pε ε2 = div µ ε(∇~uε+t∇~uε)+∇(λεdiv ~uε)+ σεa+σsε n ~ I1ε, (1.14) ε e C∂tI ε 0 + 1 n div ~I ε 1 = σεa(Bε− I0ε) , (1.15) ε e C∂t ~ I1ε+ ∇I0ε= − (σaε+ σsε) ~I1ε, where we denoted pε= P (%ε), µε= µ(%ε), σεa= σa(%ε) and so on.
In order to compute the limit system, we consider the formal expansions (1.16) I0ε= I00+ ε e CI 1 0 + O(ε2), ~ I1ε= ~I10+ ε e C ~ I11+ O(ε2), %ε= %0+ ε%1+ O(ε2), ~uε= ~u0+ ε~u1+ O(ε2).
Identifying the low order terms, we discover that %0 must be constant. Denoting by ¯%
that constant, we thus get
(1.17) div ~u0 = 0, (1.18) %∂¯ t~u0+ ¯%div (~u0⊗ ~u0) + ∇Π = µ( ¯%)∆~u0, (1.19) ∇I00= −[σa( ¯%) + σs( ¯%)]~I10, (1.20) 1 n div ~I 0 1 = σa( ¯%)[B( ¯%) − I00],
which is an incompressible Navier-Stokes system decoupled from two stationary transport equations.
We also get two equations for the radiative correctors, namely
(1.21) 1 e C∂tI 0 1 + ∇I01= − 1 e C[σa( ¯%) + σs( ¯%)]~I 1 1 − [∂%σa( ¯%) + ∂%σs( ¯%)]%1I~10, (1.22) 1 e C∂tI 0 0 + 1 n eC div ~I 1 1 = %1∂%σa( ¯%)[B( ¯%) − I00] + σa( ¯%)[%1∂%B( ¯%) − eC−1I01].
The rest of the paper is devoted to proving rigorously the convergence to the incompress-ible Navier-Stokes equations. In the next section, we reformulate the low Mach number problem, introduce the functional framework we shall work in and state our main result: global well-posedness for small perturbations of a linearly stable constant reference state (%ε, ~uε, Iε
0, ~I1ε) = ( ¯%, ~0, B( ¯%), ~0), and rigorous derivation of the above asymptotics. The
proof strongly relies on a fine analysis of the linearized equations about ( ¯%, ~0, B( ¯%), ~0), that is performed in Section 3. Next we come to the proof of the global existence result (Section 4) and, finally, to the study of the convergence when the Mach number goes to 0 (Section 5). Some useful estimates pertaining to a toy 2 × 2 linear system of ODE may be found in Appendix.
2. Results
To simplify the presentation, we shall assume from now on that the viscosity and radi-ation coefficients are independent of % (for the nonconstant case see Remark 4.1 below). We shall focus on perturbations of some constant reference state
(%ε, ~uε, I0ε, ~I1ε) = ( ¯%, ~0, B( ¯%), ~0) with P0( ¯%) > 0 and B0( ¯%) > 0.
It is thus natural to introduce the new unknowns jε0 := I0ε− B(¯%) and ~j1ε:= ~I1ε. Further-more, as we expect to have %ε= ¯% + O(ε), and as we prefer to work with linear equations for jε0 and ~j1ε, we define
bε:= 1 e ε B(%ε) − B( ¯%) ¯ %B0( ¯%) with ε :=e ε pP0( ¯%)·
Of course, as B0( ¯%) > 0, the functions depending on %ε may be expressed in terms of bε
for small enough perturbations of ¯%. The governing equations for (bε, ~uε, jε
0,~j1ε) thus read (2.1) ∂tbε+ ~uε· ∇bε+ div ~uε e ε = k1(εbe ε)div ~uε, ∂t~uε+ ~uε· ∇~uε− A~uε ¯ % + ∇bε e ε − σa+ σs n ¯% ~jε 1 = k2(εbe ε)A~uε ¯ % +k3(eεb ε) e ε ∇b ε+ σa+ σs n ¯% k4(εbe ε)~jε 1, ε e C ∂tj ε 0+n1 div~j ε 1 = σa( ¯%B0( ¯%)eεb ε− jε 0) , ε e C ∂t~j ε 1+ ∇j0ε= − (σa+ σs)~j1ε,
with A := µ∆ + (λ + µ)∇div and where k1, k2, k3 and k4 are smooth functions vanishing
at 0.
Before stating our main result, let us specify the functional framework we shall work in. Roughly, we shall adopt the same critical regularity framework as in our first paper [7]. However, we will have to set norms depending on the parameters ε and eC in order to get optimal estimates, enabling us to study the low Mach number asymptotics.
Let us first very briefly recall the definition of homogeneous Besov spaces B˙2,1s (the reader is referred to [1], Chap. 2 for more details). For simplicity, we focus on the Rn case, the adaptation to the periodic setting being quite strandard. Fix some smooth radial bump function χ : Rn→ [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
˙
∆ju := ϕ(2−jD)u = F−1(ϕ(2−jD)F u), j ∈ Z
where we denote by F the standard Fourier transform in Rn.
For any s ∈ R, the homogeneous Besov space ˙Bs2,1 is the set of tempered distributions u so that kukB˙s 2,1 := X j∈Z 2jsk ˙∆jkL2 < ∞ and (2.2) lim λ→+∞χ(λD)u = 0 in L ∞.
As pointed out in [7], scaling considerations (that neglect low order terms of System (2.1)) suggest that critical regularity is B˙
n 2−1
2,1 for ~u0, j0,0 and ~j1,0, and B˙
n 2
2,1 for b0.
However, being ‘out of scaling’ the lower order terms may hinder the proof of global-in-time estimates. To overcome this, one has to make additional assumptions for the low frequencies of some unknowns. This motivates our introducing norms where the specific behavior of low frequencies is taken into account. More precisely, for any distribution u satisfying (2.2) and any positive parameter η, we set:
kuk`,η˙ Bs 2,1 := X 2k≤2η 2ksk ˙∆kukL2 and kukh,η˙ Bs 2,1 := X 2k≥η/2 2ksk ˙∆kukL2, and also u`,η := X 2k≤η ˙ ∆ku and uh,η := X 2k>η ˙ ∆ku.
Note that ku`,ηkB˙s 2,1 ≤ Ckuk `,η ˙ Bs 2,1 and kuh,ηkB˙s 2,1 ≤ Ckuk h,η ˙ Bs 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.
From our investigation of the linearized equations in the next section, we shall find out that the ‘natural’ threshold between low and high frequencies for the radiative unknowns (j0,~j1) is at
(2.3) ρeε := ρ0
q
ν eC/ε with ν := λ + 2µ,
where ρ0 depends only on n, ¯%, P0( ¯%), B0( ¯%), σa and σs, and that the threshold between
low and high frequencies for b is at 1/(εν), exactly as in the nonradiative case (see [5]). One can now state our main result.
Theorem 2.1. Let ν0> 0 and ρeε be given by (2.3). Assume that
(2.4) 1 . eC . νε−1.
There exist three constants ε0, c and C depending only on ¯%, P, B, σa, σs, n, λ/µ and
ν0 such that if 0 < ε ≤ ε0, 0 < ν ≤ ν0 and the data satisfy
(2.5) I0ε:= k~uε0k ˙ B n 2−1 2,1 + eC−12 νk(jε 0,0,~jε1,0)k `,ρeε ˙ B n 2−1 2,1 + k(j0,0ε ,~j1,0ε )kh,ρeε ˙ B n 2−1 2,1 + kbε0k`, 1 εν ˙ B n 2−1 2,1 + ενkbε0kh, 1 εν ˙ B n 2 2,1 ≤ cν, then System (2.1) has a unique global solution (bε, ~uε, j0ε,~j1ε) with:
— bε∈ C(R+; ˙B n 2−1 2,1 ∩ ˙B n 2 2,1), (bε)`,1/(εν)∈ L1(R+; ˙B n 2+1 2,1 ), (bε)h,1/(εν) ∈ L1(R+; ˙B n 2 2,1) ; — ~uε∈ C(R+; ˙B n 2−1 2,1 ) ∩ L1(R+; ˙B n 2+1 2,1 ) ; — j0ε and ~j1ε in C(R+; ˙B n 2−1
2,1 ) and, besides (jε0)`,ρeε and (~jε
1)h,ρeε in L1(R+; ˙B
n 2−1
2,1 ) with
(2.6) jε0 := j0ε− c1εbε− c2Ce−1εdiv ~uε− c3Ce−2ε2div~j1ε and ~jε1:= ~j1ε− c4ε∇bε, where the coefficients c1, c2, c3 c4 may be computed in terms of σa, σs, ¯%, P0( ¯%),
B0( ¯%) and n.
