Existence of strong solutions with critical regularity to a polytropic model for radiating flows

Abstract

This paper is the continuation of our recent work Danchin and Ducomet (J Evol Equ 14:155–195, 2013) devoted to barotropic radiating flows. We here aim at investigating the more physically relevant situation of polytropic flows. More precisely, we consider a model arising in radiation hydrodynamics which is based on the full Navier–Stokes–Fourier system describing the macroscopic fluid motion, and a P1-approximation (see below) of the transport equation modeling the propagation of radiative intensity. In the strongly under-relativistic situation, we establish the global-in-time existence and uniqueness of solutions with critical regularity for the associated Cauchy problem with initial data close to a stable radiative equilibrium. We also justify the nonrelativistic limit in that context. For smoother (possibly) large data bounded away from the vacuum and more general physical coefficients that may depend on both the density and the temperature, the local existence of strong solutions is shown.

1 Introduction

We consider a model of radiative fluid introduced in [4, 25, 26]. It is governed by the standard field equations of classical continuum fluid mechanics: the compressible Navier–Stokes–Fourier system, describing the evolution of the mass density $\varrho = \varrho (t,x)$, the velocity field ${\vec u}= {\vec u}(t,x)$, and the absolute temperature $\vartheta = \vartheta (t,x)$ as functions of the time t and of the Eulerian spatial coordinate x. In all that follows, we assume the fluid domain to be the whole space $\mathbb {R}^n$.

The effect of radiation is incorporated in a scalar quantity: the radiative intensity $I = I(t,x, {\vec \omega }, \nu )$, depending on the direction vector $\vec {\omega } \in {{\mathcal {S}}}^{n-1}$, where ${{\mathcal {S}}}^{n-1}$ denotes the unit sphere of $\mathbb {R}^n,$ and on the frequency $\nu \ge 0$. The collective effect of radiation is expressed in terms of integral means (with respect to the variables ${\vec \omega }$ and $\nu$) of quantities depending on I. In particular, the radiation energy $S_E$, the radiation momentum $\vec {F}_R$ and the radiation tensor ${{\mathbb {P}}}_R$ are given by

$$\begin{aligned} E_R(t,x) \!= \!\frac{1}{c} \int _{{{\mathcal {S}}}^{n-1}} \int _0^\infty I(t,x,\nu , {\vec \omega }) \, \mathrm{d}\nu \,\mathrm{d} {\vec \omega },\quad \vec {F}_R(t,x) \!=\! \int _{{{\mathcal {S}}}^{n-1}} \int _0^\infty {\vec \omega }\ I(t,x,\nu , {\vec \omega }) \, \mathrm{d}\nu \,\mathrm{d} {\vec \omega }, \end{aligned}$$

and

$$\begin{aligned} {{\mathbb {P}}}_R(t,x) = \frac{1}{c} \int _{{{\mathcal {S}}}^{n-1}} \int _0^\infty {\vec \omega }\otimes {\vec \omega }\ I(t,x,\nu , {\vec \omega }) \, \mathrm{d}\nu \,\mathrm{d} {\vec \omega }, \end{aligned}$$

where c is the velocity of light.

The time evolution of I is governed by a transport equation with a source term depending on the temperature $\vartheta$, while the coupling to the macroscopic motion of the fluid is achieved by extra source terms in the momentum equation evaluated by means of I and $\vartheta$.

The corresponding system of equations (that is derived in, e.g., [4, 25, 26]) reads:

$$\begin{aligned} \partial _t \varrho + \, \mathrm{div}_x\, (\varrho {\vec u}) = 0 \ \text{ in }\ (0,T) \times \mathbb {R}^n; \end{aligned}$$

(1)

$$\begin{aligned}&\partial _t (\varrho {\vec u}) + \, \mathrm{div}_x\, (\varrho {\vec u}\otimes {\vec u}) + \nabla _xp(\varrho ,\vartheta ) = \, \mathrm{div}_x\, \mathbb {T}- {\vec S} _F \ \text{ in } \ (0,T) \times \mathbb {R}^n; \end{aligned}$$

(2)

$$\begin{aligned}&\partial _t \left( \varrho \left( \frac{1}{2} |{\vec u}|^2 + e(\varrho ,\vartheta ) \right) \right) + \, \mathrm{div}_x\, \left( \varrho \left( \frac{1}{2} |{\vec u}|^2 + e(\varrho ,\vartheta ) \right) {\vec u}\right) \nonumber \\&\quad + \, \mathrm{div}_x\, \Big ( p {\vec u}+ {\vec q} - \mathbb {T}{\vec u}\Big ) = - S_E \ \text{ in } \ (0,T) \times \mathbb {R}^n, \end{aligned}$$

(3)

$$\begin{aligned}&\frac{1}{c} \partial _t I + {\vec \omega }\cdot \nabla _xI = S \ \text{ in } \ (0,T) \times \mathbb {R}^n \times (0, \infty ) \times {{\mathcal {S}}}^{n-1}. \end{aligned}$$

(4)

The notations $p=p(\varrho ,\vartheta )$ and $e=e(\varrho ,\vartheta )$ designate the gaseous pressure and internal energy, respectively, and $\mathbb {T}$ stands for the viscous stress tensor determined by Newton’s rheological law

$$\begin{aligned} \mathbb {T}= \mu \left( \nabla _x{\vec u}+ \nabla _x^t {\vec u}\right) + \lambda \ \, \mathrm{div}_x\, {\vec u}\ {{\mathbb {I}}}_n, \end{aligned}$$

(5)

where $\mu >0$ is the shear viscosity coefficient and $\lambda =\zeta -\frac{1}{n}\ \mu .$ Here $\zeta \ge 0$ is the bulk viscosity coefficient. We assume the heat flux ${\vec q}$ to be given by Fourier’s law

$$\begin{aligned} {\vec q} = - \kappa \nabla _x\vartheta , \end{aligned}$$

(6)

with a strictly positive heat conductivity coefficient $\kappa .$ In the applications, all those coefficients may depend on both $\varrho$ and $\vartheta .$ However they will be taken temperature independent in the part of the paper dedicated to the global existence issue.

In order to simplify the presentation, we assume the internal energy e to be linear (and increasing) with respect to $\vartheta ,$ namely

$$\begin{aligned} \partial _\vartheta e =C_v\quad \hbox {for some constant }\ C_v>0. \end{aligned}$$

(7)

This leads, through Maxwell’s law:

$$\begin{aligned} \varrho ^2\partial _\varrho e=p-\vartheta \partial _\vartheta p \end{aligned}$$

(8)

to a pressure law depending linearly on $\vartheta ,$ namely

$$\begin{aligned} p(\varrho ,\vartheta )=\pi _0(\varrho )+\vartheta \pi _1(\varrho ), \end{aligned}$$

(9)

where $\pi _0$ and $\pi _1$ are smooth functions.

Finally, the radiative sources in the right-hand sides of (2) and (3) are given by

$$\begin{aligned} S_E = \int _{{{\mathcal {S}}}^{n-1}} \int _0^\infty S(\cdot , \nu , {\vec \omega }) \ \mathrm{d} \nu \ \mathrm{d} {\vec \omega }\ \ \text{ and } \ \ {\vec S}_F = \frac{1}{c} \int _{{{\mathcal {S}}}^{n-1}} \int _0^\infty {\vec \omega }S(\cdot , \nu , {\vec \omega }) \ \mathrm{d} \nu \ \mathrm{d} {\vec \omega }, \end{aligned}$$

(10)

with

$$\begin{aligned} S = S_{a,e} + S_s, \end{aligned}$$

(11)

and where

$$\begin{aligned} S_{a,e} = \sigma _a \Big ( B(\nu , \vartheta ) - I \Big ), \quad S_s = \sigma _s \left( \frac{1}{|{{\mathcal {S}}}^{n-1}|} \int _{{{\mathcal {S}}}^{n-1}} I(\cdot , {\vec \omega }) \ \mathrm{d} {\vec \omega }- I \right) . \end{aligned}$$

(12)

Above, $|{{\mathcal {S}}}^{n-1}|$ stands for the measure of the $(n-1)$-sphere, and $B(\nu , \vartheta ) \ge 0$ for the equilibrium thermal distribution of radiative intensity. A physically relevant example of function B is

$$\begin{aligned} B(\nu , \vartheta )=\frac{2h\nu ^3}{c^2}\ \frac{1}{e^{\frac{h\nu }{k\vartheta }}-1}, \end{aligned}$$

(13)

where the positive real numbers h and k are the Planck and Boltzmann’s constants. A direct computation gives

$$\begin{aligned} \int _0^{\infty }\int _{{{\mathcal {S}}}^{n-1}}B(\nu , \vartheta )\ \mathrm{d} {\vec \omega }\ d\nu =\frac{2\pi ^4}{15} \frac{k^4\vartheta ^4}{c^2 h^3}\, |{{\mathcal {S}}}^{n-1}|, \end{aligned}$$

where $|{{\mathcal {S}}}^{n-1}|=2\frac{\pi ^{n/2}}{\Gamma (n/2)}$ is the area of the $(n-1)$-sphere. In the present work however, it will be possible to consider more general laws. Finally, the absorption coefficient $\sigma _a = \sigma _a(\nu , \varrho ,\vartheta )$ and the scattering coefficient $\sigma _s = \sigma _s (\nu , \varrho ,\vartheta )$ are smooth functions that are nonnegative in the applications.

System (1–4) is supplemented with the initial conditions:

$$\begin{aligned} \varrho (0,x)=\varrho ^0(x),\quad {\vec u}(0,x)={\vec u}^0(x),\quad \vartheta (0,x)=\vartheta ^0(x)\ \ \text{ for } \ x \in \mathbb {R}^n, \end{aligned}$$

(14)

and

$$\begin{aligned} I (0,x,\nu , {\vec \omega }) = I^0 (x,\nu , {\vec \omega }) \quad \text{ for } \ x \in \mathbb {R}^n, \ {\vec \omega }\in {{\mathcal {S}}}^{n-1},\ \nu > 0. \end{aligned}$$

(15)

The boundary conditions at infinity (e.g., convergence to some stable constant state) will be implicitly given by the functional framework we shall work in.

System (1–15) has been investigated recently in astrophysics and laser applications (in the relativistic and inviscid case) by Lowrie, Morel and Hittinger [24], Buet and Després [3], with a special attention to asymptotic regimes. The global existence result of weak solutions has been established by Ducomet et al. [15]. The reader may also refer to Dubroca and Feugeas [16], Levermore [20], Lin [22], and Lin et al. [23] for related theoretical or numerical issues.

Various approximations of the radiative transfer equation have been developed in the recent past [2] for numerical purposes. In the present paper, we are going to consider the so-called P1 approximation [17] consisting in expanding I in terms of the angular variable ${\vec \omega }$ and keeping only the first two terms in the expansion. More precisely, we choose the ansatz

$$\begin{aligned} I=I_0+\vec {I}_1\cdot {\vec \omega }, \end{aligned}$$

(16)

where $I_0$ and $\vec {I}_1$ do not depend on ${\vec \omega }$.

Our work aims at extending the existence theory in critical spaces developed in [12] for the radiative barotropic Navier–Stokes system, to the more physically relevant polytropic situation. We keep on considering the P1 approximation and the so-called grey case where the transport coefficients are pure positive constants independent of the frequency $\nu$.

The rest of the paper unfolds as follows. In the next section, we write out the system for the P1 approximation of System (1–15) in the grey case, and state our main results: first local-in-time well posedness either for smooth data and quite general assumptions on the coefficients of the system, or in the “critical regularity framework” but for coefficients depending only on the density, and linear equilibrium distribution function; second, global existence for small perturbations of a stable constant equilibrium in the strongly under-relativistic situation. Section 3 is devoted to the spectral analysis of the linearized equations about a constant reference state. We shall in particular exhibit a necessary and sufficient linear stability condition in the low-frequency regime (which is fulfilled in the strongly relativistic regime), and prove optimal global-in-time estimates for the linearized equations. The next section is devoted to the proof of similar estimates for the so-called paralinearized system (1–15). Those estimates are the key to our global existence result and to the rigorous justification of the nonrelativistic limit (see Sect. 5). The last section is devoted to the proof of the local existence result for large data and rather general coefficients. We postpone in Appendix some basic material related to Fourier analysis and Besov spaces.