Moreover, the following inequality is fulfilled: (2.7) k~uεk L∞( ˙Bn2−1 2,1 ) + eC−12 νk(jε 0,~j1ε)k `,ρeε L∞( ˙Bn2−1 2,1 ) + k(j0ε,~j1ε)kh,ρeε L∞( ˙Bn2−1 2,1 ) + kbεk`, 1 εν L∞( ˙Bn2−1 2,1 ) + ενkbεkh, 1 εν L∞( ˙Bn2 2,1) + νk~uεk L1( ˙Bn2+1 2,1 ) + νkbεk`, 1 εν L1( ˙B n 2+1 2,1 ) +1 εkb εkh,εν1 L1( ˙B n 2 2,1) +ν eC 1/2 ε k(j ε 0,~jε1)k `,eρε L1( ˙B n 2−1 2,1 ) +Ce 1/2 ε k(j ε 0,~j1ε)k h,eρε L1( ˙B n 2−1 2,1 ) ≤ CI0ε· Finally, if the family of data (b0ε, ~uε0, j0ε,~j1ε)0<ε≤ε0 satisfies (2.5) and the divergence free part
P~uε0 of ~uε0 converges to ~v0 in ˙B
n 2−1
2,1 weak ∗, then (bε, ~uε, j0ε,~j1ε) converges to (0, ~v, 0, ~0)
in L∞(R+; ˙B n 2−1 2,1 ) weak ∗, where ~v ∈ C(R+; ˙B n 2−1 2,1 ) ∩ L1(R+; ˙B n 2+1
2,1 ) stands for the unique
global solution of the incompressible Navier-Stokes equations (1.17)–(1.18), supplemented with initial velocity ~v0.
A few remarks are in order:
(1) Note that (2.4) corresponds to ε−1 . C . νε−2. We could treat smaller values of C. However, as pointed out in the introduction, this would be unphysical.
(2) The decay estimate for ~jε1 is the key to handling the nonlinear term k4(eεb
ε)~jε 1 in
the velocity equation of (2.1).
(3) More accurate convergence results are available (see Theorems 5.1 and 5.2 below). (4) We have a similar statement if the coefficients σa, σs, λ and µ depend smoothly
on ρ (see Remark 4.1).
(5) We expect local results for large data, in the spirit of [5]. In particular, the lifes-pan of the limit system (1.17)–(1.18) should be a lower bound of the lifeslifes-pan of the corresponding solution to (2.1), for small enough ε. In order to avoid wild computations however, we preferred to concentrate on the small data case.
We end this section with a short description of the method leading to the above state-ment. Proving the global a priori estimate (2.7) is the main step. Recall that a rougher inequality has been established in our recent paper [7]. Unfortunately, we did not keep track of the physical coefficients of the system therein. As specifying the dependency with respect to ε is fundamental in the study of the low Mach number asymptotics, we will have to refine our previous analysis. In fact, as in the nonradiative case, it is convenient to perform a rescaling so as to avoid terms of order ε−1 in the system. This naturally leads to the following change of unknowns:
(2.8) b(t, x) :=εbeε(eε2t,eεx), ~u(t, x) :=eε~uε(eε2t,εx),e j0(t, x) :=eε ζ0j ε 0(eε 2t, e εx), ~j1(t, x) :=εe ζ0 √ n~j ε 1(εe 2t, e εx), with ε := (Pe 0( ¯%))−1/2ε and ζ0 := 1 ¯ % 1 q e CB0( ¯%) P0( ¯%) n 14r σ a+ σs σa · We eventually get (2.9) ∂tb + ~u · ∇b + div ~u = k1(b)div ~u, ∂t~u + ~u · ∇~u − ν eA~u + ∇b −eς eC 1/2ε2~j 1= νk2(b) eA~u + k3(b)∇b +eς eC 1/2ε2k 4(b)~j1, ∂tj0+α eeCdiv~j1+ eβ eCεj0−ς eeC 1/2ε2b = 0, ∂t~j1+α eeC∇j0+γ eeCε~j1= 0, with ν := λ+2µ%¯ , eA := A/ν, e α := 1 pnP0( ¯%), β :=e σa P0( ¯%), e γ := σa+ σs P0( ¯%) , ς :=e √ σa(σa+σs)B0( ¯%) n1/4(P0( ¯%))5/4 ·
According to [7], the linear stability condition for (2.9) reads e C eβ2eγ 2ν > e αeς 2( eβ + e γ)ε,
and is thus fulfilled for ε going to 0. Hence System (2.9) is globally well posed for small enough data and ε. However, [7] does not specify the dependency of the smallness condition
with respect to ε and eC. In order to prove that it is harmless, we will have to make a very careful analysis of the linearized equations associated to (2.9). In Fourier variables (with respect to x ∈ Rn), those equations may be seen as a linear system of ODEs with constant coefficients depending on the frequency size ρ. The overall linear system is of mixed type : second order parabolic, first order hyperbolic and zero order dissipative, which complicates the task of finding optimal estimates (see the next section). However, once this system is completely understood, proving (2.7) is rather easy : it is only a matter of paralinearizing the system (to keep the convection terms under control) and use appropriate nonlinear estimates.
Next, Inequality (2.7) combined with weak compactness arguments ensures that there exists εn → 0 so that (bεn, ~uεn, j0εn,~j
εn
1 ) * (b, ~u, j0,~j1). That b, j0 and ~j1 are zero, and
that ~u is divergence free may be seen directly by passing to the limit in (2.1). Proving additional uniform estimates for the time derivative of the incompressible part of ~uε com-bined with Ascoli theorem (Aubin-Lions type argument) allows to pass to the limit in the velocity equation. We eventually find out that ~v satisfies the incompressible Navier-Stokes equations.
Finally, if Ω = Rn then, as in the nonradiative case [5], one may take advantage of the dispersive properties of the acoustic wave equation, to upgrade the weak convergence to strong convergence.
3. Linear analysis
This section is devoted to the study of the linearization of (2.9) about (b, ~u, j0,~j1) =
(0, ~0, 0, ~0), namely, (3.1) ∂tb + div ~u = f, ∂t~u − ν eA~u + ∇b − ς~j1= ~g, ∂tj0+ αdiv~j1+ βj0− ηb = 0, ∂t~j1+ α∇j0+ γ~j1 = ~0, with (3.2) α :=α eeC, β := eβ eCε, γ :=eγ eCε, η = ς :=eς eC 1/2ε2.
The study of the evolution of the divergence free parts P~u and P~j1 of ~u and ~j1 is obvious
as we just have
(3.3) ∂tP~u − µ∆P~u = ςP~j1+ P~g and ∂tP~j1+ γP~j1 = ~0.
So, as in [7], we focus on the linearized system fulfilled by b, j0 and the potential parts
of ~u and of ~j1. To work with scalar unknowns, we set d := Λ−1div ~u and j1 := Λ−1div~j1
(with Λs:= (−∆)s2). We eventually get the following system (if f = 0 and ~g = ~0 ):
(3.4) ∂tb + Λd = 0, ∂td − Λb − ν∆d − ςj1 = 0, ∂tj0+ βj0+ αΛj1− ηb = 0, ∂tj1+ γj1− αΛj0= 0.
Denoting ρ := |ξ|, in Fourier variables, the above system rewrites: (3.5) d dt bb b d b j0 b j1 + 0 ρ 0 0 −ρ νρ2 0 −ς −η 0 β αρ 0 0 −αρ γ bb b d bj0 bj1 = 0 0 0 0 ·
3.1. Estimates for small ρ . Making the change of unknown
(3.6) U :=b 1 0 0 0 0 1 0 ςγ −ηβ 0 1 0 0 0 0 1 bb b d bj0 bj1 ,
and setting α0 := α +βγςη, we observe that bU = bU (t, ρ) satisfies (E) ∂tU + Ab 0U + ρ (Ab 1+ B1) bU + νρ2A2U = 0,b with A0 := 0 0 0 0 0 0 0 0 0 0 β 0 0 0 0 γ , A1 := 0 1 0 0 −1 − αςηβγ 0 0 0 0 0 0 α0 0 0 −α 0 , B1 := − 0 0 0 γς 0 0 αςγ 0 0 βη 0 0 αη β 0 0 0 and A2 := 0 0 0 0 0 1 0 −ς γ 0 0 0 0 0 0 0 0 ·
Note that both A0 and A2 are diagonal (up to a small coefficient as regards A2) and that
the diagonal is nonnegative, but degenerate. As for matrix A1, it is antisymmetric and still
tractable by adapting the analysis of the linearized barotropic Navier-Stokes equations. In fact the main difficulty comes from matrix B1.
We claim that the change of unknown bV := (I + ρP ) bU for some suitable matrix P will enable us to annihilate the bad term ρB1U if ρ is small enough. Indeed, we observe thatb
∂tV + (I + ρP )Ab 0(I + ρP )−1V + ρ(I + ρP )(Ab 1+ B1)(I + ρP )−1Vb
+νρ2(I + ρP )A2(I + ρP )−1V = 0.b Because (I + ρP )−1 = I − ρP (I + ρP )−1 = I − ρP + ρ2P2(I + ρP )−1 = I − ρP + ρ2P2− ρ3P3(I + ρP )−1, we discover that ∂tV + Ab 0V + ρ Ab 1+ B1+ [P, A0] b V + ρ2 [A0, P ]P + [P, A1] + [P, B1] + νA2 b V +ρ3(I + ρP ) (A1+ B1)P2− A0P3− νA2P(I + ρP )−1V = 0.b Therefore, if one can choose P so that
then we end up with
(3.8) ∂tV + Ab 0V + ρAb 1V + ρb 2(νA2+ P B1+ [P, A1]) bV = ρ3(I + ρP ) A3(I + ρP )−1V ,b where A3 := (P A0− A1)P2+ νA2P. Note that the matrix B1 now appears in the second
order term instead of a first order term before the change of unknown. In order to determine P , let us rewrite A0, B1 and P in block form:
A0= 0 0 0 D , B1= 0 B1 1 B12 0 , P = P11 P12 P21 P22 · Computing the commutator
(3.9) [A0, P ] = 0 −P12D DP21 [D, P22] , we see that a convenient choice for P is
P11:= 0, P22:= 0, P12:= −B11D−1, P21:= D−1B12, that is to say, (3.10) P = 0 0 0 γς2 0 0 βγας 0 0 −βη2 0 0 −αηβγ 0 0 0 ·
Remembering (3.6), we thus find out that
(3.11) V =b b b b d bj0 bj1 := 1 0 0 γς2ρ −βαςη2γρ 1 βγαςρ γς −βη −βη2ρ 1 − ςη β2γρ −αηβγρ 0 0 1 bb b d bj0 bj1 ·
In terms of coefficients α, ee β, eγ and eς, the above change of variables writes b b bd bj0 bj1 := 1 0 0 eς e γ2Ce3/2ρ −αeeς 2ε e β2 e γ eCρ 1 e αςe e βeγ eC1/2ρ e ςε e γ eC1/2 − ςεe e β eC1/2 − e ς e β2Ce3/2ρ 1 − e ς2ε e β2 e γ eC2ρ − αeςe e βeγ eC 1/2ρ 0 0 1 bb b d bj0 bj1 ·
Hence, assuming that α, ee β, γ ande ς are of order 1, and that ee C & 1, we deduce that the constants appearing when changing (bb, bd, bj0, bj1) to (bb,bd,bj0,bj1), and conversely, may be
uniformly bounded for (ε, ρ) ∈ [0, R]2 (for any fixed R > 0 ).
Next, let us use the explicit form of P to rewrite (3.8). To this end, we have to compute the matrices P B1, [P, A1] and A3. We easily get
P B1 = −βγαςη2 0 0 0 0 −αςηβ2γ 0 0 0 0 αςηβ2γ 0 0 0 0 βγαςη2 and
[P, A1] = 0 0 −αςγ(β1+γ1) 0 0 0 0 ααβγ0ς +γς2(1+ αςη βγ) αα0η βγ + η β2(1+ αςη βγ) 0 0 0 0 −αηβ (β1+1γ) 0 0 ·
Finally, A3 := (P A0− A1)P2+ νA2P, and thus
(3.12) A3= αςη βγ 0 β12 0 − ς γ3 ν γ − 1 γ2 1+ αςη βγ 0 νη − ας β2γ 0 0 0 0 γα20 0 0 −α β2 0 · Hence (3.13) |A3| ≤ C(1 + ν)
with C depending only on α, ee β, eγ, and ς (if assuming that ee C & 1). Let us focus on the system satisfied by (bb,bd) for a while. We have
(3.14) d dt b b b d + ρ 0 1 −1 − αςηβγ 0 bb bd + ρ2 − αςη βγ2 0 0 ν − αςηβ2γ ! bb bd = ρ2 ας γ ( 1 β + 1 γ) 0 0 ςνγ −αα0ς βγ − ς γ2(1 + αςη βγ ) ! bj0 bj1 ! + (1+ν)O(ρ3). For small enough ρ, optimal estimates may be proved by taking advantage of the results of Appendix D. Indeed, we see from (A.7) and (A.8) that if we set
Uρ2 := 1 +αςη βγ |bb|2+ |bd|2− ρ ν + αςη βγ 1 γ − 1 β Re (bbbd) then, under the following stability condition3 :
(3.15) ˜ν := ν −αςη βγ 1 β + 1 γ > 0, we have for small enough ε (see (A.5) and (A.6)):
(3.16) 1 2Uρ≤ |(bb,bd)| ≤ 2Uρ and d dtU 2 ρ+ ν 4ρ 2U2 ρ ≤ CUρ| bFρ|,
where bFρ stands for the r.h.s. of (3.14) and C is an absolute constant, whenever
(3.17) ρ ≤
q 1 +αςηβγ ν + αςηβγ (1γ− 1β)·
Note that in the case we are interested in, γ ≥ β, hence (3.17) is fulfilled if ρν ≤ 1.
So finally, we get for small enough ε and some appropriate constant C = C(α, ee β,eγ,eς) : |(bb,bd)(t)| + νρ 2 Z t 0 |(bb,bd)| dτ ≤ C |(bb,bd)(0)| + (1 + εν)ρ 2 Z t 0 |(bj0,bj1)| dτ + (1+ν)ρ3 Z t 0 |(bb,bd,bj0,bj1)| dτ , which, if in addition (1+ν)ρ ν, may be simplified into
(3.18) |(bb,bd)(t)| + νρ2 Z t 0 |(bb,bd)| dτ ≤ C |(bb,bd)(0)| + (1+εν)ρ2 Z t 0 |(bj0,bj1)| dτ · Next we see that the modified radiative modes j0 and j1 satisfy:
(3.19) d dt bj0 bj1 ! + ρ 0 α0 −α 0 bj0 bj1 ! + β + αςη β2γρ2 0 0 γ +αςηβγ2ρ2 ! bj0 bj1 ! = ρ2 − αα0η βγ − η β2(1 + αςη βγ) 0 0 αηβ (β1 + 1γ) ! bb bd + (1 + ν)O(ρ3). Therefore, there exists some constant C = C(α, ee β,eγ,ς) so thate
1 2 d dt |bj0|2+α 0 α|bj1| 2 + β + αςη β2γρ 2 |bj0|2+α 0 α γ +αςη βγ2ρ 2 |bj1|2 ≤ C (1 + eC1/2ε)ρ2|(bb,bd)| + (1+ν)ρ 3|(bb, b d,bj0,bj1)|.
Then, integrating in time, remembering that min(β, γ) = min( eβ,eγ) eCε and assuming that e C1/2ε . 1, (1 + ν)ρ 1 and (1 + ν)ρ3 ε eC yields : (3.20) |(bj0,bj1)(t)| + ε eC Z t 0 |(bj0,bj1)| dτ ≤ C |(bj0,bj1)(0)| + ρ2 Z t 0 |(bb,bd)| dτ ·
Combining with (3.18), we can conclude that if ε is small enough then there exists some positive constants ρ0 and C depending only on (α, ee β,eγ,ς) so that for alle
(3.21) 0 ≤ ρ ≤ ρ0min ν, ν−1, (εν eC)12, ε eC 1 + ν 13 , we have (3.22) |(bb,bd)(t)| + ν|(bj0,bj1)(t)| + νρ2 Z t 0 |(bb,bd)| dτ + νε eC Z t 0 |(bj0,bj1)| dτ ≤ C |(bb,bd)(0)| + ν|(bj0,bj1)(0)|.
3.2. Estimates for large ρ . The case of large ρ ’s (high frequency regime) is tractable by means of a more standard approach, as the coupling between (b, d) and (j0, j1) is low
order. In fact, considering the term in j1 in the equation for d as a source term, the
‘classical’ estimate for the linearized barotropic equations gives4 : |(bb, νρbb, bd)(t)|+min(1, νρ) Z t 0 ρ|bb| dτ +νρ2 Z t 0 | bd| dτ ≤ C |(bb, νρbb, bd)(0)|+ eC1/2ε2 Z t 0 |bj1| dτ ,
4That may be proved by considering |νρbb|2
while the equations for (bj0, bj1) give us |(bj0, bj1)(t)| + eCε Z t 0 |(bj0, bj1)| dτ ≤ |(bj0, bj1)(0)| + eC1/2ε2 Z t 0 |bb| dτ. Adding up those two inequalities, we discover that
(3.23) |(bb, νρbb, bd, bj0, bj1)(t)| + min(1, νρ) Z t 0 ρ|bb| dτ + νρ2 Z t 0 | bd| dτ + ε eC Z t 0 |(bj0, bj1)| dτ ≤ C|(bb, νρbb, bd, bj0, bj1)(0)|,
provided (still assuming 1 . C . ε−2)
(3.24) ρ ε eC
1/4
ν1/2 ·
Note that there is some overlap between the low frequency condition given by (3.21), and (3.24).
For simplicity, we shall assume from now on that (2.4) is fulfilled. This will imply that for any ν0 > 0 there exist ε0, ρ0 and ρ1 depending only on α, ee β, eγ, eς and ν0 so that (3.22) and (3.23) are fulfilled whenever ε ≤ ε0, ν ≤ ν0 and
(3.25) ρ ≤ ρ0
p
εν eC and ρ ≥ ρ1ν−1/2Ce1/4ε, respectively. 4. Proof of the global existence
This section is dedicated to the proof of the first part of Theorem 2.1. Set ρε:= ρ0
p
εν eC and introduce the space Es
ε,ν, eC of quadruplets of functions (b, ~u, j0,~j1) with ~ u ∈ C(R+; ˙B2,1s ) ∩ L1(R+; ˙B2,1s+2), b`,ν−1 ∈ C(R+; ˙B2,1s ) ∩ L1(R+; ˙B2,1s+2), bh,ν −1 ∈ C(R+; ˙B2,1s ) ∩ L1(R+; ˙B2,1s ), (j`,ρε 0 ,~j `,ρε 1 ) ∈ C(R+; ˙B s 2,1) ∩ L1(R+; ˙B2,1s+2), (jh,ρε 0 ,~j h,ρε 1 ) ∈ C(R+; ˙B2,1s ) ∩ L1(R+; ˙Bs2,1),
such that the following norm is finite: k(b, ~u, j0,~j1)kEs ε,ν, eC := k~ukL∞( ˙Bs2,1)+ νk(j0,~j1)k `,ρε L∞( ˙Bs 2,1) + k(j0,~j1)kh,ρL∞ε( ˙Bs 2,1) +kbk`,ν−1 L∞( ˙Bs 2,1) + νkbkh,ν−1 L∞( ˙Bs+1 2,1 ) + Z R+ νk~ukB˙s+2 2,1 + νkbk `,ν−1 ˙ B2,1s+2 + kbk h,ν−1 ˙ B2,1s+1 dt +ε eC Z R+ νk(j0,~j1)k`,ρB˙sε 2,1 + k(j0,~j1)kh,ρB˙sε 2,1 dt. with (4.1) j0 := j0− e ςε e β eC1/2b − e ς e β2Ce3/2div ~u − e ς2ε e β2 e γ eC2div~j1 and ~j1 := ~j1− e αςe e βeγ eC 1/2∇b.