2 Main results

We focus on the “grey” case where the transport coefficients are independent of the frequency $\nu ,$ and assume that the radiative quantities I,  B and S have all been integrated on frequencies. Keeping the ansatz (16) for I,  denoting by b the integrated thermal distribution and taking advantage of the identity

$$\begin{aligned} \int _{{{\mathcal {S}}}^{n-1}} {\vec \omega }\otimes {\vec \omega }\ \mathrm{d} {\vec \omega }=\frac{1}{n}\ {{\mathbb {I}}}_n, \end{aligned}$$

we see that one may replace the transport equation (4) for I by the following system for $(I_0,{\vec I}_1)$:

$$\begin{aligned}&\frac{1}{c} \partial _t I_0 + \frac{1}{n} \, \mathrm{div}_x\, {\vec I}_1 = \sigma _a(b-I_0), \end{aligned}$$

(17)

$$\begin{aligned}&\frac{1}{c} \partial _t {\vec I}_1+ \nabla _xI_0 = -(\sigma _a+\sigma _s){\vec I}_1. \end{aligned}$$

(18)

Here we used the fact that the averaged radiative source is given by

$$\begin{aligned} s(\varrho ,\vartheta ,I)\,{:=}\,\sigma _a(b-I)+\sigma _s(<I>-I)\quad \hbox {with}\quad <I>\,{:=}\, \frac{1}{|{{\mathcal {S}}}^{n-1}|}\int _{{{\mathcal {S}}}^{n-1}}I(t,x, {\vec \omega }) \ \mathrm{d} {\vec \omega }. \end{aligned}$$

Next, because the integrated radiative energy and momentum are given by

$$\begin{aligned} s_E=\int _{{{\mathcal {S}}}^{n-1}} s \ \mathrm{d} {\vec \omega }\quad \hbox {and}\quad {\vec s}_F=\int _{{{\mathcal {S}}}^{n-1}} {\vec \omega }\, s \ \mathrm{d} {\vec \omega }, \end{aligned}$$

we get, remembering (7) and (8),

$$\begin{aligned} \partial _t \varrho + {\vec u}\cdot \nabla _x\varrho +\varrho \ \, \mathrm{div}_x\, {\vec u}= 0, \end{aligned}$$

(19)

$$\begin{aligned}&\varrho (\partial _t {\vec u}+ {\vec u}\cdot \nabla _x{\vec u}) +\nabla _xp = 2\, \mathrm{div}_x\, (\mu D_x{\vec u})+\nabla _x(\lambda \, \mathrm{div}_x\, {\vec u}) - \Bigl (\frac{\sigma _a+\sigma _s}{n}\Bigr ) {\vec I}_1, \end{aligned}$$

(20)

$$\begin{aligned}&\varrho C_v(\partial _t \vartheta + {\vec u}\cdot \nabla _x\vartheta ) -\, \mathrm{div}_x\, (\kappa \nabla _x\vartheta ) \nonumber \\&\quad = 2\mu D_x{\vec u}:D_x{\vec u}+\lambda (\, \mathrm{div}_x\, {\vec u})^2 -\vartheta \,\partial _\vartheta p\,\, \mathrm{div}_x\, {\vec u}-\sigma _a(b(\vartheta )-I_0) + \Bigl (\frac{\sigma _a+\sigma _s}{n}\Bigr ){\vec I}_1\cdot {\vec u},\nonumber \\ \end{aligned}$$

(21)

where $(D_x{\vec u})_{ij}\,{:=}\,\frac{1}{2}(\partial _iu_j+\partial _ju_i)$ and p is given by (9).

Recall that a physically relevant example of thermal distribution B is given by (13) which, after suitable renormalization, recasts in

$$\begin{aligned} B(\nu ,\vartheta )=\frac{15}{|{{\mathcal {S}}}^{n-1}|\,\pi ^4}\ \frac{1}{e^{\frac{\nu }{\vartheta }}-1}\cdot \end{aligned}$$

In this new setting, the integral on frequencies is $b(\vartheta )= \vartheta ^4$. Our approach enables us to consider much more general functions b,  though.

2.1 Local results

The local-in-time well-posedness theory does not require much assumptions on the coefficients nor on the functions in System (17–21) (apart from enough smoothness). We do not even need the radiation coefficients to be positive. In fact, we will establish in any dimension $n\ge 1$ the following basic local-in-time existence result:¹

Theorem 2.1

Assume that b depends smoothly on the temperature, and that $\lambda ,$ $\mu ,$ $\kappa ,$ $\sigma _a$ and $\sigma _s$ are smooth functions of the density and of the temperature. If in addition

$$\begin{aligned} \mu >0,\quad \lambda +2\mu >0\ \mathrm{and } \kappa >0, \end{aligned}$$

(22)

then for any data $\varrho ^0,$ ${\vec u}^0,$ $\vartheta ^0,$ $I_0^0$ and ${\vec I}^0_1$ satisfying

• (1) $\varrho ^0$ and $\vartheta ^0$ are bounded, and bounded away from 0,

• (2) ${\vec I}_1^0,$ $\nabla \varrho ^0$ and ${\vec u}^0$ are in $B^{\frac{n}{2}}_{2,1}$,

• (3) $\Theta ^0\,{:=}\,\vartheta ^0-{\bar{\vartheta }}$ is in $B^{\frac{n}{2}}_{2,1}$ for some positive constant ${\bar{\vartheta }},$

• (4) $I^0_0=b({\bar{\vartheta }})+j^0_0$ with $j_0^0$ in $B^{\frac{n}{2}}_{2,1}$,

there exists $T>0$ so that System (17–21) with pressure law (9) and data $(\varrho ^0,{\vec u}^0,\vartheta ^0, I^0_0,{\vec I}_1^0)$ admits a unique local solution $(\varrho ,{\vec u},\vartheta ,I_0,{\vec I}_1)$ on $[0,T]\times \mathbb {R}^n$ with

• (1) $\vartheta ^{\pm 1}$ and $\varrho ^{\pm 1}$ in ${{\mathcal {C}}}_b\left( [0,T]\times \mathbb {R}^n\right)$ and $\nabla \varrho \in {{\mathcal {C}}}\left( [0,T];B^{\frac{n}{2}}_{2,1}\right) ,$

• (2) ${\vec u}$ and $\Theta \,{:=}\,\vartheta -{\bar{\vartheta }}$ in ${{\mathcal {C}}}\left( [0,T];B^{\frac{n}{2}}_{2,1}\right) \cap L^1(0,T;B^{\frac{n}{2}+2}_{2,1})$,

• (3) $j_0=I_0-b({\bar{\vartheta }})$ and ${\vec I}_1$ in ${{\mathcal {C}}}\left( [0,T];B^{\frac{n}{2}}_{2,1}\right) .$

Remark 2.1

Resorting to more elaborate arguments (like Proposition 6 of [9]), it should be possible to consider density with the same regularity as the other data, instead of one more derivative. In fact, having more regular density is helpful in parabolic estimates as the second-order terms have coefficients depending on both $\varrho$ and $\vartheta ,$ and no gain of smoothness is obtained for $\varrho ,$ through the evolution. Of course, it is possible to propagate higher Besov (or Sobolev) regularity, provided it is related to the $L^2$ space.

Although a bit technical, the proof of Theorem 2.1 relies on completely standard arguments: basic estimates for the transport equation (as regards the density), parabolic equations or systems (temperature and velocity) and hyperbolic symmetric systems with constant coefficients (radiative equations). High regularity is needed just to handle the dependency of the coefficients of the system on both $\varrho$ and $\vartheta .$ Roughly speaking, composition lemmas are nicer in spaces embedded in the set of bounded functions (see the Appendix), which in the Besov spaces scale $B^s_{2,1},$ is equivalent to $s\ge \frac{n}{2}\cdot$

Let us now go to the case where the coefficients depend only on $\varrho ,$ and where b depends linearly on $\vartheta .$ Then the critical regularity framework becomes relevant for solving System (17–21). As in the nonradiative case studied in [6, 7], critical norms for $(\varrho ,{\vec u},\vartheta )$ are invariant for all $\ell >0$ by the following scaling transformation:

$$\begin{aligned} \bigl (\varrho (t,x),{\vec u}(t,x),\vartheta (t,x)\bigr )\leadsto \bigl (\varrho (\ell ^2 t,\ell x),\ell {\vec u}(\ell ^2 t,\ell x),\ell ^2\vartheta (\ell ^2 t,\ell x)\bigr ) \end{aligned}$$

(23)

which leaves the density, velocity and temperature equations invariant, up to a suitable change of the pressure law.

Although the radiative unknowns do not have any natural scaling invariance, the coupling between hydrodynamic and radiative unknowns forces us to work at the same level of regularity as for the velocity.

To be more specific, let us fix some reference positive constant density ${\bar{\varrho }}$ and temperature ${\bar{\vartheta }},$ and set $\bar{b}\,{:=}\,b({\bar{\vartheta }}).$ Keeping in mind that the pressure is given by (9), the system for $a\,{:=}\,\varrho -{\bar{\varrho }},$ ${\vec u},$ $\Theta \,{:=}\,\vartheta -{\bar{\vartheta }},$ $j_0\,{:=}\,I_0-\bar{b}$ and ${\vec j}_1\,{:=}\,{\vec I}_1$ reads:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a + {\vec u}\cdot \nabla _xa+({\bar{\varrho }}+a) \, \mathrm{div}_x\, {\vec u}= 0,\\ ({\bar{\varrho }}+a)(\partial _t {\vec u}+ {\vec u}\cdot \nabla _x{\vec u}) + \nabla _x\bigl (p({\bar{\varrho }}+a,{\bar{\vartheta }}+\Theta )\bigr )\\ \quad = 2\, \mathrm{div}_x\, (\mu D_x{\vec u})+\nabla _x(\lambda \, \mathrm{div}_x\, {\vec u}) - \bigl (\frac{\sigma _a+\sigma _s}{n}\bigr ) {\vec j}_1,\\ C_v({\bar{\varrho }}+a)(\partial _t \vartheta + {\vec u}\cdot \nabla _x\vartheta ) -\, \mathrm{div}_x\, (\kappa \nabla _x\vartheta )= 2\mu D_x{\vec u}:D_x{\vec u}\\ \quad +\,\lambda (\, \mathrm{div}_x\, {\vec u})^2 -\vartheta \,\pi _1({\bar{\varrho }}+a)\,\, \mathrm{div}_x\, {\vec u}-\sigma _a(b(\vartheta )-\bar{b}-j_0)+ \bigl (\frac{\sigma _a+\sigma _s}{n}\bigr ){\vec j}_1\cdot {\vec u},\\ \frac{1}{c} \partial _t j_0 + \frac{1}{n} \, \mathrm{div}_x\, {\vec j}_1 = \sigma _a(b-\bar{b}-j_0),\\ \frac{1}{c} \partial _t {\vec j}_1+ \nabla _xj_0 = -(\sigma _a+\sigma _s){\vec j}_1.\end{array}\right. \end{aligned}$$

(24)

To state our results, let us associate to any tempered distribution z its low- and high-frequency parts denoted by $z^\ell$ and $z^h,$ respectively [see the definition in (136)]. Then we have:

Theorem 2.2

Let $n\ge 3.$ Assume that the data $a^0,$ ${\vec u}^0,$ $\vartheta ^0,$ $j_0^0$ and ${\vec j}_0^1$ satisfy

$$\begin{aligned} (a^0)^\ell , \, {\vec u}^0,\, (\vartheta ^0)^\ell , \, j_0^0, \, {\vec j}_1^0 \in \dot{B}^{\frac{n}{2}-1}_{2,1},\quad (a^0)^h\in \dot{B}^{\frac{n}{2}}_{2,1} \ \hbox { and }\ (\vartheta ^0)^h\in \dot{B}^{\frac{n}{2}-2}_{2,1}.\end{aligned}$$

If in addition ${\bar{\varrho }}+a^0$ is bounded away from 0 then there exists $T>0$ such that System (24) with data $(a^0,{\vec u}^0,\vartheta ^0,j^0_0,{\vec j}_1^0)$ and pressure law (9) admits a unique local solution $(a,{\vec u},\vartheta ,j_0,{\vec j}_1)$ on $[0,T]\times \mathbb {R}^n$ with

• (1) $a^\ell \in {{\mathcal {C}}}\left( [0,T];\dot{B}^{\frac{n}{2}-1}_{2,1}\right) ,$ $a^h\in {{\mathcal {C}}}\left( [0,T];\dot{B}^{\frac{n}{2}}_{2,1}\right) ,$ and $1+a$ bounded away from 0, 

• (2) ${\vec u}\in {{\mathcal {C}}}\left( [0,T];\dot{B}^{\frac{n}{2}-1}_{2,1}\right) \cap L^1\left( 0,T;\dot{B}^{\frac{n}{2}+1}_{2,1}\right)$,

• (3) $\vartheta ^\ell \in {{\mathcal {C}}}\left( [0,T];\dot{B}^{\frac{n}{2}-1}_{2,1}\right) \cap \, L^1\left( 0,T;\dot{B}^{\frac{n}{2}+1}_{2,1}\right)$ and $\vartheta ^h\in {{\mathcal {C}}}\left( [0,T];\dot{B}^{\frac{n}{2}-2}_{2,1}\right) \cap \, L^1\left( 0,T;\dot{B}^{\frac{n}{2}}_{2,1}\right) ,$

• (4) $j_0$ and ${\vec j}_1$ in ${{\mathcal {C}}}\left( [0,T];\dot{B}^{\frac{n}{2}-1}_{2,1}\right) .$

Let us emphasize that in contrast with the nonradiative case studied in [6], whether one may adapt the above statement to the critical $L^p$ framework (that is to critical Besov spaces $\dot{B}^s_{p,1}(\mathbb {R}^n)$) is unclear. The reason why is that the unknowns $(j_0,{\vec j}_1)$ satisfy a symmetric hyperbolic system (the coupling with the other two equations being lower order), and solving hyperbolic systems in spaces which are not related to $L^2$ is not possible in general.

2.2 Global results

Let us now present our global well-posedness result in the critical regularity framework, in the case where the coefficients of the system depend only on the density. To find out sufficient conditions for the global well posedness, it is convenient to work with the non-dimensional form of System (17–21). To this end, following [2], we fix

$$\begin{aligned} L_{ref}, \ T_{ref}, \ U_{ref}, \ \varrho _{ref}, \ \vartheta _{ref}, \ p_{ref}, \ e_{ref}, \ \mu _{ref}=\lambda _{ref}, \ \kappa _{ref}, \end{aligned}$$

some reference hydrodynamical quantities (length, time, velocity, density, temperature, pressure, energy, viscosity, conductivity), and

$$\begin{aligned} I_{ref}, \ \sigma _{a,ref}, \ \sigma _{s,ref}, \end{aligned}$$

the reference radiative quantities (radiative intensity, absorption and scattering coefficients). Then we put

$$\begin{aligned} {\widehat{x}}= & {} \frac{x}{L_{ref}},\quad {\widehat{t}}=\frac{t}{T_{ref}}, \quad {\widehat{\varrho }}=\frac{\varrho }{\varrho _{ref}},\quad {\widehat{\vartheta }}=\frac{\vartheta }{\vartheta _{ref}}, \quad {\widehat{p}}=\frac{p}{\varrho _{ref}U_{ref}^2},\\ {\widehat{I}}= & {} \frac{kI}{a_r ch\vartheta _{ref}^3} \quad \hbox { with }\ a_r=\frac{2\pi ^4|{{\mathcal {S}}}^{n-1}|k^4}{15c^3h^3} \cdot \end{aligned}$$

Set $C_p\,{:=}\,C_v+\frac{\vartheta (\partial _{\vartheta }p)^2}{\varrho ^2\partial _{\varrho }p}\cdot$ Let

$$\begin{aligned} Sr=\frac{L_{ref}}{T_{ref}U_{ref}}, \quad Ma=\frac{U_{ref}}{\sqrt{p_{ref}/\varrho _{ref}}}, \quad Re=\frac{U_{ref}\varrho _{ref}L_{ref}}{\mu _{ref}}, \quad Pr=\frac{{C_p}_{ref}\mu _{ref}}{\kappa _{ref}} \end{aligned}$$

be the Strouhal, Mach, Reynolds, Prandtl (dimensionless) numbers corresponding to hydrodynamics, and

$$\begin{aligned} {{\mathcal {C}}}=\frac{c}{U_{ref}}, \quad {{\mathcal {L}}}=L_{ref}\sigma _{a,ref}, \quad {{\mathcal {L}}}_s=\frac{\sigma _{s,ref}}{\sigma _{a,ref}}, \quad {{\mathcal {P}}}=\frac{a_r\vartheta _{ref}^4}{ \varrho _{ref}U_{ref}^2}, \end{aligned}$$

be dimensionless numbers corresponding to radiation.

In the following, we suppose for simplicity that $Sr={{\mathcal {P}}}=1$ (that is the radiative energy is comparable to the kinetic energy). We also have to keep in mind that only the situation where ${{\mathcal {C}}}\gg 1$ is relevant in our model, for the matter is treated classically.

Considering the reference equilibrium $\varrho ={\varrho }_{ref},\,{\vec u}=\vec {0}$, $\vartheta ={\vartheta }_{ref},$ $I_0=b({\vartheta }_{ref}),$ $\vec {I}_1=\vec {0}$, that corresponds, after rescaling, to

$$\begin{aligned} {\widehat{\varrho }}= 1,\ \widehat{{\vec u}}=\vec {0},\ {\widehat{\vartheta }}=1,\ {\widehat{I}}_0=b(\overline{\vartheta }),\ \widehat{{\vec I}_1}=\vec {0}, \end{aligned}$$

we set (omitting the carets for notational simplicity)

$$\begin{aligned} a\,{:=}\, \varrho -1,\quad \Theta \,{:=}\,\vartheta -1, \quad j_0\,{:=}\, I_0-b(1),\quad {\vec j}_1\,{:=}\,{\vec I}_1, \end{aligned}$$

and eventually get the following system:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a + {\vec u}\cdot \nabla _x a=-(1+a)\, \mathrm{div}_x\, {\vec u},\\ \partial _t {\vec u}+ {\vec u}\cdot \nabla _x {\vec u}- \frac{1}{Re}\frac{1}{1+a}\bigl (\, \mathrm{div}_x\, (2\mu D_x{\vec u})+\nabla _x(\lambda \, \mathrm{div}_x\, {\vec u})\bigr ) + \frac{1}{Ma^2}\frac{1}{1+a}\nabla _xp\\ \quad = \frac{1}{n}\frac{1}{1+a}{{\mathcal {L}}}(\sigma _a+{{\mathcal {L}}}_s\sigma _s){\vec j}_1,\\ \partial _t \Theta + {\vec u}\cdot \nabla _x\Theta -\frac{1}{ Pr Re} \frac{1}{1+a}\, \mathrm{div}_x\, (\kappa \nabla _x\Theta ) = \frac{Ma^2}{ Re}\frac{1}{1+a}\left( 2\mu D_x{\vec u}:D_x{\vec u}+\lambda (\, \mathrm{div}_x\, {\vec u})^2\right) \\ \quad -\,\frac{1+\Theta }{1+a}\,\partial _\vartheta p\, \, \mathrm{div}_x\, {\vec u}-\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr}\frac{\sigma _a}{1+a}\,(b(1+\Theta )-b(1)-j_0) +\frac{{{\mathcal {L}}}}{n\,Pr}\frac{1}{1+a}\left( \sigma _a+{{\mathcal {L}}}_s\sigma _s\right) {\vec j}_1\cdot {\vec u},\\ \partial _t j_0 + \frac{1}{n}\ {{\mathcal {C}}} \, \mathrm{div}_x\, {\vec j}_1 = {{\mathcal {C}}}{{\mathcal {L}}}\sigma _a\bigl (b(1+\Theta )-b(1)-j_0\bigr ),\\ \partial _t {\vec j}_1+ {{\mathcal {C}}} \nabla _xj_0 = -{{\mathcal {C}}}{{\mathcal {L}}}(\sigma _a+{{\mathcal {L}}}_s\sigma _s){\vec j}_1, \end{array}\right. \end{aligned}$$

which rewrites, omitting the dependency with respect to x in the differential operators from now on, and using (9),

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a +\mathrm{div}{\vec u}=F-{\vec u}\cdot \nabla a,\\ \partial _t {\vec u}- \frac{1}{Re}\ \underline{{{\mathcal {A}}}}{\vec u}+ \frac{1}{Ma^2}\ \underline{\alpha }_1\nabla a+ \frac{1}{Ma^2}\ \underline{\alpha }_2\nabla \Theta -\frac{1}{n}\ {{\mathcal {L}}}(\underline{\sigma }_a+{{\mathcal {L}}}_s\underline{\sigma }_s){\vec j}_1=\vec {G}-{\vec u}\cdot \nabla {\vec u},\\ \partial _t \Theta -\frac{1}{ Pr Re}\,\underline{\kappa }\,\Delta \Theta +\underline{\alpha }_2\mathrm{div}{\vec u}-\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr} \ \underline{\sigma }_a\left( j_0-\underline{\alpha }'\Theta \right) =H-{\vec u}\cdot \nabla \Theta ,\\ \partial _t j_0 + \frac{1}{n}\ {{\mathcal {C}}} \mathrm{div}{\vec j}_1 + {{\mathcal {C}}}{{\mathcal {L}}}\underline{\sigma }_a\left( j_0-\underline{\alpha }'\Theta \right) =J_0,\\ \partial _t {\vec j}_1+ {{\mathcal {C}}} \nabla j_0 +{{\mathcal {C}}}{{\mathcal {L}}}(\underline{\sigma }_a+{{\mathcal {L}}}_s\underline{\sigma }_s){\vec j}_1=\vec {J}_1, \end{array} \right. \end{aligned}$$

(25)

with the notations $\underline{{{\mathcal {A}}}}\,{:=}\, \underline{\mu }\Delta +(\underline{\lambda }+\underline{\mu })\nabla \mathrm{div},$

$$\begin{aligned} \begin{array}{l}\underline{\sigma }_a\,{:=}\,\sigma _a(1),\quad \underline{\sigma }_**s\,{:=}\,\sigma _s(1), \quad \underline{\lambda }\,{:=}\,\lambda (1),\quad \underline{\mu }\,{:=}\,\mu (1),\quad \underline{\kappa }=\kappa (1),\quad \underline{b}\,{:=}\,b(1),\\ \underline{\alpha }_1\,{:=}\,\partial _{\varrho }p(1,1)=\pi '_0(1)+\pi '_1(1),\ \ \underline{\alpha }_2\,{:=}\,\partial _{\vartheta }p(1,1)=\pi _1(1),\quad \underline{\alpha }'=\partial _{\vartheta }b(1), \end{array} \end{aligned}$$

(26)

and the right-hand sides

$$\begin{aligned} F\, {:=} -a\ \mathrm{div}{\vec u}, \end{aligned}$$

$$\begin{aligned}&\vec {G} := \frac{1}{Re}\left[ \frac{1}{1+a}\bigl (\mathrm{div}(2\mu D{\vec u})+\nabla (\lambda \mathrm{div}{\vec u})\bigr )-\underline{{{\mathcal {A}}}}{\vec u}\right] \\&\qquad +\, \frac{1}{Ma^2}\left[ \partial _{\varrho }p(1,1)-\frac{1}{1+a}\ \partial _{\varrho }p(1+a,1+\Theta )\right] \nabla a\\&\qquad +\, \frac{1}{Ma^2}\left[ \partial _{\vartheta }p(1,1)-\frac{1}{1+a}\ \partial _{\vartheta }p(1+a,1+\Theta )\right] \nabla \Theta \\&\qquad -\frac{1}{n}\ {{\mathcal {L}}}\left[ \underline{\sigma }_a-\frac{\sigma _a}{1+a}+{{\mathcal {L}}}_s\biggl (\underline{\sigma }_s -\frac{\sigma _s}{1+a}\biggr )\right] \vec {j}_1,\\&H :=\frac{1}{ Pr Re}\,\frac{1}{1+a}\mathrm{div}\bigl ((\kappa -\underline{\kappa })\nabla \Theta \bigr ) -\left[ (1+\Theta )\frac{\pi _1(1+a)}{1+a}-\pi _1(1)\right] \mathrm{div}{\vec u}\\&\qquad +\, \frac{{{\mathcal {L}}}}{n Pr}\frac{1}{1+a}\left( \sigma _a+{{\mathcal {L}}}_s\sigma _s\right) \vec {j}_1\cdot {\vec u}\\&\qquad +\,\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr} \left( \frac{\sigma _a}{1+a}-\underline{\sigma }_a\right) \bigl (j_0-\underline{\alpha }'\Theta \bigr )+ \frac{Ma^2}{ Re}\frac{1}{1+a} \left( 2\mu D\vec {u}:D\vec {u}+ \lambda (\mathrm{div}\vec {u})^2\right) \\&J_0:={\mathcal {C}}{\mathcal {L}}\left[ \sigma _a\left( b(1+\Theta )-\underline{b})-\underline{\alpha }'\Theta \right) +(\underline{\sigma _a}-\sigma _a)(j_0-\underline{\alpha }'\Theta )\right] ,\\&\vec {J}_1\,{:=}\, -{{\mathcal {C}}}{{\mathcal {L}}}\left[ \sigma _a-\underline{\sigma }_a+{{\mathcal {L}}}_s(\sigma _s-\underline{\sigma }_s)\right] \vec {j}_1. \end{aligned}$$