Through the rescaling (2.8), the first part of Theorem 2.1 is a straightforward consequence of the following statement.
Theorem 4.1. Let ν0 > 0 and n ≥ 2. Assume (2.4). There exist three constants ε0, c
and C depending only on α, ee β, eγ, eς, ν0 and λ/µ such that if 0 < ε ≤ ε0, 0 < ν ≤ ν0
and the data satisfy (4.2) I0 := k ~u0k˙ B n 2−1 2,1 + νk(j0,0,~j1,0)k`,ρε ˙ B n 2−1 2,1 + k(j0,0,~j1,0)kh,ρε ˙ B n 2−1 2,1 + kb0k`,ν −1 ˙ B n 2−1 2,1 + νkb0kh,ν −1 ˙ B n 2 2,1 ≤ cν
then System (2.9) has a unique global solution (b, ~u, j0,~j1) in E
n 2−1 ε,ν, eC satisfying k(b, ~u, j0,~j1)k E n 2−1 ε,ν, eC ≤ CI0.
Proof of Theorem 2.1: Granted with the above statement, making the change of un-knowns (2.8) gives the first part of Theorem 2.1. Indeed, we notice that, up to some irrelevant constant depending only on σa, σs, ¯%, P0( ¯%) and n the term I0ε is equal
to the quantity I0 of Theorem 4.1. Besides, computing k(b, ~u, j0,~j1)k E n 2−1 ε,ν, eC in terms of (bε, ~uε, jε0,~j1ε) gives (2.7).
The rest of this section is devoted to the proof of Theorem 4.1. We focus on the proof of global a priori estimates for smooth solutions to (2.9), and refer to [7] for existence and uniqueness.
At a first trial, one may tempt to apply the analysis of the previous section to System (3.1) with
f = −~u · ∇b + k1(b)div ~u and ~g = −~u · ∇~u + νk2(b) eA~u + k3(b)∇b +eς eC
1/2ε2k 4(b)~j1.
Indeed, localizing System (3.1) in the Fourier space by means of the Littlewood-Paley operator ˙∆k, and combining the inequalities that we proved in Section 3 with
Fourier-Plancherel theorem, it is easy to deduce estimates in any Besov space related to L2. Nonzero terms f and ~g may be included in our analysis, by taking advantage of Duhamel formula. However, one cannot treat the convection term ~u · ∇b as a source, as it would cause a loss of one derivative (exactly as for the standard compressible Navier-Stokes equations). At the L2 level, this may be avoided by an energy method, after integrating by parts in that term. In our case, the idea is more or less the same, except that the energy argument has to be performed on the localized convection term, namely ˙∆k(~u · ∇b). A nice (and nowadays
classical) way of performing this computation is to paralinearize the system: we consider
(4.3) ∂tb + T~v· ∇b + div ~u = f, ∂t~u + T~v· ∇~u + ∇b − ν eA~u − eC1/2eςε 2~j 1= ~g, ∂tj0+ eβ eCεj0+α eeCdiv~j1− eC 1/2 e ςε2b = 0, ∂t~j1+eγ eCε~j1+α eeC∇j0 = ~0,
4.1. Estimates for the paralinearized system. Before stating the estimates, let us introduce one more notation: we shall need the following quantity involving medium fre-quencies: kzkm,η,ι˙ Bs 2,1 := X η/2≤2k≤2ι 2ksk ˙∆kzkL2.
Proposition 4.1. Let ν0> 0. Assume that (2.4) is fulfilled. There exist positive constants
ε0 = ε0(α, ee β,eγ,eς, ν0) and C = C(α, ee β,eγ,eς, λ/µ, ν0) such that if 0 < ν ≤ ν0 and 0 < ε ≤ ε0 then the solutions to (4.3) satisfy the following a priori estimates (for all s ∈ R) :
• Low frequencies: k(b, ~u)(t)k`,ρε ˙ Bs 2,1 + νk(j0,~j1)(t)k`,ρB˙sε 2,1 + ν Z t 0 k(b, ~u)k`,ρε ˙ B2,1s+2+ ε eCk(~j0,~j1)k `,ρε ˙ Bs 2,1 dτ ≤ C k(b, ~u)(0)k`,ρε ˙ Bs 2,1 + νk(j0,~j1)(0)k`,ρB˙sε 2,1 + Z t 0 k(f − T~v· ∇b, ~g − T~v· ∇~u)k`,ρε ˙ Bs 2,1 dτ · • Middle frequencies: k(b, ~u, j0,~j1)(t)km,ρB˙s ε,1/ν 2,1 + Z t 0 νk(b, ~u)km,ρε,1/ν ˙ B2,1s+2 +ε eCk(j0,~j1)k m,ρε,1/ν ˙ Bs 2,1 dτ ≤ C k(b, ~u, j0,~j1)(0)km,ρB˙s ε,1/ν 2,1 + Z t 0 k(f, ~g)km,ρε,1/ν ˙ Bs 2,1 dτ + Z t 0 k∇~vkL∞k(b, ~u, j0,~j1)k˙ Bs 2,1dτ · • High frequencies: k(ν∇b, ~u, j0,~j1)(t)k h,1/ν ˙ Bs 2,1 + Z t 0 νk~ukh,1/ν˙ B2,1s+2+ k∇bk h,1/ν ˙ Bs 2,1 + eCεk(j0,~j1)k h,1/ν ˙ Bs 2,1 dτ ≤ C k(ν∇b, ~u, j0,~j1)(t)kh,1/νB˙s 2,1 + Z t 0 k(ν∇f, ~g)kh,1/ν˙ Bs 2,1 dτ + Z t 0 k∇~vkL∞k(ν∇b, ~u, j0,~j1)k˙ Bs 2,1dτ · Proof: The main idea is to localize System (4.3) according to Littlewood-Paley decom-position, and to use Fourier Plancherel theorem to evaluate the L2 norm of each localized part of the solution. Concretely, applying ˙∆k to (4.3) and denoting bk:= ˙∆kb, ~uk:= ˙∆k~u,
etc, gives (4.4) ∂tbk+ ˙∆k(T~v· ∇b) + div ~uk= fk, ∂t~uk+ ˙∆k(T~v· ∇~u) + ∇bk− ν eA~uk−ς eeC 1/2ε2~j 1,k = ~gk, ∂tj0,k+ eβ eCεj0,k +α eeCdiv~j1,k−eς eC 1/2ε2b k= 0, ∂t~j1,k+eγ eCε~j1,k+ eCα∇je 0,k = ~0.
To handle low frequencies (i.e. 2k ≤ ρε), we put the paraconvection terms in the
right-hand side of (4.4) and follow the computations leading to (3.22). We thus have to consider bk:= ˙∆kb, ~uk := ˙∆ku, j0,k := ˙∆kj0, ~j1,k := ˙∆k~j1 with j0, ~j1 defined in (4.1), (4.5) b:= b + eς e γ2Ce3/2div~j1 and ~u := ~u − e αeς2ε e β2 e γ eC∇b + e αeς e βeγ eC1/2∇j0+ e ςε e γ eC1/2~j1.
Using eventually Fourier-Plancherel theorem, we get for all positive t and 2k≤ ρ ε k(bk, ~uk)(t)kL2+ νk(j0,k,~j1,k)(t)kL2+ 22kν Z t 0 k(bk, ~uk)kL2dτ + νε eC Z t 0 k(j0,k,~j1,k)kL2dτ ≤ C k(bk, ~uk)(0)kL2 + νk(j0,k,~j1,k)(0)kL2 + Z t 0 kfk− ˙∆k(T~v· ∇b)kL2+ k~gk− ˙∆k(T~v· ∇~u)kL2 dτ · Note that, as 2k ≤ ρε, the first two terms of the l.h.s. and of the r.h.s. may be changed to the similar ones with (bk, ~uk, j0,k,~j1,k) (up to a harmless change of the constant C of
course). Hence multiplying by 2ks and summing up over all k ∈ Z with 2k≤ ρε yields the desired inequality.
To handle the regime corresponding to 2k ≥ ρ1ν−1/2Ce1/4ε, we introduce the Lyapunov functional
L2k:= 2k(bk, ~uk)k2L2 + kν∇bkk2L2+ 2ν(∇bk|~uk),
which is obviously equivalent to k(bk, ν∇bk, ~uk)k2L2. We observe that
1 2 d dtL 2 k+ νk(∇bk, Q~uk)k2L2 + 2µk∇P~ukk2L2 = 2ε2Ce1/2eς(~uk|~j1,k) +ε2Ce1/2ς(ν∇be k|~j1,k) + ε2Ce1/2eς(bk|j0,k) + 2(fk|bk) + 2(~gk|~uk) + (ν∇fk|ν∇bk) + (ν∇fk|~uk) +(~gk|ν∇bk) + 2 ˙∆k(T~v· ∇b)|bk + 2 ˙∆k(T~v· ∇~u)|~uk + ν∇ ˙∆k(T~v· ∇b)|ν∇bk + ν∇ ˙∆k(T~v· ∇b)|~uk + ˙∆k(T~v· ∇~u)|ν∇bk.