Constructing global strong solutions for (25) in the case of small data with critical regularity is the second (and main) purpose of the present paper. Before giving the statement, let us introduce the solution space: we denote by $E^s$ the set of functions $(a,{\vec u},\Theta ,j_0,{\vec j}_1)$ so that²

$$\begin{aligned}&a^\ell \in {{\mathcal {C}}}_b(\mathbb {R}^+;\dot{B}^{s}_{2,1})\cap L^1(\mathbb {R}^+;\dot{B}^{s+2}_{2,1})\quad \hbox {and}\quad a^h\in {{\mathcal {C}}}_b(\mathbb {R}^+;\dot{B}^{s+1}_{2,1})\cap L^1(\mathbb {R}^+;\dot{B}^{s+1}_{2,1})\\&\quad j_0^\ell ,\, \vec {j}_1^\ell ,\, \vec {u}\in {{\mathcal {C}}}_b(\mathbb {R}^+;\dot{B}^{s}_{2,1})\cap L^1(\mathbb {R}^+;\dot{B}^{s+2}_{2,1})\\&\quad j_0^h,\, \vec {j}_1^h \in {{\mathcal {C}}}_b(\mathbb {R}^+;\dot{B}^{s}_{2,1})\cap L^1(\mathbb {R}^+;\dot{B}^{s}_{2,1}),\\&\quad \Theta ^\ell \in {{\mathcal {C}}}_b(\mathbb {R}^+;\dot{B}^{s}_{2,1})\cap L^1(\mathbb {R}^+;\dot{B}^{s+2}_{2,1}) \quad \hbox {and}\quad \Theta ^h\in {{\mathcal {C}}}_b(\mathbb {R}^+;\dot{B}^{s-1}_{2,1})\cap L^1(\mathbb {R}^+;\dot{B}^{s+1}_{2,1}). \end{aligned}$$

The following result states that for strongly under-relativistic fluids and small data, global existence of strong critical solutions to (25) holds true.

Theorem 2.3

Let $n\ge 3.$ Assume that $\lambda ,$ $\mu ,$ $\kappa$ and $\sigma _s$ depend smoothly on $\varrho$ [with $\lambda ,$ $\mu$, $\kappa$ satisfying (22)], and that $\sigma _a$ is a positive constant. Suppose that the thermal distribution function b depends linearly on $\vartheta .$

There exist a large constant ${{\mathcal {C}}}_0>0$ and a small constant $c>0$ depending only on the dimension n and on the rescaled parameters of the system, such that if

$$\begin{aligned} {{\mathcal {C}}}\ge {{\mathcal {C}}}_0, \end{aligned}$$

(27)

and the initial data $a^0,$ $\vec {u}^0,$ $\Theta ^0,$ $j_{0}^0$ and $\vec {j}_{1}^0$ satisfy the smallness condition

$$\begin{aligned} X_0\,{:=}\,\Vert (\vec {u}^0,j_{0}^0,\vec {j}_{1}^0)\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}+ \Vert (\Theta ^0,a^0)\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^\ell +\Vert a^0\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}^h +\Vert \Theta ^0\Vert _{\dot{B}^{\frac{n}{2}-2}_{2,1}}^h \le c, \end{aligned}$$

(28)

then System (25) has a unique global solution $(a,{\vec u},\Theta ,j_0,\vec {j}_1)$ in $E^{\frac{n}{2}-1}.$ Besides,

$$\begin{aligned} \Vert (a,{\vec u},\Theta ,j_0,{\vec j}_1)\Vert _{E^{\frac{n}{2}-1}}\le KX_0, \end{aligned}$$

(29)

with K depending only on n and on the coefficients of the system, and we have the following decay estimates:

$$\begin{aligned} {{\mathcal {C}}}\int _{\mathbb {R}^+}\Bigl (\Vert \zeta _0\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^\ell + \Vert {\vec j}_1\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}+\Vert j_0\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^h +\Vert \Theta \Vert _{\dot{B}^{\frac{n}{2}-2}_{2,1}}^h\Bigr )\mathrm{d}t \le { KX }_0 \end{aligned}$$

(30)

with $\displaystyle \zeta _0\,{:=}\,j_0-\underline{\alpha }'\Theta -\frac{{\underline{\alpha }_2}\,\underline{\alpha }'}{{{\mathcal {C}}}{{\mathcal {L}}}\,\underline{\alpha }_1\underline{\sigma }_a\bigl (1+\frac{1}{Pr}\,\underline{\alpha }'\bigr )}\mathrm{div}\,{\vec u}.$

The proof is based on a fine analysis of the linearized equations about the reference state and on paralinearization arguments similar to those of, e.g., [1], Chap. 10 or [12], to avoid the loss of one derivative that may cause the convection terms. Let us make another comments:

• (1) Exhibiting the low-frequency decay properties in (30) is absolutely essential for the proof of the global existence as it allows to get a quadratic control on the radiative terms in H (defined above).

• (2) The stability condition (27) comes from our analysis of the linearized equations. In fact, in the low-frequency regime, we proved that the necessary and sufficient stability condition reads as follows:³

  $$\begin{aligned} \begin{array}{lc}\displaystyle &{}\digamma \,{:=}\, \frac{{{\mathcal {C}}}{{\mathcal {L}}}\sigma _a Ma^2}{Re}\biggl ({\widetilde{\alpha }}\nu +\frac{\kappa }{Pr}\biggr ) +\frac{\alpha '\alpha _2}{{\widetilde{\alpha }}}\biggl (\frac{\alpha _2}{Pr}-\frac{Ma^2}{n}\biggr )>0 \quad \hbox {and}\\ &{} \quad \displaystyle \biggl (1+\frac{\alpha ' Ma^2}{n\alpha _2}\biggr )\digamma +\frac{\nu {\widetilde{\alpha }}^2\sigma _a\alpha _1^2{{\mathcal {C}}}{{\mathcal {L}}}Ma^2}{\alpha _2^2Re}+\frac{\alpha '\alpha _1}{Pr} >\frac{\alpha _1\alpha 'Ma^2}{n\alpha _2}\biggl (1+\frac{{\widetilde{\alpha }}}{1+{{\mathcal {L}}}_s\sigma _s/\sigma _a}\biggr )\cdot \end{array} \end{aligned}$$

  (31)

  We also established that for high frequencies, linear stability is true whenever (22) is satisfied and all the other coefficients entering in (26) are positive.

     Unfortunately, in contrast with the barotropic case that we treated in [12], the computations for middle frequencies are so wild that we have not been able to check whether (31) does ensure linear stability, unless ${{\mathcal {C}}}$ is very large. We strongly believe however that it is the correct necessary and sufficient stability conditions for all frequencies.

• (3) For any set of parameters for which the linear system given by the l.h.s. of (25) is strongly stable (i.e., the eigenvalues of the matrix of the system in Fourier variables have positive real parts), one may reproduce the decay estimates of Sect. 3. Indeed linear stability implies (51) and (22) (as we found out the necessary and sufficient stability condition in low and high frequency) and the decay estimates we proved in Sect. 3 are thus valid. Now as the set of medium frequency is compact, strong stability implies uniform exponential stability in that range (see, e.g., [12]). Then going through the computations of Sect. 5, we get another global existence statement for small enough data (with a smallness condition depending on all the parameters of the system), and we can even afford to have some density dependent $\sigma _a,$ as we do not care if the smallness condition depends on ${{\mathcal {C}}}.$

• (4) We strongly believe that if we take smoother data (like in, e.g., Theorem 2.2), then one can get a global-in-time statement under the same stability Condition (27), even if the coefficients all depend on both the density and on the temperature, and for more general distribution function b (like $\theta ^4$ for example). The reason why we refrained from addressing that physically relevant question here is to keep the paper a reasonable size.

As a corollary of the estimates (29) and (30) pointed out in the above theorem, we get the following result for the nonrelativistic limit of (25).

Corollary 2.1

Let $(a^0_\varepsilon ,{\vec u}^0_\varepsilon ,\Theta ^0_\varepsilon ,j^0_{0,\varepsilon },{\vec j}^0_{1,\varepsilon })$ be a family of data fulfilling (28). Consider the corresponding family of solutions $(a_\varepsilon ,{\vec u}_\varepsilon ,\Theta _\varepsilon ,j_{0,\varepsilon },{\vec j}_{1,\varepsilon })$ to (25) with ${{\mathcal {C}}}=\varepsilon ^{-1},$ provided by Theorem 2.3.

If we assume in addition that $(a^0_\varepsilon ,{\vec u}^0_\varepsilon )\rightharpoonup (a^0,{\vec u}^0)$ in the sense of distributions when $\varepsilon \rightarrow 0,$ then we have

$$\begin{aligned} (a_\varepsilon ,{\vec u}_\varepsilon )\rightharpoonup (a,{\vec u})\quad \hbox {and}\quad (\Theta _\varepsilon ,j_{0,\varepsilon },{\vec j}_{1,\varepsilon })\rightharpoonup (0,0,\vec {0}), \end{aligned}$$

where $(a,{\vec u})\in \bigl ({{\mathcal {C}}}(\mathbb {R}_+;\dot{B}^{\frac{n}{2}-1}_{2,1}\cap \dot{B}^{\frac{n}{2}}_{2,1}) \cap L^1(\mathbb {R}_+;\dot{B}^{\frac{n}{2}}_{2,1}+\dot{B}^{\frac{n}{2}+1}_{2,1})\bigr )\times \bigl ({{\mathcal {C}}}(\mathbb {R}_+;\dot{B}^{\frac{n}{2}-1}_{2,1}) \cap L^1(\mathbb {R}_+;\dot{B}^{\frac{n}{2}+1}_{2,1})\bigr )^n$ is the unique solution of the following (isothermal) compressible Navier–Stokes equations:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a + {\vec u}\cdot \nabla a=-(1+a)\, \mathrm{div}_x\, {\vec u},\\ \partial _t {\vec u}+ {\vec u}\cdot \nabla {\vec u}- \frac{1}{Re}\frac{1}{1+a}\bigl (\mathrm{div}(2\mu D{\vec u})+\nabla (\lambda \mathrm{div}{\vec u})\bigr ) + \frac{1}{Ma^2}\frac{1}{1+a}\nabla (\pi _0+\pi _1)=\vec {0}, \end{array}\right. \end{aligned}$$

(32)

supplemented with initial data $(a_0,{\vec u}_0).$

3 Linear analysis of the P1 approximation system

This section is devoted to the linear analysis of System (25) about the null equilibrium. After noticing that the divergence free parts of ${\vec u}$ and ${\vec j}_1$ are uncoupled from the rest of the system, we concentrate on the linear stability of the other unknowns. The main difficulty is that the linearized equations, although with constant coefficients, involve a great number of parameters. To reduce that number to “only” nine, we perform a suitable rescaling. Then, the most intricate part of the analysis concerns the low frequencies (the high-frequency regime turns out to be easier since the radiative part of the system tends to uncouple from the hydrodynamics one). In fact, the linearized system does not enter in any standard class of partial differential equations : it does not have much structure and has terms of three different orders. We shall nevertheless succeed in implementing the method we introduced recently in [13, 14] in the simpler context of barotropic radiative flows so as to reduce our study to that of a nicer system up to error terms that are small in the low-frequency regime.

3.1 The linearized P1 system

It corresponds to the l.h.s. of (25), looking at the r.h.s. as given source terms. We shall concentrate on the case where the source terms are zero, keeping in mind that the general case may be deduced afterward from Duhamel’s formula.

Let ${\mathcal {P}}$ and ${\mathcal {Q}}$ denote the Helmholtz projectors on solenoidal and potential vector fields, respectively. We notice that ${\mathcal {P}}{\vec u}$ and ${\mathcal {P}}{\vec j}_1$ satisfy a linear heat equation, and a damped equation, namely denoting $\underline{\beta }=\frac{1}{n}\ (\underline{\sigma }_a+{{\mathcal {L}}}_s\underline{\sigma }_s),$

$$\begin{aligned} \partial _t{\mathcal {P}}{\vec u}-\frac{1}{Re}\,\underline{\mu }\Delta {\mathcal {P}}{\vec u}=\underline{\beta }{{\mathcal {L}}}{\mathcal {P}}{\vec j}_1\quad \hbox {and}\quad \partial _t{\mathcal {P}}{\vec j}_1+n\underline{\beta }{{\mathcal {C}}}{{\mathcal {L}}}{\mathcal {P}}{\vec j}_1=\vec {0}. \end{aligned}$$

(33)

The system satisfied by $(a,{\mathcal {Q}}{\vec u}, \Theta ,j_0,{\mathcal {Q}}{\vec j}_1)$ is much more involved. To work with scalar unknowns, one sets

$$\begin{aligned} d\,{:=}\,\Lambda ^{-1}\mathrm{div}\vec {u}\quad \hbox {and}\quad j_1\,{:=}\,\Lambda ^{-1}\mathrm{div}\vec {j}_1\quad \hbox {with}\quad \Lambda ^{\pm 1}\,{:=}\,(-\Delta )^{\pm 1/2}. \end{aligned}$$

From the point of view of a priori estimates, working with $({\mathcal {Q}}{\vec u},{\mathcal {Q}}{\vec j}_1)$ or $(d,j_1)$ is equivalent, since ${\mathcal {Q}}{\vec u}=-\Lambda ^{-1}\nabla d$ and ${\mathcal {Q}}{\vec j}_1=-\Lambda ^{-1}\nabla j_1,$ and 0-th order Fourier multipliers are self-maps on homogeneous Besov spaces.

Now the $5\times 5$ system for $(a,d,\Theta ,j_0,j_1)$ reads, putting $\underline{\nu }={\underline{\lambda }}+2{\underline{\mu }}$:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a + \Lambda d = 0,\\ \partial _t d - \frac{1}{Re}\,{\underline{\nu }}\Delta d- \frac{1}{Ma^2}\ \underline{\alpha }_1\Lambda a- \frac{1}{Ma^2}\ \underline{\alpha }_2\Lambda \Theta - {{\mathcal {L}}} \underline{\beta }j_1=0,\\ \partial _t \Theta -{\underline{\kappa }}\,\frac{1}{ Pr Re}\, \Delta \Theta +\underline{\alpha }_2\Lambda d+\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr} \ \underline{\sigma }_a(\underline{\alpha }'\Theta -j_0)=0,\\ \partial _t j_0 +\frac{1}{n}\ {{\mathcal {C}}} \Lambda j_1 - {{\mathcal {C}}}{{\mathcal {L}}}\underline{\sigma }_a(\underline{\alpha }' \Theta -j_0)=0,\\ \partial _t j_1- {{\mathcal {C}}} \Lambda j_0+n{{\mathcal {C}}}{{\mathcal {L}}}\underline{\beta }j_1=0. \end{array}\right. \end{aligned}$$

Taking the Fourier transform with respect to the space variable, the above system recasts in

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} \left( \begin{array}{cc} {\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\Theta }}\\ {\widehat{j_0}}\\ {\widehat{j_1}} \end{array} \right) =A(\xi ) \left( \begin{array}{cc} {\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\Theta }}\\ {\widehat{j_0}}\\ {\widehat{j_1}} \end{array} \right) , \end{aligned}$$

(34)

where, omitting the underlines from now on for better readability,

$$\begin{aligned} A(\xi )\,{:=}\, \left( \begin{array}{ccccc} 0 &{} \quad -|\xi | &{} \quad 0 &{} \quad 0 &{} \quad 0\\ \frac{\alpha _1}{Ma^2} |\xi | &{} \quad -\frac{\nu }{Re}|\xi |^2 &{} \quad \frac{\alpha _2}{Ma^2}\,|\xi | &{} \quad 0 &{} \quad {{\mathcal {L}}} \beta \\ 0 &{} \quad -\alpha _2|\xi | &{} \quad -\frac{\kappa }{Pr Re}|\xi |^2-\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr}{\sigma }_a\alpha ' &{} \quad \frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr}{\sigma }_a &{} \quad 0\\ 0 &{} \quad 0 &{} \quad {{\mathcal {C}}}{{\mathcal {L}}}{\sigma }_a\alpha ' &{} \quad -{{\mathcal {C}}}{{\mathcal {L}}}\sigma _a &{} \quad -\frac{1}{n}\ {{\mathcal {C}}} |\xi | \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad {{\mathcal {C}}} |\xi | &{} \quad -n{{\mathcal {C}}}{{\mathcal {L}}}\beta \end{array} \right) . \end{aligned}$$

3.2 The rescaled system in Fourier variables

System (34) enters in the following class of linear systems:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _ta+\Lambda d=0,\\ \partial _td-b_1\Lambda a-b'_1\Lambda \Theta -b_2\Delta d-b_3j_1=0,\\ \partial _t\Theta +c_1\Lambda d+c_2\Theta -c_3\Delta \Theta -c_4j_0=0,\\ \partial _tj_0+b_5j_0+b_6\Lambda j_1-b_4\Theta =0,\\ \partial _tj_1+b_8j_1-b_7\Lambda j_0=0. \end{array}\right. \end{aligned}$$

(35)

The case we are interested in corresponds to

$$\begin{aligned} b_1= & {} \frac{\alpha _1}{Ma^2},\quad b_2=\frac{\nu }{Re},\quad b'_1=\frac{\alpha _2}{Ma^2},\quad b_3=\beta {{\mathcal {L}}},\quad b_4={{\mathcal {C}}}{{\mathcal {L}}}{\sigma }_a\alpha ',\quad b_5={{\mathcal {C}}}{{\mathcal {L}}}{\sigma }_a,\quad b_6=\frac{{{\mathcal {C}}}}{n},\\ b_7= & {} {{\mathcal {C}}},\quad b_8=n\beta {{\mathcal {C}}}{{\mathcal {L}}},\quad c_1=\alpha _2,\quad c_2=\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr}\ {\sigma }_a\alpha ',\quad c_3=\frac{\kappa }{Re Pr},\quad c_4=\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr}\ {\sigma }_a. \end{aligned}$$

In order to simplify the analysis, we shall first reduce the number of parameters in the above system by performing a convenient rescaling of the unknowns $a,d,\Theta ,j_0,j_1,$ and of the time and space variables. More precisely, setting

$$\begin{aligned} \begin{aligned}&a(t,x)={\widetilde{a}}(\tau t,\chi x),\quad d(t,x)=\delta {\widetilde{d}} (\tau t,\chi x),\quad \Theta (t,x)=\delta ' {\widetilde{\Theta }} (\tau t,\chi x),\\&j_0(t,x)=\zeta _0{\widetilde{j}}_0(\tau t,\chi x),\quad j_1(t,x)=\zeta _1{\widetilde{j}}_1(\tau t,\chi x), \end{aligned}\end{aligned}$$

(36)

with

$$\begin{aligned} \tau \,{:=}\,b_1,\quad \chi \,{:=}\,\sqrt{b_1},\quad \delta \,{:=}\,\sqrt{b_1},\quad \delta '\,{:=}\,\frac{b_1}{b'_1} ,\quad \zeta _0\,{:=}\,\frac{b_1}{b_3}\sqrt{\frac{b_6 b_1}{ b_7}},\quad \zeta _1\,{:=}\,\frac{ b_1^{3/2}}{b_3}, \end{aligned}$$

allows to reduce the number of parameters to nine.⁴ We eventually get the following system for $({\widetilde{a}},{\widetilde{d}},{\widetilde{\Theta }},{\widetilde{j}}_0,{\widetilde{j}}_1)$:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t{\widetilde{a}}+\Lambda {\widetilde{d}}=0,\\ \partial _t{\widetilde{d}}-\Lambda {\widetilde{a}}-\Lambda {\widetilde{\Theta }}-\overline{\nu }\Delta {\widetilde{d}}-{\widetilde{j}}_1=0,\\ \partial _t{\widetilde{\Theta }}+\eta _2\Lambda {\widetilde{d}}+\eta _3{\widetilde{\Theta }}-\overline{\kappa }\Delta {\widetilde{\Theta }}-\eta _4{\widetilde{j}}_0=0,\\ \partial _t{\widetilde{j}}_0+\eta _5 {\widetilde{j}}_0+\eta _6\Lambda {\widetilde{j}}_1-\eta _7 {\widetilde{\Theta }}=0,\\ \partial _t{\widetilde{j}}_1+\eta _8 {\widetilde{j}}_1-\eta _6\Lambda {\widetilde{j}}_0=0, \end{array}\right. \end{aligned}$$

(37)

with

$$\begin{aligned} \left\{ \begin{array}{l} \overline{\nu }=\frac{\nu }{Re},\ \ \overline{\kappa }=\frac{\kappa }{Re Pr},\ \ \eta _2=\frac{c_1b'_1}{b_1}=\frac{\alpha _2^2}{\alpha _1},\ \ \eta _3=\frac{c_2}{b_1}=\frac{{{\mathcal {C}}}{{\mathcal {L}}}{\sigma }_a Ma^2\alpha '}{\alpha _1 Pr},\\ \ \eta _4=\frac{c_4b'_1}{ b_3}\sqrt{\frac{b_6}{b_1b_7}}=\frac{{{\mathcal {C}}}{\sigma }_a \alpha _2}{\sqrt{n\alpha _1}\beta Pr Ma},\ \ \eta _5=\frac{b_5}{b_1}=\frac{{{\mathcal {C}}}{{\mathcal {L}}}{\sigma }_a Ma^2}{\alpha _1},\ \ \eta _6=\sqrt{\frac{b_6 b_7}{b_1}}=\frac{{{\mathcal {C}}}Ma}{\sqrt{n\alpha _1} },\\ \ \eta _7=\frac{b_3b_4}{b_1 b'_1}\sqrt{\frac{b_7}{b_1b_6}}=\frac{\sqrt{n}{{\mathcal {C}}}{{\mathcal {L}}}^2\beta {\sigma }_a\alpha 'Ma^5}{\alpha _1^{3/2}\alpha _2},\ \ \eta _8=\frac{b_8}{b_1}=\frac{n\beta {{\mathcal {C}}}{{\mathcal {L}}}Ma^2}{\alpha _1}, \end{array}\right. \end{aligned}$$

(38)

where

$$\begin{aligned} \beta =\frac{1}{n}({\sigma }_a+{{\mathcal {L}}}_s{\sigma }_s),\quad \alpha _1=\partial _{\varrho }p(1,1),\quad \alpha _2=\partial _{\vartheta }p(1,1),\quad \alpha '=\partial _{\vartheta }b(1). \end{aligned}$$

Let us point out that the coefficients $\eta _3,$ $\eta _4,$ $\eta _5$ and $\eta _7$ are interrelated through

$$\begin{aligned} \eta _3\eta _5-\eta _4\eta _7=0. \end{aligned}$$

(39)

This will be of importance in some of the computations that follow.

Setting $\rho \,{:=}\,|\xi |,$ System (37) in Fourier variables reads (omitting tildes from now on)

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} \left( \begin{array}{c} {\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\Theta }}\\ {\widehat{j}}_{0}\\ {\widehat{j}}_{1} \end{array} \right) +\left( \begin{array}{ccccc} 0&{} \quad \rho &{} \quad 0&{} \quad 0&{} \quad 0\\ -\rho &{} \quad \overline{\nu }\rho ^2&{} \quad -\rho &{} \quad 0&{} \quad 1\\ 0&{} \quad \eta _2\rho &{} \quad \eta _3+\overline{\kappa }\rho ^2&{} \quad -\eta _4&{} \quad 0\\ 0&{} \quad 0&{} \quad -\eta _7&{} \quad \eta _5&{} \quad \eta _6\rho \\ 0&{} \quad 0&{} \quad 0&{} \quad -\eta _6\rho &{} \quad \eta _8 \end{array} \right) \left( \begin{array}{c} {\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\Theta }}\\ {\widehat{j}}_{0}\\ {\widehat{j}}_{1} \end{array}\right) =\left( \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right) , \end{aligned}$$

(40)

that is to say

$$\begin{aligned} \partial _t{\widehat{W}}+A{\widehat{W}}+\rho B{\widehat{W}}+\rho ^2C{\widehat{W}}=0 \end{aligned}$$

(41)

where

$$\begin{aligned}&{\widehat{W}}{:=}\left( \begin{array}{c} {\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\Theta }}\\ {\widehat{j}}_{0}\\ {\widehat{j}}_{1} \end{array} \right) ,\quad A\,{:=}\, \left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad -1\\ 0&{} \quad 0&{} \quad \eta _3&{} \quad -\eta _4&{} \quad 0\\ 0&{} \quad 0&{} \quad -\eta _7&{} \quad \eta _5&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \eta _8 \end{array} \right) ,\\&B{:=}\left( \begin{array}{ccccc} 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0\\ -1&{} \quad 0&{} \quad -1&{} \quad 0&{} \quad 0\\ 0&{} \quad \eta _2&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \eta _6\\ 0&{} \quad 0&{} \quad 0&{} \quad -\eta _6&{} \quad 0 \end{array} \right) ,\ \ C\,{:=}\,\left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad \overline{\nu }&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad \overline{\kappa }&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array} \right) \cdot \end{aligned}$$