Convection terms may be handled according to Lemma 4.1 in [7]: for example, we have for some universal integer N :
∇ ˙∆k(T~v· ∇b)|∇bk ≤ Ck∇~vkL∞k∇bkkL2 X |k0−k|≤N k∇bk0kL2.
We thus eventually get: 1 2 d dtL 2 k+min(ν22k, 1)L2k≤ CLk k(fk, ~gk, ν∇fk)kL2+ eC1/2eςε2k~j1,kkL2+k∇~vkL∞ X |k0−k|≤N Lk0
whence, integrating in time, (4.6) Lk(t) + min(ν22k, 1) Z t 0 Lk(τ ) dτ ≤ Lk(0) + C Z t 0 k(fk, ~gk, ν∇fk)kL2dτ + C eC1/2ςεe2 Z t 0 k~j1,kkL2dτ + C X |k0−k|≤N Z t 0 k∇~vkL∞Lk0dτ.
From the last two equations of (4.4), we easily get (remembering that eγ ≥ eβ ): k(j0,k,~j1,k)(t)kL2 + eCε eβ Z t 0 k(j0,k,~j1,k)kL2dτ ≤ k(j0,k,~j1,k)(t)kL2+ eC1/2 e ςε2 Z t 0 kbkkL2dτ.
Combining with (4.6) and taking ρ1 large enough, we conclude that for small enough ε and any ν ≤ ν0, (4.7) k(bk, ν∇bk, ~uk, j0,k,~j1,k)(t)kL2 + min(ν22k, 1) Z t 0 k(bk, ν∇bk, ~uk)kL2dτ + eCε Z t 0 k(j0,k,~j1,k)kL2dτ ≤ C k(bk, ν∇bk, ~uk, j0,k,~j1,k)(0)kL2 + Z t 0 k(fk, ~gk, ν∇fk)kL2dτ + C X |k0−k|≤N Z t 0 k∇~vkL∞k(bk0, ν∇bk0, ~uk0, j0,k0,~j1,k0)kL2dτ ·
In order to exhibit the parabolic gain of regularity for ~u, we use the fact that ∂t~uk− ν eA~uk= ~gk− ∇bk+eς eC
1/2ε2~j
1,k− ˙∆k(T~v· ∇~u),
which implies by energy method, Bernstein inequality and Lemma 4.1 in [7], 1 2 d dtk~ukk 2 L2+ ν22kk~ukk2L2 ≤ Ck~ukkL2 k~gkkL2 + ε2Ce1/2k~j1,kkL2 +k∇bkkL2 + X |k0−k|≤N k∇~vkL∞k~uk0kL2 ·
Integrating with respect to time, and keeping (4.7) in mind, one may bound the second and third term of the r.h.s. in terms of the data. We thus conclude that
k(bk, ν∇bk, ~uk, j0,k,~j1,k)(t)kL2 + min(1, ν2k) Z t 0 k∇bkkL2dτ + ν22k Z t 0 k~ukkL2dτ + eCε Z t 0 k(j0,k,~j1,k)kL2dτ ≤ C k(bk, ν∇bk, ~uk, j0,k,~j1,k)(0)kL2 + Z t 0 k(fk, ~gk, ν∇fk)kL2dτ + C X |k0−k|≤N Z t 0 k∇~vkL∞k(bk0, ν∇bk0, ~uk0, j0,k0,~j1,k0)kL2dτ ·
Multiplying by 2ks and summing up over all the integers k ∈ Z such that 2k≥ ρ1ν−1/2Ce1/4ε yields the wanted inequalities in the middle and high frequencies range. 4.2. Uniform estimates. It is now easy to prove global estimates for smooth solutions to (2.9) : it suffices to apply Proposition 4.1 with s = n/2 − 1,
f = −T∇b0 · ~u + k1(b)div ~u and ~g = −T∇~0u· ~u + νk2(b) eA~u + k3(b)∇b +ς eeC
1/2ε2k 4(b)~j1.
As regards f, standard product, paraproduct and composition estimates in Besov spaces lead to kf k L1( ˙Bn2−1 2,1 ∩ ˙B n 2 2,1) . k∇bk L∞( ˙Bn2−2 2,1 ∩ ˙B n 2−1 2,1 ) k~uk L1( ˙Bn2+1 2,1 ) +kbk L∞( ˙Bn2−1 2,1 ∩ ˙B n 2 2,1) kdiv ~uk L1( ˙Bn2 2,1) . (4.8)
Therefore, if ν = 1 then Inequality (4.8) provides a control of the left-hand side by k(b, ~u, j0,~j1)k2 E n 2−1 ε,1, eC
. The general case follows from the case ν = 1 after change of vari-ables, and we eventually get
kf k`,ν−1 L1( ˙Bn2−1 2,1 ) +νkf kh,ν−1 L1( ˙Bn2 2,1) . kbk`,ν−1 L∞( ˙Bn2−1 2,1 ) +νkbkh,ν−1 L∞( ˙Bn2 2,1) kdiv ~uk L1( ˙B n 2 2,1) . ν−1k(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC . (4.9)
Next, we need a quadratic estimate for ~g in L1(R +; ˙B
n 2−1
2,1 ). The first three terms may be
handled by means of usual product, paraproduct and composition estimates: we get kT∇~0 u~uk L1( ˙Bn2−1 2,1 ) . k∇~uk L1( ˙Bn2 2,1) k~uk L∞( ˙Bn2−1 2,1 ) , νkk2(b) eA~uk L1( ˙B n 2−1 2,1 ) . νkbk L∞( ˙Bn2 2,1) k∇2~uk L1( ˙B n 2−1 2,1 ) , kk3(b)∇bk L1( ˙Bn2−1 2,1 ) . kbk2 L2( ˙Bn2 2,1) . Because kbk`,ν−1 B n 2−1 2,1 + νkbkh,ν−1 B n 2 2,1 ≈ kbk ˙ B n 2−1 2,1 + νkbk ˙ B n 2 2,1 and, by interpolation, (4.10) ν1/2kbk L2( ˙Bn2 2,1) . k(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC ,
we deduce that the above three terms may be bounded by ν−1k(b, ~u, j0,~j1)k2 E
n 2−1 ε,ν, eC
. For the last term, we just write that
k4(b)~j1 = k4(b)~j1+ e αeς e βeγ eC1/2k4(b)∇b. Hence (4.11) kk4(b)~j1k L1( ˙Bn2−1 2,1 ) . kbk L∞( ˙Bn2 2,1) k~j1k L1( ˙Bn2−1 2,1 ) + kbk2 L2( ˙Bn2 2,1) .
Finally, we need to bound the low frequencies of the paraconvection terms T~u· ∇b and
T~u· ∇~u in L1(R+; ˙B
n 2−1
2,1 ). We observe (by product laws and interpolation) that
kT~u· ∇bk L1( ˙Bn2−1 2,1 ) . k~uk L2( ˙Bn2 2,1) k∇bk L2( ˙Bn2−1 2,1 ) . ν−1k(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC , kT~u· ∇~uk L1( ˙B n 2−1 2,1 ) . k~uk L2( ˙B n 2 2,1) k∇~uk L2( ˙B n 2−1 2,1 ) . ν−1k(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC .
Putting together the above inequalities and Proposition 4.1 with s = n/2 − 1 and using the embedding ˙B n 2 2,1 ,→ L∞, we end up with k(b, ~u, j0,~j1)k E n 2−1 ε,ν, eC ≤ C I0+ ν−1k(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC ·
Assuming that (4.2) is fulfilled with a sufficiently small constant c, it is clear that the above inequality implies that
k(b, ~u, j0,~j1)k E n 2−1 ε,ν, eC ≤ 2CI0,
which is exactly what we wanted. Remark 4.1. The above theorem and the following corollary extend to the case where λ, µ, σa and σs depend smoothly on %. Compared to the constant case, the main difference is
that we have to bound in L1(R+; ˙B
n 2−1
2,1 ) extra nonlinear terms like ∇(K(b))⊗∇u, K1(b)j0,
K2(b)~j1 and K3(b)b for some explicit smooth functions K, K1, K2 and K3 vanishing at
0. All those terms may be handled by taking advantage of the damped modes j0 and ~j1.
However, as we do not know how to treat nonlinear terms of the type K(b)b if they occur in the equation for j0, one has to change the definition of b into, say,
eb :=
σa(%) B(%) − B( ¯%)
σa( ¯%) ¯% B0( ¯%)
· The details are left to the reader.
5. The proof of convergence
This section is devoted to the rigorous justification of the low Mach number asymptotics pointed out in the introduction. We here propose two approaches. The first one is based on the uniform estimates of Theorem 2.1 and compactness arguments, and is thus valid indistinctly in Tn or Rn. In contrast, the second approach combines the estimates of Theorem 2.1 with Strichartz type inequalities, and thus works only in the Rn case with n ≥ 2. At the same time, the result is more accurate: we get strong convergence for explicit norms and with explicit decay rate.
5.1. Weak convergence results. Here we establish a general weak convergence result which holds true both in the periodic and the whole space cases.
Theorem 5.1. Consider a family of data (bε0, ~uε0, j0,0ε ,~j1,0ε ) satisfying the assumptions of Theorem 2.1 with ε → 0. Let (bε, ~uε, j0ε,~j1ε) be the corresponding family of global solutions to (2.1). Then, for suitable norms, %ε:= B−1(B( ¯%) + ¯%B0( ¯%)eεb
ε) converges strongly to ¯%,
(j0ε,~j1ε) → (0, ~0) with the rate of convergence O(ε), and (bε, div ~uε) * (0, ~0) in S0. If we suppose in addition that
(5.1) P~uε0 * ~v0 in S0
then ~uε converges to ~v in S0 when ε → 0, where ~v ∈ C(R +; ˙B n 2−1 2,1 ) ∩ L1(R+; ˙B n 2+1 2,1 ) stands
for the unique global solution of
(5.2) ∂t~v + ~v · ∇~v + ∇Π − ¯µ∆~v = ~0, div ~v = 0. ~ v|t=0= ~v0, with µ := µ/ ¯¯ %.