3.3 Low-frequency decay estimates

The starting point is the observation that, keeping (39) in mind, the eigenvalues of the matrix of System (40) for $\rho =0$ are 0 (with multiplicity 3), $\eta _3+\eta _5$ and $\eta _8.$ The corresponding modes are

$$\begin{aligned} {\widehat{a}},\quad {\widehat{d}}+ \frac{1}{\eta _8}\,{\widehat{j}}_{1},\quad {\widehat{\Theta }}+\frac{\eta _4}{\eta _5}\, {\widehat{j}}_{0},\quad {\widehat{j}}_{0}-\frac{\eta _3}{\eta _4}\,{\widehat{\Theta }},\quad {\widehat{j}}_{1}. \end{aligned}$$

Changing unknowns accordingly, System (41) rewrites

$$\begin{aligned} \partial _t{\widehat{U}}+A_0{\widehat{U}}+\rho B_0{\widehat{U}}+\rho ^2 C_0 {\widehat{U}}=0 \qquad \hbox {with}\qquad {\widehat{U}}\,{:=}\,\left( \begin{array}{c} {\widehat{a}} \\ {\widehat{d}}+\frac{1}{\eta _8}{\widehat{j}}_{1}\\ {\widehat{\Theta }}+\frac{\eta _4}{\eta _5}{\widehat{j}}_{0}\\ {\widehat{j}}_{0}-\frac{\eta _3}{\eta _4}{\widehat{\Theta }}\\ {\widehat{j}}_{1} \end{array} \right) =\Pi ^{-1}{\widehat{W}}. \end{aligned}$$

Remembering that $\eta _3\eta _5=\eta _4\eta _7,$ we have

$$\begin{aligned} \Pi = \left( \begin{array}{ccccc} 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad -\frac{1}{\eta _8}\\ 0&{} \quad 0&{} \quad {\widetilde{\eta }}_5&{} \quad -{\widetilde{\eta }}_4&{} \quad 0\\ 0&{} \quad 0&{} \quad {\widetilde{\eta }}_7&{} \quad {\widetilde{\eta }}_5&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1 \end{array} \right) \quad \hbox {and}\quad \Pi ^{-1}= \left( \begin{array}{ccccc} 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad \frac{1}{\eta _8}\\ 0&{} \quad 0&{} \quad 1&{} \quad \frac{\eta _4}{\eta _5}&{} \quad 0\\ 0&{} \quad 0&{} \quad -\frac{\eta _3}{\eta _4}&{} \quad 1&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1 \end{array} \right) , \end{aligned}$$

with the rescaled coefficients ${\widetilde{\eta }}_i$ ($i\ge 3$) defined by

$$\begin{aligned} {\widetilde{\eta }}_i\,{:=}\,\frac{\eta _i}{\eta _3+\eta _5}\hbox { for } i\in \{3,4,5,7\},\quad \hbox {and}\quad {\widetilde{\eta }}_6\,{:=}\,\frac{\eta _6}{\eta _8}\cdot \end{aligned}$$

(42)

Note that the coefficients ${\widetilde{\eta }}_i$ are of order 1 and $\Pi$ is thus nicely conditioned in the asymptotics ${{\mathcal {C}}}\rightarrow +\infty .$

Let us compute the matrices $A_0,$ $B_0$ and $C_0.$ We have

$$\begin{aligned} A_0= & {} \Pi ^{-1}A\Pi =\left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad \eta _3+\eta _5&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \eta _8 \end{array} \right) ,\\ B_0= & {} \Pi ^{-1}B\Pi =\left( \begin{array}{ccccc} 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad -\frac{1}{\eta _8}\\ -1&{} \quad 0&{} \quad -{\widetilde{\eta }}_5-{\widetilde{\eta }}_6{\widetilde{\eta }}_7&{} \quad {\widetilde{\eta }}_4-{\widetilde{\eta }}_5{\widetilde{\eta }}_6&{} \quad 0\\ 0&{} \quad \eta _2&{} \quad 0&{} \quad 0&{} \quad \frac{\eta _4\eta _6}{\eta _5}-\frac{\eta _2}{\eta _8}\\ 0&{} \quad -\frac{\eta _2\eta _3}{\eta _4}&{} \quad 0&{} \quad 0&{} \quad \frac{\eta _2\eta _3}{\eta _4\eta _8}+\eta _6\\ 0&{} \quad 0&{} \quad -\eta _6{\widetilde{\eta }}_7&{} \quad -\eta _6{\widetilde{\eta }}_5&{} \quad 0 \end{array} \right) , \end{aligned}$$

and

$$\begin{aligned} C_0=\Pi ^{-1}C\Pi =\left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad \overline{\nu }&{} \quad 0&{} \quad 0&{} \quad -\frac{\overline{\nu }}{\eta _8}\\ 0&{} \quad 0&{} \quad {\widetilde{\eta }}_5\overline{\kappa }&{} \quad -{\widetilde{\eta }}_4\overline{\kappa }&{} \quad 0\\ 0&{} \quad 0&{} \quad -{\widetilde{\eta }}_7\overline{\kappa }&{} \quad {\widetilde{\eta }}_3\overline{\kappa }&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array} \right) \cdot \end{aligned}$$

One splits $B_0$ into $A_1+B_1$, where

$$\begin{aligned} A_1 =\left( \begin{array}{ccccc} 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0\\ -1&{} \quad 0&{} \quad -{\widetilde{\eta }}_5-{\widetilde{\eta }}_6{\widetilde{\eta }}_7&{} \quad 0&{} \quad 0\\ 0&{} \quad \eta _2&{} \quad 0&{} \quad 0&{} \quad \frac{\eta _4\eta _6}{\eta _5}-\frac{\eta _2}{\eta _8}\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \frac{\eta _2\eta _3}{\eta _4\eta _8}+\eta _6\\ 0&{} \quad 0&{} \quad -\eta _6{\widetilde{\eta }}_7&{} \quad -\eta _6{\widetilde{\eta }}_5&{} \quad 0 \end{array} \right) , \end{aligned}$$

and

$$\begin{aligned} B_1 =\left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad -\frac{1}{\eta _8}\\ 0&{} \quad 0&{} \quad 0&{} \quad {\widetilde{\eta }}_4-{\widetilde{\eta }}_5{\widetilde{\eta }}_6&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad -\frac{\eta _2\eta _3}{\eta _4}&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array} \right) . \end{aligned}$$

On the one hand, $A_1$ can be antisymmetrized by some positive diagonal matrix and is thus harmless from the point of view of decay estimates. On the other hand, $B_1$ does not have much structure and is likely to spoil our analysis as it cannot be completely counterbalanced by the matrix $A_0$ which is degenerate. As observed in [13, 14] in a simpler context, the bad contribution of $B_1$ may be somewhat weakened if performing a second change of unknowns

$$\begin{aligned} V\,{:=}\,(\mathbb {I}_d+\rho P){\widehat{U}} \end{aligned}$$

for a $5\times 5$ matrix P such that

$$\begin{aligned}{}[P,A_0]+B_1=0. \end{aligned}$$

(43)

Indeed setting $A_3\,{:=}\,(PA_0-A_1)P+C_0,$ we get the identity

$$\begin{aligned} \partial _tV+A_0V+\rho A_1 V+\rho ^2\left( C_0+PB_1+[P,A_1]\right) V =\rho ^3[A_3,P](\mathbb {I}_d+\rho P)^{-1}V. \end{aligned}$$

(44)

Let us rewrite the previous matrices in block form $M=\left( \begin{array}{cc} M^{11}&{} \quad M^{12}\\ M^{21}&{} \quad M^{22} \end{array}\right) ,$ where $M^{11}$ is a $3\times 3$ block and $M^{22}$ is a $2\times 2$ block:

$$\begin{aligned} B_1= \left( \begin{array}{cc} 0&{} \quad B_1^{12}\\ B_1^{21}&{} \quad 0 \end{array} \right) ,\ \ \ A_0= \left( \begin{array}{cc} 0&{} \quad 0\\ 0&{} \quad \Delta \end{array} \right) ,\ \ \ A_1= \left( \begin{array}{cc} A^{11}_1&{} \quad 0\\ 0&{} \quad A^{22}_1 \end{array} \right) ,\ \ \ P= \left( \begin{array}{cc} P^{11}&{} \quad P^{12}\\ P^{21}&{} \quad P^{22} \end{array} \right) \cdot \end{aligned}$$

Hence

$$\begin{aligned}{}[P,A_0]= \left( \begin{array}{cc} 0&{} \quad P^{12}\Delta \\ -\Delta P^{21}&{} \quad [P^{22},\Delta ] \end{array} \right) \end{aligned}$$

and one can thus ensure (43) if taking

$$\begin{aligned} P^{11}=0,\quad P^{22}=0,\quad P^{12}=-B_1^{12}\Delta ^{-1}\ \hbox { and }\ P^{21}=\Delta ^{-1}B_1^{21}, \end{aligned}$$

that is to say,

$$\begin{aligned}P=\left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad p_1\\ 0&{} \quad 0&{} \quad 0&{} \quad p_2&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad p_4&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array}\right) \quad \hbox {with}\quad p_1\,{:=}\,\frac{1}{\eta _8^2},\quad p_2\,{:=}\,\frac{{\widetilde{\eta }}_5{\widetilde{\eta }}_6-{\widetilde{\eta }}_4}{\eta _3+\eta _5}\ \hbox { and }\ p_4\,{:=}\,-\frac{\eta _2{\widetilde{\eta }}_3}{\eta _4}\cdot \end{aligned}$$

So we end up with

$$\begin{aligned} V=\left( \begin{array}{c}{\widehat{\mathfrak {a}}}\\ {\widehat{\mathfrak {d}}}\\ {\widehat{\Theta }}\\ {\widehat{\mathfrak {j}}}_0\\ {\widehat{\mathfrak {j}}}_1\end{array}\right) =(\mathbb {I}_d+\rho P)\Pi ^{-1}{\widehat{W}} =\left( \begin{array}{ccccc} 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \rho p_1\\ 0&{} \quad 1&{} \quad -\frac{\rho \eta _3p_2}{\eta _4}&{} \quad \rho p_2&{} \quad \frac{1}{\eta _8}\\ 0&{} \quad 0&{} \quad 1&{} \quad \frac{\eta _4}{\eta _5}&{} \quad 0\\ 0&{} \quad \rho p_4&{} \quad -\frac{\eta _3}{\eta _4}&{} \quad 1&{} \quad \frac{\rho p_4}{\eta _8}\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1 \end{array} \right) \left( \begin{array}{c} {\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\Theta }}\\ {\widehat{j}}_{0}\\ {\widehat{j}}_{1} \end{array} \right) . \end{aligned}$$

We notice that $\det \Pi ^{-1}=1+\frac{\eta _3}{\eta _5}$ and that

$$\begin{aligned} \text{ det }\ (\mathbb {I}_d+\rho P) = \text{ det }\ \left( \begin{array}{ccccc} 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \rho p_1\\ 0&{} \quad 1&{} \quad 0&{} \quad \rho p_2&{} \quad 0\\ 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0\\ 0&{} \quad \rho p_4&{} \quad 0&{} \quad 1&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1 \end{array} \right) =1-\rho ^2p_2p_4. \end{aligned}$$

Given that $\eta _3/\eta _5$ is of order 1 and that all the $p_i$’s are of order at most ${{\mathcal {C}}}^{-1},$ we deduce that

$$\begin{aligned} |{\widehat{W}}|\approx |V|\quad \hbox {whenever}\quad \rho \ll {{\mathcal {C}}}. \end{aligned}$$

(45)

Let us also emphasize that all the coefficients of $C_0$ are of order at most 1,  and that the coefficients of $A_0$ and of $A_1$ are of order at most ${{\mathcal {C}}}.$ Therefore the matrix $(PA_0-A_1)P+C_0$ is ${\mathcal {O}}(1)$ and the commutator $[A_3,P]$ in (44) is thus of order ${{\mathcal {C}}}^{-1}.$ So finally, one can write that if $\rho \ll {{\mathcal {C}}}$ then

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} V+A_0V+\rho A_1V+\rho ^2B_2V={\mathcal {O}}({{\mathcal {C}}}^{-1}\rho ^3)\quad \hbox {with } B_2\,{:=}\,C_0+PB_1+[P,A_1]. \end{aligned}$$

(46)