Proof: Let us first observe that
(5.3) kbε0kh, 1 εν ˙ B n 2−1 2,1 . ενkbε0k h,εν1 ˙ B n 2 2,1 . Hence the data (bε0, ~uε0, j0,0ε ,~j1,0ε ) are uniformly bounded in ˙B
n 2−1
2,1 and one thus have, up to
some omitted extraction,
(bε0, ~uε0, j0,0ε ,~j1,0ε ) * (b0, ~u0, j0,0,~j1,0) in B˙
n 2−1
2,1 weak ∗ .
Likewise, the corresponding sequence of solutions (bε, ~uε, j0ε,~j1ε) is bounded in the space Cb(R+; ˙B
n 2−1
2,1 ) hence we have, up to another omitted extraction,
(bε, ~uε, j0ε,~j1ε) * (b, ~u, j0,~j1) in L∞(R+; ˙B
n 2−1
2,1 ) weak ∗ .
Let us first focus on (j0,~j1). The above bounds and convergence result imply that one may
pass to the limit in the last two equations of (2.1) in the distributional meaning: we get
(5.4) 1
ndiv~j1+ σa(j0− ¯%B
0
( ¯%)b) = 0 and ∇j0+ (σa+ σs)~j1 = ~0.
We claim that (j0,~j1) ≡ (0, ~0). Indeed, using the decompositions
j0ε = (j0ε)h,ρeε + (jε
0)`,ρeε+ c1ε(bε)`,ρeε+ c2Ce−1ε(div ~uε)`,eρε + c3Ce−2ε2(div~jε 1)`,ρeε,
~jε
1 = (j1ε)`,eρε + c4ε(∇bε)`,ρeε,
the uniform bounds of Theorem 2.1, and (4.10), we deduce that there exists some positive real number M0 depending only on the coefficients of the system (other than ε and eC )
such that (5.5) kj0εk L1( ˙Bn2−1 2,1 + ˙B n 2 2,1)+L2( ˙B n 2 2,1)+L∞( ˙B n 2−2 2,1 ) + k~j1εk (L1+L2)( ˙Bn2−1 2,1 ) ≤ M0ε.
Hence one may conclude that j0 ≡ 0 and ~j1 ≡ ~0.
The strong convergence of the density to ¯% is obvious: we have %ε= B−1 B( ¯%) + ¯%B0( ¯%)εbe
ε,
hence, given that bε is bounded in L2(R+; ˙B
n 2 2,1), we have %ε → ¯% in L2(R+; ˙B n 2 2,1), with
rate ε. We even have a stronger result: because j0 = div~j1 = 0, the first equality in (5.4)
implies that b ≡ 0.
To see that div ~u = 0, we use the mass equation: div ~uε= k1(εbe
ε)div ~uε−
e
ε~uε· ∇bε−ε∂e tbε.
Given that bε and ~uε are bounded in L2(R+; ˙B
n 2
2,1), the first two terms in the right-hand side
are O(ε) in L1(R+; ˙B
n 2−1
2,1 ). As for the last term, it tends to 0 in the sense of distributions,
for bε is bounded in L2(R+; ˙B
n 2
2,1).
In order to complete the proof of the statement, it is only a matter of establishing that P~uε converges in the sense of distributions to the solution ~v of (5.2). To achieve it, we
project the velocity equation onto divergence-free vector fields: (5.6) ∂tP~uε− µ∆P~uε= −P(~uε· ∇~uε) + ¯%−1P k2(εbe ε)A~uε + (n¯%)−1 (σa+ σs)P (1 + k4(eεb ε))~jε 1.
Because Q~u = 0, the left-hand side converges to ∂t~u − µ∆~u. Next, using that bε (resp.
~ uε) is bounded in L∞(R+; ˙B n 2−1 2,1 ) (resp. L1(R+; ˙B n 2+1
2,1 ) ), we see that the second term in
the right-hand side is O(ε) in L1(R+; ˙B
n 2−2
2,1 ) (or rather in L1(R+× Rn) if d = 2 owing
to endpoint product estimates in Besov spaces). The above considerations also show that the last term is O(ε) in the space (L1+ L2)(R+; ˙B
n 2−1
Let us finally study the convergence of P(~uε· ∇~uε). We note that
~
uε· ∇~uε= 1 2∇|Q~u
ε|2+ P~uε· ∇~uε+ Q~uε· ∇P~uε.
Projecting the first term onto divergence free vector fields gives 0, so we just have to study the convergence of P(P~uε· ∇~uε) and P(Q~uε· ∇P~uε). Now, a further glance at (5.6) shows that the r.h.s. is bounded in, say, L43(R+; ˙B
n 2− 3 2 2,1 ) + L∞(R+; ˙B n 2−1 2,1 ).
Indeed, by interpolation, we see that ∇~uε is bounded in L43(R+; ˙B n 2− 1 2 2,1 ). As ~uε and εbε are bounded in L∞(R +; ˙B n 2−1 2,1 ) and L∞(R+; ˙B n 2
2,1), respectively, product laws in Besov
spaces ensure that the first two terms of the r.h.s. of (5.6) are bounded in L43(R+; ˙B n 2−
3 2
2,1 ).
The last term is bounded in L∞(R+; ˙B
n 2−1 2,1 ) because εbε is bounded in L∞(R+; ˙B n 2 2,1) and ~j1ε is bounded in L∞(R+; ˙B n 2−1
2,1 ). This finally means that ∂tP~uε is bounded in
L43(R+; ˙B n 2− 3 2 2,1 ) + L∞(R+; ˙B n 2−1 2,1 ), whence P~uε is bounded in C 1 4 loc(R+; B n 2− 3 2 2,1,loc). As P~uε is also bounded in Cb(R+; ˙B n 2−1
2,1 ), and as the embedding of B
n 2−1 2,1 in B n 2− 3 2 2,1 is locally
com-pact (see e.g. [1], page 108), we conclude that, up to extraction, for all φ ∈ S(Rd) and
T > 0, φP~uε −→ φP~u in C([0, T ]; ˙B n 2− 3 2 2,1 ).
Combining with the uniform bounds in Cb(R+; ˙B
n 2−1
2,1 ) ∩ L1(R+; ˙B
n 2+1
2,1 ), this allows to
con-clude that P(P~uε· ∇~uε) and P(Q~uε· ∇P~uε) converge to P(P~u · ∇~u) and P(Q~u · ∇P~u), respectively, in the sense of distributions. As P~u = ~u, this completes the proof that ~u
satisfies (5.2) for some ∇Π.
5.2. Strong convergence results. This part is devoted to the proof of a more precise result, in the whole space case, that does take advantage of the dispersive properties of the acoustic wave equation.
Theorem 5.2. Assume that the fluid domain is Rn ( n ≥ 2 ) and consider a family of data (bε0, ~uε0, j0,0ε ,~j1,0ε ) as in Theorem 2.1. Let (bε, ~uε, j0ε,~j1ε) be the corresponding solution of System (2.1). Then (5.5) is fulfilled by (jε
0,~j1ε). Furthermore, (bε, Q~uε) → 0 and P~uε→ ~v
(with ~v solution to (5.2)) in the following sense : • Case n ≥ 4: √νk(bε, Q~uε)k e L2(R +; ˙B n p −12 p,1 ) ≤ Cε1/2Iε
0 for all p ∈ [pc, ∞] with pc :=
(2n − 2)/(n − 3), and (5.7) kP~uε− ~vk L1(R +; ˙B n p +12 p,1 + ˙B n 2+1 2,1 )∩L∞(R+; ˙B n p −32 p,1 + ˙B n 2−1 2,1 ) ≤ Cν ε1/2I0ε+ kP~uε0− ~v0k ˙ B n p −32 p,1 + ˙B n 2−1 2,1 . • Case n = 3: √νk(bε, Q~uε)k e L2(R +; ˙B 4 q −12 q,1 ) ≤ Cε12− 1 qIε
0 for all q ∈ [2, ∞), and
(5.8) kP~uε− ~vk L1(R +; ˙B 4 q +12 q,1 + ˙B 5 2 2,1)∩L∞(R+; ˙B 4 q −32 q,1 + ˙B 1 2 2,1) ≤ Cν ε12− 1 qIε 0 + kP~uε0− ~v0k ˙ B 4 q −32 p,1 + ˙B 1 2 2,1 .
• Case n = 2: √νk(bε, Q~uε)k e L2(R +; ˙B 5 2q− 14 q,1 ) ≤ Cε14− 1 2qIε
0 for all q ∈ [2, 6], and
(5.9) kP~uε− ~vk L1(R +; ˙B 5 2q+ 34 q,1 + ˙B22,1)∩L∞(R+; ˙B 5 2q− 54 q,1 + ˙B02,1) ≤ Cν ε 1 4− 1 2qIε 0+ kP~uε0− ~v0k ˙ B 5 2q− 54 p,1 + ˙B2,10 . Proof: We have already established (5.5) in the previous subsection. In order to prove the strong convergence of (bε, Q~uε) to 0, it is convenient to rescale the system : defining (b, ~u, j0,~j1) according to (2.8), we see that
(5.10) ( ∂tb + div Q~u = F ∂tQ~u + ∇b = ~G +ς eeC 1/2ε2Q (1+k 4(b))~j1
where the solution (b, ~u, j0,~j1) is given by Theorem 4.1,
F := Q k1(b)div ~u − ~u · ∇b
and G := Q −~~ u · ∇~u + ν (1 + k2(b)) ˜A~u + k3(b)∇b.