To go further into our analysis, computing ${ PB }_1$ and $[P,A_1]$ is required. We find that

$$\begin{aligned} { PB }_1= & {} \left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad -\gamma &{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad \gamma &{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array} \right) \quad \hbox {with}\quad \gamma \,{:=}\,\frac{\eta _2{\widetilde{\eta }}_3}{\eta _4}({\widetilde{\eta }}_5{\widetilde{\eta }}_6-{\widetilde{\eta }}_4)=\frac{\eta _2}{\eta _3+\eta _5}({\widetilde{\eta }}_6{\widetilde{\eta }}_7-{\widetilde{\eta }}_3),\\ { PA }_1= & {} \left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad -\frac{{\widetilde{\eta }}_6{\widetilde{\eta }}_7}{\eta _8}&{} \quad -\frac{{\widetilde{\eta }}_6{\widetilde{\eta }}_5}{\eta _8}&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \eta _6\bigl (1+\frac{\eta _2\eta _3}{\eta _4\eta _6\eta _8}\bigr )\bigl (\frac{{\widetilde{\eta }}_5{\widetilde{\eta }}_6-{\widetilde{\eta }}_4}{\eta _3+\eta _5}\bigr )\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ \frac{\eta _2{\widetilde{\eta }}_3}{\eta _4}&{} \quad 0&{} \quad \frac{\eta _2{\widetilde{\eta }}_3}{\eta _4}({\widetilde{\eta }}_5+{\widetilde{\eta }}_6{\widetilde{\eta }}_7)&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} A_1P=\left( \begin{array}{ccccc} 0&{} \quad 0&{} \quad 0&{} \quad \frac{{\widetilde{\eta }}_5{\widetilde{\eta }}_6-{\widetilde{\eta }}_4}{\eta _3+\eta _5}&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad -\frac{1}{\eta _8^2}\\ 0&{} \quad 0&{} \quad 0&{} \quad \eta _2\bigl (\frac{{\widetilde{\eta }}_5{\widetilde{\eta }}_6-{\widetilde{\eta }}_4}{\eta _3+\eta _5}\bigr )&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad \frac{\eta _2{\widetilde{\eta }}_3{\widetilde{\eta }}_5\eta _6}{\eta _4}&{} \quad 0&{} \quad 0&{} \quad 0\end{array}\right) \cdot \end{aligned}$$

Therefore $B_2=\left( \begin{array}{cc} B_2^{11}&{}\quad B_2^{12}\\ B_2^{21}&{}\quad B_2^{22}\end{array} \right)$ with

$$\begin{aligned} B_2^{11}= & {} \left( \begin{array}{ccc} 0&{} \quad 0&{} \quad -\frac{{\widetilde{\eta }}_6{\widetilde{\eta }}_7}{\eta _8}\\ 0&{} \quad \overline{\nu }-\gamma &{} \quad 0\\ 0&{} \quad 0&{} \quad {\widetilde{\eta }}_5\overline{\kappa }\end{array} \right) ,\quad B_2^{12}=\left( \begin{array}{cc} \frac{{\widetilde{\eta }}_4}{\eta _3+\eta _5}-{\widetilde{\eta }}_5{\widetilde{\eta }}_6\bigl (\frac{1}{\eta _8}+\frac{1}{\eta _3+\eta _5}\bigr )&{}0\\ 0&{}\eta _6\bigl (1+\frac{\eta _2\eta _3}{\eta _4\eta _6\eta _8}\bigr )\bigl (\frac{{\widetilde{\eta }}_5{\widetilde{\eta }}_6-{\widetilde{\eta }}_4}{\eta _3+\eta _5}\bigr )+\frac{1}{\eta _8^2}-\frac{\overline{\nu }}{\eta _8}\\ -{\widetilde{\eta }}_4\overline{\kappa }+\eta _2\bigl (\frac{{\widetilde{\eta }}_4-{\widetilde{\eta }}_5{\widetilde{\eta }}_6}{\eta _3+\eta _5}\bigr )&{}0\end{array}\right) \\ B_2^{21}= & {} \left( \begin{array}{ccc} \frac{\eta _2{\widetilde{\eta }}_3}{\eta _4}&{}0&{}\frac{\eta _2{\widetilde{\eta }}_3}{\eta _4}({\widetilde{\eta }}_5+{\widetilde{\eta }}_6{\widetilde{\eta }}_7)-{\widetilde{\eta }}_7\overline{\kappa }\\ 0&{}-\frac{\eta _2{\widetilde{\eta }}_3{\widetilde{\eta }}_5\eta _6}{\eta _4}&{}0\end{array}\right) ,\qquad B_2^{22}=\left( \begin{array}{cc} \gamma +{\widetilde{\eta }}_3\overline{\kappa }&{}0\\ 0&{}0\end{array}\right) . \end{aligned}$$

Let us observe that $A_0$ is a nonnegative (degenerate) diagonal matrix of order ${{\mathcal {C}}},$ that $A_1$ is also of order ${{\mathcal {C}}}$ but anti-symmetrizable through a diagonal matrix of order 1,  and is thus likely to have no influence in the energy-type estimates. The leading order terms of $B_2$ are of order 1. They are located either on the diagonal (and are positive if $\overline{\nu }>\gamma >0$ and $\overline{\kappa }>0$) or in the blocks $B^{12}_2$ and $B^{21}_2$ that correspond to interactions between the (modified) fluid unknowns $^tV_1\,{:=}\,({\widehat{\mathfrak {a}}}, {\widehat{\mathfrak {d}}}, {\widehat{\Theta }})$ and radiative unknowns $^tV_2\,{:=}\,( {\widehat{j}}_{0},{\widehat{j}}_{1}).$ Therefore an important part of the stability analysis will be dedicated to the $3\times 3$ subsystem with matrix $A_1^{11}\rho +B_2^{11}\rho ^2$ satisfied by $V_1,$ and to the $2\times 2$ subsystem with matrix $\Delta +\rho A_1^{22}+\rho ^2B_2^{22}$ fulfilled by $V_2.$ For both sub-systems, interactions between the fluid unknowns $V_1$ and radiative unknowns $V_2$ will be considered as error terms in the right-hand side, that may be eliminated for small enough $\rho .$ More concretely:

$$\begin{aligned} \partial _tV_1+\rho A_1^{11}V_1+\rho ^2 B_2^{11}V_1=-\rho ^2B_2^{12}V_2+{\mathcal {O}}({{\mathcal {C}}}^{-1}\rho ^3), \end{aligned}$$

(47)

and

$$\begin{aligned} \partial _tV_2+(\Delta +\rho ^2B_2^{22})V_2 +\rho A_1^{22}V_2=-\rho ^2B_2^{21}V_1+{\mathcal {O}}({{\mathcal {C}}}^{-1}\rho ^3). \end{aligned}$$

(48)

Let us first investigate the system fulfilled by the (modified) hydrodynamic unknowns $V_1,$ looking at the coupling with $V_2$ as a source term. Denoting

$$\begin{aligned} \textstyle \nu \,{:=}\,\overline{\nu }-\gamma ,\quad \kappa \,{:=}\,{\widetilde{\eta }}_5\overline{\kappa },\quad \varepsilon \,{:=}\,\frac{{\widetilde{\eta }}_6{\widetilde{\eta }}_7}{\eta _8},\quad {\widetilde{\alpha }}\,{:=}\,{\widetilde{\eta }}_5+{\widetilde{\eta }}_6{\widetilde{\eta }}_7\ \hbox { and }\ \alpha \,{:=}\,\eta _2, \end{aligned}$$

(49)

that system reads

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left( \begin{array}{c} {\widehat{\mathfrak {a}}}\\ {\widehat{\mathfrak {d}}}\\ {\widehat{\mathfrak {t}}} \end{array}\right) +\rho \left( \begin{array}{ccc} 0&{} \quad 1&{} \quad 0\\ -1&{} \quad 0&{} \quad -{\widetilde{\alpha }}\\ 0&{}\alpha &{}0 \end{array}\right) \left( \begin{array}{c} {\widehat{\mathfrak {a}}}\\ {\widehat{\mathfrak {d}}}\\ {\widehat{\mathfrak {t}}} \end{array}\right) +\rho ^2\left( \begin{array}{ccc} 0&{} \quad 0&{} \quad -\varepsilon \\ 0&{} \quad \nu &{} \quad 0\\ 0&{} \quad 0&{} \quad \kappa \end{array}\right) \left( \begin{array}{c} {\widehat{\mathfrak {a}}}\\ {\widehat{\mathfrak {d}}}\\ {\widehat{\mathfrak {t}}} \end{array}\right) =\left( \begin{array}{c} {\widehat{f}}\\ {\widehat{g}}\\ {\widehat{h}} \end{array}\right) \cdot \end{aligned}$$

(50)

The associated characteristic polynomial reads

$$\begin{aligned} P_\rho (\lambda )=-\lambda ^3+a_1(\rho )\lambda ^2-a_2(\rho )\lambda +a_3(\rho ) \end{aligned}$$

with $a_1(\rho )=(\nu +\kappa )\rho ,$ $a_2(\rho )=1+\alpha {\widetilde{\alpha }}+\nu \kappa \rho ^2$ and $a_3(\rho )=(\kappa +\alpha \varepsilon )\rho .$

Note that (49) ensures that the coefficients $\kappa ,$ $\varepsilon ,$ $\alpha$ and ${\widetilde{\alpha }}$ are positive. As regards $\nu ,$ a necessary condition for all the real parts of the roots of the polynomial $P_\rho$ to be positive is that $a_1(\rho )\ge 0,$ and thus $\nu +\kappa \ge 0.$ Now if that latter condition is fulfilled then Liénard-Chipart criterion [21] guarantees that the real parts of all the roots of the polynomial $P_\rho$ $(\rho >0$) are positive if and only if

$$\begin{aligned} a_1(\rho )a_2(\rho )-a_3(\rho )=\bigl ((\nu +\kappa )(1+\alpha {\widetilde{\alpha }})-\kappa -\alpha \varepsilon \bigr )\rho +\nu \kappa (\nu +\kappa )\rho ^2>0. \end{aligned}$$

We thus find out the following necessary and sufficient stability condition for (50):

$$\begin{aligned} \varepsilon <{\widetilde{\alpha }}\kappa +\Bigl (\frac{1}{\alpha }+{\widetilde{\alpha }}\Bigr )\nu \quad \hbox {and}\quad \nu +\kappa >0. \end{aligned}$$

(51)

In terms of the coefficients of System (37), Condition (51) reads

$$\begin{aligned}&\biggl (\frac{\eta _5\eta _8+\eta _6\eta _7}{\eta _3+\eta _5}\biggr )\biggl ((\eta _3+\eta _5)\overline{\nu }+\eta _5\overline{\kappa }- \frac{\eta _2}{\eta _3+\eta _5}\biggl (\frac{\eta _6\eta _7}{\eta _8}-\eta _3\biggr )\biggr ) +\frac{\eta _8}{\eta _2}(\eta _3+\eta _5)\overline{\nu }\nonumber \\&\qquad +\,\frac{\eta _3\eta _8}{\eta _3+\eta _5}>\eta _6\eta _7\biggl (\frac{1}{\eta _8}+\frac{1}{\eta _3+\eta _5}\biggr )\quad \hbox {and}\quad (\eta _3+\eta _5)\overline{\nu }+\eta _5\overline{\kappa }>\biggl (\frac{\eta _2}{\eta _3+\eta _5}\biggr )\biggl (\frac{\eta _6\eta _7}{\eta _8}-\eta _3\biggr )\cdot \nonumber \\ \end{aligned}$$

(52)

Resuming to the definition of coefficients $\eta _i,$ we find (31). In particular, this implies that (51) is satisfied for ${{\mathcal {C}}}\rightarrow +\infty$.

We aim at recovering that stability condition, supplemented with explicit decay estimates in terms of $\rho$. We claim that if ${\widehat{f}}={\widehat{g}}={\widehat{h}}\equiv 0$ and (51) is fulfilled then there exists some positive threshold $\rho _0,$ and two positive constants c and C (depending continuously on $\varepsilon ,$ $\alpha ,$ ${\widetilde{\alpha }},$ $\kappa$ and $\nu$) so that

$$\begin{aligned} |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|\le Ce^{-c\rho ^2t} |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)| \quad \hbox {for all }t>0\hbox { and }\rho \in [0,\rho _0]. \end{aligned}$$

(53)

To prove our claim, let us introduce:

$$\begin{aligned} {{\mathcal {L}}}_{\varepsilon _1,\varepsilon _2}^2(\rho )\,{:=}\,|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}})|^2+\frac{{\widetilde{\alpha }}}{\alpha }|{\widehat{\mathfrak {t}}}|^2-2\varepsilon _1\rho \mathfrak {R}({\widehat{\mathfrak {a}}}\,\overline{{\widehat{\mathfrak {d}}}})+2\varepsilon _2\rho \mathfrak {R}({\widehat{\mathfrak {d}}}\,\overline{{\widehat{\mathfrak {t}}}}), \end{aligned}$$

for suitable parameters $\varepsilon _1>0$ and $\varepsilon _2>0$ to be chosen hereafter.