Note that the terms F and ~G are exactly those that appear when dealing with the standard low Mach number limit for the barotropic Navier-Stokes equations. Because ~u is bounded in L∞(R+; ˙B n 2−1 2,1 ), ∇~u, in L1(R+; ˙B n 2 2,1), and b, in L∞(R+; ˙B n 2 2,1) ∩ L2(R+; ˙B n 2 2,1), it is easy
to show that F and G are bounded in L1(R+; ˙B
n 2−1
2,1 ). More precisely, we have (see [5] for
more details) k(F, ~G)k L1(R +; ˙B n 2−1 2,1 ) ≤ Ck(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC . Next, according to (4.11), kk4(b)~j1k L1(R +; ˙B n 2−1 2,1 ) ≤ Ck(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC .
As we do not have any control in L1(R+; ˙B
n 2−1
2,1 ) for the low frequencies of the term Q~j1, we
proceed differently in low and high frequencies. More precisely, projecting System (5.10) on frequencies larger than ρε = ρ0
p
εν eC (we use a smooth cut-off of course), and taking advantage of Strichartz estimates for the acoustic wave operator (see e.g. [1], Chap. 10), we get in dimension n ≥ 4, for all p ∈ [pc, +∞],
(5.11) √νk(b, Q~u)kh,ρε e L2(R +; ˙B n p −12 p,1 ) ≤ C k(b0, Q~u0)kh,ρε ˙ B n 2−1 2,1 + k(F, ~G)kh,ρε L1( ˙B n 2−1 2,1 ) + eC1/2ε2kQ(k4(b)~j1)kh,ρε L1( ˙B n 2−1 2,1 ) + ε2Ce1/2kQ~j1k h,ρε L1( ˙B n 2−1 2,1 ) . To handle the low frequencies, we write
∂tb + div Q~u = F, ∂tQ~u + 1 +eαeς 2ε2 e βeγ ∇b = ~G +ς eeC1/2ε2Q(k 4(b)~j1) +ς eeC 1/2ε2Q~j 1.
The left-hand side reduces to the acoustic wave operator with velocity 1. Indeed, it suffices to change b to q1 +αeςe
2ε2
e βeγ
b and to rescale the time variable by a factor q1 +αeςe
2ε2
e βeγ
. Thus we get the same Strichartz estimates as for the previous acoustic wave operator, up to
some harmless constant tending to the usual one when ε goes to 0. If n ≥ 4 then we thus get for all p ∈ [pc, +∞],
√ νk(b, Q~u)k`,ρε e L2(R +; ˙B n p −12 p,1 ) ≤ C k(b0, Q~u0)k`,ρε ˙ B n 2−1 2,1 +k(F, ~G)k`,ρε L1( ˙Bn2−1 2,1 ) + eC1/2ε2kQ(k4(b)~j1)k`,ρε L1( ˙Bn2−1 2,1 ) + eC1/2ε2kQ~j1k`,ρε L1( ˙Bn2−1 2,1 ) . Combining that inequality with (5.11), we thus get
√ νk(b, Q~u)k e L2(R +; ˙B n p −12 p,1 ) ≤ Ck(b0, Q~u0)k˙ B n 2−1 2,1 +k(b, ~u, j0,~j1)k2 E n 2−1 ε,ν, eC + ε e C1/2νk(b, ~u, j0,~j1)kE n 2−1 ε,ν, eC . Setting I0 := k~u0, j0,0,~j1,0k˙ B n 2−1 2,1 + kb0k`,ν −1 ˙ B n 2−1 2,1 + νkb0kh,ν −1 ˙ B n 2 2,1
and using the estimates of The-orem 4.1, we conclude that
(5.12) √νk(b, Q~u)k e L2(R +; ˙B n p −12 p,1 ) ≤ CI0 for all p ∈ [pc, ∞].
Resuming to the original variables gives the announced rate of convergence for (bε, Q~uε) if n ≥ 4.
In the case n = 2, 3 the above arguments have to be slightly modified as less Strichartz inequalities are available. At the same time, the bounds for F, ~G Q~j1 and so on, in
L1(R +; ˙B
n 2−1
2,1 ) remain unchanged.
More precisely, if n = 3 then we get instead of (5.12):
(5.13) ν12− 1 pk(b, Q~u)k e L 2p p−2(R +; ˙B 2 p −12 p,1 ) ≤ CI0 for all p ∈ [2, ∞).
Recall that the bounds of Theorem 4.1 imply that
(5.14) νk(b, Q~u)k`,ν−1 L1(R +; ˙B 5 2 2,1) ≤ CI0.
From interpolation, we discover that k(b, Q~u)k`,ν−1 L2(R +; ˙B 4 q −12 q,1 ) . k(b, Q~u)k`,ν−1 L1(R +; ˙B 5 2 2,1) p+22 k(b, Q~u)k`,ν−1 L 2p p−2(R +; ˙B 2 p −12 p,1 ) p+2p with q := 1 + p/2.
Hence putting (5.13) and (5.14) together gives (5.15) √νk(b, Q~u)k`,ν−1 L2(R +; ˙B 4 q −12 q,1 ) ≤ CI0 for all q ∈ [2, ∞).
To handle high frequencies, we just have to notice that, because 3/q ≥ 4/q − 1/2 for q ≥ 2, we have the following chain of inequalities:
(5.16) k(b, Q~u)kh,ν−1 ˙ B 4 q −12 q,1 . ν12− 1 qk(b, Q~u)kh,ν −1 ˙ B 3 q q,1 . ν12− 1 qk(b, Q~u)kh,ν −1 ˙ B 3 2 2,1 . Remember that Theorem 4.1 yields
√ νk(b, Q~u)kh,ν−1 L2(R +; ˙B 3 2 2,1) ≤ CI0
which, together with (5.15) and (5.16) implies, if ν ≤ ν0, that √ νk(b, Q~u)k L2(R +; ˙B 4 q −12 q,1 ) ≤ CI0 for all q ∈ [2, ∞), whence the desired result of convergence for (bε, Q~uε), after rescaling.
Let us finally bound (b, Q~u) in the case n = 2. Then Strichartz estimates combined with the above bounds in L1(R+; ˙B2,10 ) imply that for all p ∈ [2, +∞],
(5.17) k(b, Q~u)k e L 4p p−2(R +; ˙B 3 2p− 34 p,1 ) + k(b, Q~u)k`,ν−1 L1(R +; ˙B2,12 ) ≤ CI0. From interpolation, we discover that
k(b, Q~u)k`,ν−1 L2(R +; ˙B 5 2q−14 q,1 ) . k(b, Q~u)k`,νL1(R−1 +; ˙B2,12 ) p+2 3p+2 k(b, Q~u)k`,ν −1 L 4p p−2(R +; ˙B 3 2p− 34 p,1 ) 2p 3p+2 with q := (6p + 4)/(p + 6) ∈ [2, 6]. Therefore (5.17) implies that (5.18) √νk(b, Q~u)kh,ν−1 L2(R +; ˙B 5 2q− 14 q,1 ) ≤ CI0 for all q ∈ [2, 6].
Next, because q ≥ 2, we have k(b, Q~u)kh,ν−1 ˙ B 5 2q− 14 q,1 . ν14− 1 2qk(b, Q~u)kh,ν −1 ˙ B 2 q q,1 . ν14− 1 2qk(b, Q~u)kh,ν −1 ˙ B1 2,1 . Then combining with the bounds of Theorem 4.1 eventually leads to
√ νk(b, Q~u)k L2(R +; ˙B 5 2q− 14 q,1 ) ≤ CI0 for all q ∈ [2, 6],
whence the desired result of convergence for (bε, Q~uε), after rescaling.
Let us finally go to the proof of the convergence of P~uε. Subtracting (5.2) from the velocity equation (projected onto divergence free vector fields) of (2.1), and setting δ~vε := P~uε− v, we get (5.19) ∂tδ~vε− µ ¯ %∆δ~v ε= −P(P~uε· ∇δ~vε+ δ~vε· ∇~v + ~uε· ∇Q~uε+ Q~uε· ∇P~uε) +1 ¯ %P k2(eεb ε)A~uε + σa+σs n ¯% P (1 + k4(eεb ε))~jε 1.
Except for the last term, the proof is exactly the same as for the barotropic case (see [1, 5]). More precisely, the first two terms of the r.h.s. satisfy linear estimates (with small coefficient) with respect to δ~vε, while the next three terms are expected to be of order εα for some α > 0, owing to the convergence of (bε, Q~uε) and uniform estimates. Let us give more details. To simplify the presentation, we do not keep track of the dependency of the estimates, on ν.
Let us first consider the case n ≥ 4. Then we have for all p ∈ [pc, +∞]
k~uε·∇Q~uε+ Q~uε·∇P~uεk L1(R +; ˙B n p −32 p,1 ) . k~uεk L2(R +; ˙B n 2 2,1) kQ~uεk L2(R +; ˙B n p −12 p,1 ) . ε1/2(I0ε)2
and, because (5.20) kbεk ˙ B n 2−α 2,1 . εα−1 kbεk`, 1 εν ˙ B n 2−1 2,1 + εkbεkh, 1 εν ˙ B n 2 2,1 for all α ∈ [0, 1], we have kk2(eεbε)A~uεk L1(R +; ˙B n p −32 p,1 ) . kk2(εbe ε)A~uεk L1(R +; ˙B n 2− 32 2,1 ) . keεbεk L∞(R +; ˙B n 2− 12 2,1 ) k∇2~uεk L1(R +; ˙B n 2−1 2,1 ) . ε1/2(I0ε)2.
Keeping in mind the regularity estimates for the heat equation, it seems thus natural to bound δ~vε in L1(R+; ˙B n p+ 1 2 p,1 ) ∩ L∞(R+; ˙B n p− 1 2
p,1 ). In contrast with the nonradiative
situa-tion however, this is not possible owing to the term P~jε1 which is only damped, and not dispersed. More precisely, we have
P (1 + k4(eεbε))~j1ε = P~jε
1+ P(k4(εbe
ε)~jε 1).
By following the argument leading to (4.11) and using the estimates supplied by Theorem 2.1, it is easy to bound the last term as follows:
(5.21) kP(k4(εbe ε)~jε 1)kL1(R +; ˙B n 2−1 2,1 ) ≤ C eC−1/2εI0ε.
For the first term, we just have to notice that P~j1ε= P~jε1, hence (5.21) is also fulfilled. So finally the r.h.s. of (5.19) has to be bounded in L1(R+; ˙B
n p− 3 2 p,1 + ˙B n 2−1 2,1 ) and thus δ~vε, in L∞(R+; ˙B n p− 3 2 p,1 + ˙B n 2−1 2,1 ) ∩ L1(R+; ˙B n p+ 1 2 p,1 + ˙B n 2+1
2,1 ) (if δ~v0ε satisfies suitable assumptions
of course). Now, because kP~uε· ∇δ~vε+ δ~vε· ∇~vk L1(R +; ˙B n p −32 p,1 + ˙B n 2−1 2,1 ) . kP~uε, ~vk L2(R +; ˙B n 2 2,1) kδ~vεk L2(R +; ˙B n p −12 p,1 + ˙B n 2 2,1) ≤ CIε 0kδ~vεk L2(R +; ˙B n p −12 p,1 + ˙B n 2 2,1)
it is easy to conclude to (5.7), if I0ε is small enough. In the three-dimensional case, we claim that
δ~vε→ 0 in L∞(R+; ˙B 4 q− 3 2 q,1 + ˙B 1 2 2,1) ∩ L1(R+; ˙B 4 q− 3 2 q,1 + ˙B 1 2 2,1), for all q ∈ [2, ∞).
To achieve this result, it suffices to prove suitable estimates for the r.h.s. of (5.19) in L1(R+; ˙B 4 q− 3 2 q,1 + ˙B 1 2 2,1). We have k~uε·∇Q~uε+ Q~uε·∇P~uεk L1(R +; ˙B 4 q −32 q,1 ) . k~uεk L2(R +; ˙B 3 2 2,1) kQ~uεk L2(R +; ˙B 4 q −12 q,1 ) . ε12− 1 qIε 0 kP~uε· ∇δ~vε+ δ~vε· ∇~vk L1(R +; ˙B 4 q −32 q,1 + ˙B 1 2 2,1) . kP~uε, ~vk L2(R +; ˙B 3 2 2,1) kδ~vεk L2(R +; ˙B 4 q −12 q,1 + ˙B 3 2 2,1) ≤ CIε 0kδ~vεk L2(R +; ˙B 4 q −12 q,1 + ˙B 3 2 2,1)
and, thanks to (5.20) with α = 1/2 − 1/q, kk2(eεb ε)A~uεk L1(R +; ˙B 4 q −32 q,1 ) . kk2(eεb ε)A~uεk L1(R +; ˙B 1 q 2,1) . keεbεk L∞(R +; ˙B 1+ 1q 2,1 ) k∇2~uεk L1(R +; ˙B 1 2 2,1) . ε12− 1 qIε 0.
Keeping (5.21) in mind, we can thus conclude to (5.8).
In the two-dimensional case, we just have to prove suitable estimates for the r.h.s. of (5.19) in L1(R+; ˙B 5 2q− 5 4 q,1 + ˙B2,10 ). We have k~uε·∇Q~uε+ Q~uε·∇P~uεk L1(R +; ˙B 5 2q− 54 q,1 ) . k~uεkL2(R +; ˙B2,11 )kQ~u εk L2(R +; ˙B 5 2q− 14 q,1 ) . ε14− 1 2qIε 0 kP~uε· ∇δ~vε+ δ~vε· ∇~vk L1(R +; ˙B 5 2q− 54 q,1 + ˙B 1 2 2,1) . kP~uε, ~vkL2(R +; ˙B2,11 )kδ~v εk L2(R +; ˙B 5 2q− 14 q,1 + ˙B2,11 ) ≤ CIε 0kδ~vεk L2(R +; ˙B 5 2q− 14 q,1 + ˙B2,11 )
and, thanks to (5.20) with α = 1/4 − 1/(2q), kk2(εbeε)A~uεk L1(R +; ˙B 5 2q− 54 q,1 ) . kk2(eεb ε)A~uεk L1(R +; ˙B 1 2q− 14 2,1 ) . keεbεk L∞(R +; ˙B 3 4+ 12q 2,1 ) k∇2~uεk L1(R +; ˙B2,10 ) . ε14− 1 2qIε 0.
Keeping (5.21) in mind, we can thus conclude to (5.8).
Appendix A
We here provide exponential decay estimates for the following linear system of ordinary differential equations with X and Y complex-valued:
(A.1)
∂tX + aρY − bρ2X = A
∂tY − cρX + dρ2Y = B.
Above, ρ stands for a given nonnegative small parameter and a, b, c and d are four real numbers satisfying the stability condition
(A.2) a > 0, c > 0 and d − b > 0.
Even though (A.1) may be solved explicitly, thus giving the desired (and optimal) decay results, we here aim at recovering such results by means of an energy type method. We start with the following identities:
1 2
d dt c|X|
2+ a|Y |2 − bcρ2|X|2+ adρ2|Y |2 = Re (cA ¯X + aB ¯Y ),
d
dtRe (X ¯Y ) + aρ|Y |
2− cρ|X|2+ (d − b)ρ2Re (X ¯Y ) = Re (B ¯X + A ¯Y )
from which we easily get for any real number η, (A.3) 1 2 d dt c|X| 2+ a|Y |2− 2ρηRe (X ¯Y ) + (η − b)cρ2|X|2 + (d − η)aρ2|Y |2+ η(b − d)ρ3Re (X ¯Y ) = Re cA ¯X + aB ¯Y − 2ρη(B ¯X + A ¯Y ).
Choosing η so that d − η = η − b, that is to say η := b+d2 , we discover that the Lyapunov functional L2ρ:= c|X|2+ a|Y |2− ρ(d + b)Re (X ¯Y ) satisfies
(A.4) 1 2 d dtL 2 ρ+ d − b 2 ρ2 c|X|2+ a|Y |2 + b 2− d2 2 ρ3Re (X ¯Y ) = Re cA ¯X + aB ¯Y − ρ(b+d)(B ¯X +A ¯Y ). Now, from the observation that
|Re (X ¯Y )| ≤ 1
2√ac c|X|
2+ a|Y |2,
we gather that, whenever ρ ≤
√ ac |b+d|, we have b2− d2 2 ρ3Re (X ¯Y ) ≤ d − b 4 ρ2 c|X|2+ a|Y |2 and (A.5) 1 2 c|X| 2+ a|Y |2 ≤ L2 ρ≤ 3 2 c|X| 2+ a|Y |2.
If A ≡ B ≡ 0 then resuming to (A.4) leads to
(A.6) d dtL 2 ρ+ d − b 3 L2 ρ≤ 0
and thus, for any t ≥ 0,
(A.7) Lρ(t) ≤ e−(d−b6 )ρ
2t
Lρ(0), which yields, according to (A.5),
(A.8) pc|X(t)|2+ a|Y (t)|2 ≤√3p c|X(0)|2+ a|Y (0)|2e− d−b6 ρ2t for ρ ≤ √ ac |b + d|· For nonzero source terms, Inequality (A.7) leads through Duhamel formula to
(A.9) pc|X(t)|2+ a|Y (t)|2≤√3 e−(d−b6 )ρ 2tp c|X(0)|2+ a|Y (0)|2 + Z t 0 e(d−b6 )τpc|A|2+ a|B|2dτ · References
[1] H. Bahouri, J.-Y. Chemin and R. Danchin: Fourier Analysis and Nonlinear Partial Differential Equa-tions, Grundlehren der mathematischen Wissenschaften, 343, Springer (2011).
[2] X. Blanc and B. Despr´es: Numerical methods for inertial confinement fusion, Lecture Notes of CEMRACS-10 Summer School: smai.emath.fr/cemracs/cemracs10/fr courses.html.
[3] C. Buet and B. Despr´es, Asymptotic analysis of fluid models for the coupling of radiation and hydro-dynamics, J. Quant. Spectroscopy Rad. Transf., 85:385–480, 2004.
[4] S. Chandrasekhar, Radiative transfer. Dover Publications, Inc., New York, 1960.
[5] R. Danchin: Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. ´Ecole Norm. Sup., 35(1), pages 27–75 (2002).
[6] R. Danchin: On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 12(1), 111–128 (2005).
[7] R. Danchin and B. Ducomet: On a simplified model for radiating flows, Journal of Evolution Equations, 14:155–195, 2014.