From (50), we readily get

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Bigl (|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}})|^2+\frac{{\widetilde{\alpha }}}{\alpha }|{\widehat{\mathfrak {t}}}|^2\Bigr )+\nu \rho ^2|{\widehat{\mathfrak {d}}}|^2+\kappa \frac{{\widetilde{\alpha }}}{\alpha }\rho ^2|{\widehat{\mathfrak {t}}}|^2-\varepsilon \rho ^2\mathfrak {R}({\widehat{\mathfrak {a}}}\,\overline{{\widehat{\mathfrak {t}}}})=0,\\&\frac{\hbox {d}}{\hbox {d}t}\mathfrak {R}({\widehat{\mathfrak {d}}}\,\overline{{\widehat{\mathfrak {a}}}})+\rho |{\widehat{\mathfrak {d}}}|^2-\rho |{\widehat{\mathfrak {a}}}|^2-{\widetilde{\alpha }}\rho \mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {a}}}})=\rho ^2\mathfrak {R}\bigl ((\varepsilon {\widehat{\mathfrak {t}}}-\nu {\widehat{\mathfrak {a}}})\overline{{\widehat{\mathfrak {d}}}}\bigr ),\\&\frac{\hbox {d}}{\hbox {d}t}\mathfrak {R}({\widehat{\mathfrak {d}}}\,\overline{{\widehat{\mathfrak {t}}}})+\alpha \rho |{\widehat{\mathfrak {d}}}|^2-{\widetilde{\alpha }}\rho |{\widehat{\mathfrak {t}}}|^2-\rho \mathfrak {R}({\widehat{\mathfrak {a}}}\,\overline{{\widehat{\mathfrak {t}}}}) =-(\nu +\kappa )\rho ^2\mathfrak {R}({\widehat{\mathfrak {d}}}{\widehat{\mathfrak {t}}}). \end{aligned}$$

Hence for all small enough $\rho ,$

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}{{\mathcal {L}}}^2_{\varepsilon _1,\varepsilon _2}+\rho ^2{\mathcal {H}}^2_{\varepsilon _1,\varepsilon _2}={\mathcal {O}}(\rho ^3) \end{aligned}$$

(54)

with

$$\begin{aligned} {\mathcal {H}}^2_{\varepsilon _1,\varepsilon _2}({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})=\varepsilon _1|{\widehat{\mathfrak {a}}}|^2+(\nu -\varepsilon _1+\alpha \varepsilon _2)|{\widehat{\mathfrak {d}}}|^2 +{\widetilde{\alpha }}\Bigl (\frac{\kappa }{\alpha }-\varepsilon _2\Bigr )|{\widehat{\mathfrak {t}}}|^2+({\widetilde{\alpha }}\varepsilon _1-\varepsilon -\varepsilon _2)\mathfrak {R}({\widehat{\mathfrak {a}}}\,\overline{{\widehat{\mathfrak {t}}}}). \end{aligned}$$

Note that we have

$$\begin{aligned} {\mathcal {H}}^2_{\varepsilon _1,\varepsilon _2}({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})= & {} \varepsilon _1\bigg |{\widehat{\mathfrak {a}}}+\biggl (\frac{{\widetilde{\alpha }}\varepsilon _1-\varepsilon -\varepsilon _2}{2\varepsilon _1}\biggr ){\widehat{\mathfrak {t}}}\bigg |^2+(\nu -\varepsilon _1+\alpha \varepsilon _2)|{\widehat{\mathfrak {d}}}|^2\\&+\,{\widetilde{\alpha }}\biggl (\frac{\kappa }{\alpha }-\varepsilon _2 -\frac{({\widetilde{\alpha }}\varepsilon _1-\varepsilon -\varepsilon _2)^2}{4{\widetilde{\alpha }}\varepsilon _1}\biggr )|{\widehat{\mathfrak {t}}}|^2. \end{aligned}$$

Therefore ${\mathcal {H}}^2_{\varepsilon _1,\varepsilon _2}$ is a positive definite quadratic form if and only if

$$\begin{aligned} \varepsilon _1>0,\quad \varepsilon _1<\nu +\alpha \varepsilon _2\ \hbox { and }\ ({\widetilde{\alpha }}\varepsilon _1-\varepsilon -\varepsilon _2)^2<4{\widetilde{\alpha }}\varepsilon _1\Bigl (\frac{\kappa }{\alpha }-\varepsilon _2\Bigr )\cdot \end{aligned}$$

(55)

We claim that if (51) is fulfilled then one can always find some $\varepsilon _1$ and $\varepsilon _2$ fulfilling (55).

In order to justify our claim, it is convenient to change $(\varepsilon _1,\varepsilon _2,\varepsilon )$ into $({\widetilde{\varepsilon }}_{1},{\widetilde{\varepsilon }}_{2},{\widetilde{\varepsilon }})$ as follows:

$$\begin{aligned} {\widetilde{\varepsilon }}_{1}\,{:=}\,\frac{{\widetilde{\alpha }}\varepsilon _1}{{\widetilde{\varepsilon }}_{max}}, \quad {\widetilde{\varepsilon }}_{2}\,{:=}\,\frac{1}{{\widetilde{\varepsilon }}_{max}}\Bigl (\frac{\kappa }{\alpha }-\varepsilon _2\Bigr ),\quad {\widetilde{\varepsilon }}\,{:=}\,\frac{1}{{\widetilde{\varepsilon }}_{max}}\Bigl (\varepsilon +\frac{\kappa }{\alpha }\Bigr )\ \hbox { with }\ {\widetilde{\varepsilon }}_{max}\,{:=}\,\Bigl (\frac{1}{\alpha }+{\widetilde{\alpha }}\Bigr )(\nu +\kappa ), \end{aligned}$$

and, assuming that $\nu +\kappa >0,$ to set

$$\begin{aligned} \varepsilon '_1\,{:=}\,\frac{{\widetilde{\varepsilon }}_{1}}{{\widetilde{\varepsilon }}}\quad \hbox {and}\quad \varepsilon '_2\,{:=}\,\frac{{\widetilde{\varepsilon }}_{2}}{{\widetilde{\varepsilon }}}\cdot \end{aligned}$$

Then, denoting $A\,{:=}\,\alpha {\widetilde{\alpha }},$ Condition (55) translates into

$$\begin{aligned} \varepsilon '_1>0,\quad \varepsilon '_1+A\varepsilon '_2<\frac{1}{{\widetilde{\varepsilon }}}\frac{A}{1+A}\quad \hbox { and }\ \ (\varepsilon _1'+\varepsilon _2'-1)^2<4\varepsilon '_1\varepsilon '_2. \end{aligned}$$

(56)

The latter condition is equivalent to

$$\begin{aligned} L(\varepsilon '_1,\varepsilon '_2)\,{:=}\,(\varepsilon '_1-\varepsilon '_2)^2-2(\varepsilon '_1+\varepsilon '_2)+1<0. \end{aligned}$$

It is obvious that L does not have any minimum in the interior of the domain D defined by the first two conditions in (56), and that $L\ge 0$ for $\varepsilon '_1=0.$ For $\varepsilon '_2={\widetilde{\varepsilon }}^{-1}/(1+A)-\varepsilon '_1/A,$ we have

$$\begin{aligned} L(\varepsilon '_1,\varepsilon '_2)=\biggl (\frac{A+1}{A}\varepsilon '_1-\frac{{\widetilde{\varepsilon }}^{-1}}{A+1}\biggr )^2-2\biggl (\frac{A-1}{A}\varepsilon '_1+\frac{{\widetilde{\varepsilon }}^{-1}}{A+1}\biggr )+1, \end{aligned}$$

the minimum of which corresponds to $(\varepsilon '_1,\varepsilon '_2)=(\varepsilon _1^*,\varepsilon _2^*)$ with

$$\begin{aligned} \varepsilon _1^*=\frac{A({\widetilde{\varepsilon }}^{-1}+A-1)}{(A+1)^2}\quad \hbox {and}\quad \varepsilon _2^*=\frac{A{\widetilde{\varepsilon }}^{-1}-A+1}{(A+1)^2}\cdot \end{aligned}$$

The value of L at $(\varepsilon _1^*,\varepsilon _2^*)$ is

$$\begin{aligned} L(\varepsilon _1^*,\varepsilon _2^*)=\frac{4A}{(A+1)^2}(1-{\widetilde{\varepsilon }}^{-1}). \end{aligned}$$

Hence there exists $(\varepsilon _1^*,\varepsilon _2^*)\in D$ satisfying $L(\varepsilon _1^*,\varepsilon _2^*)<0$ if and only if ${\widetilde{\varepsilon }}^{-1}>1,$ which is equivalent to the first part of the stability condition (51).

Let us recap. On the one hand, resuming to the initial parameters, we thus found some $\varepsilon _1$ and $\varepsilon _2$ satisfying (55). Taking such $\varepsilon _1$ and $\varepsilon _2,$ the quadratic form ${\mathcal {H}}^2_{\varepsilon _1,\varepsilon _2}$ is definite positive, and thus ${\mathcal {H}}^2_{\varepsilon _1,\varepsilon _2}\approx |{\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|^2.$ On the other hand, for small enough $\rho ,$ we have

$$\begin{aligned} {{\mathcal {L}}}_{\varepsilon _1,\varepsilon _2}^2\approx |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|^2. \end{aligned}$$

Therefore (54) implies that there exists some $\rho _0>0$ and a constant $c>0$ depending only on the parameters of the system and such that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}{{\mathcal {L}}}_{\varepsilon _1,\varepsilon _2}^2+c\rho ^2{{\mathcal {L}}}_{\varepsilon _1,\varepsilon _2}^2\le 0\quad \hbox {if }\ 0\le \rho \le \rho _0. \end{aligned}$$

(57)

This clearly implies (53). Now for general source terms ${\widehat{f}},$ ${\widehat{g}}$ and ${\widehat{h}}$ in (50), taking advantage of Duhamel’s formula and integrating with respect to time gives for some $K=K(\varepsilon ,\alpha ,{\widetilde{\alpha }},\kappa ,\nu ),$

$$\begin{aligned} |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+c\rho ^2\int _0^t|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K\biggl (|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|+\int _0^t|({\widehat{f}},{\widehat{g}},{\widehat{h}})|\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(58)

In the case we are interested in, the source terms ${\widehat{f}},$ ${\widehat{g}}$ and ${\widehat{h}}$ are given by the right-hand side of (47) and we thus have for small enough $\rho ,$ $|({\widehat{f}},{\widehat{g}},{\widehat{h}})|\le C\rho ^2|V_2|+C\rho ^3|V|.$ Hence we deduce from (58) that

$$\begin{aligned} |V_1(t)|+c\rho ^2\int _0^t|V_1|\,\hbox {d}\tau \le KV_1(0)+K\rho ^2\int _0^t|V_2|\,\hbox {d}\tau . \end{aligned}$$

(59)

Handling the (modified) radiative unknowns $({\widehat{\mathfrak {j}}}_0,{\widehat{\mathfrak {j}}}_1)$ is much simpler. Indeed denoting $\varsigma \,{:=}\,1+\frac{\eta _2\eta _3}{\eta _4\eta _6\eta _8},$ we have

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left( \begin{array}{c} {\widehat{\mathfrak {j}}}_0\\ {\widehat{\mathfrak {j}}}_1 \end{array}\right) +\left( \begin{array}{cc} \eta _3+\eta _5+\rho ^2(\gamma +{\widetilde{\eta }}_3\overline{\kappa })&{}0\\ 0&{}\eta _8 \end{array}\right) \left( \begin{array}{c} {\widehat{\mathfrak {j}}}_0\\ {\widehat{\mathfrak {j}}}_1 \end{array}\right) +\rho \eta _6\left( \begin{array}{cc} 0&{} \quad \varsigma \\ -1&{} \quad 0 \end{array}\right) \left( \begin{array}{c} {\widehat{\mathfrak {j}}}_0\\ {\widehat{\mathfrak {j}}}_1 \end{array}\right) =\left( \begin{array}{c} {\widehat{k}}\\ {\widehat{\ell }}\end{array}\right) \cdot \end{aligned}$$

Therefore multiplying the second equation by $\varsigma$ and taking the inner product in $\mathbb {C}^2$ with $({\widehat{\mathfrak {j}}}_0,{\widehat{\mathfrak {j}}}_1),$ we easily find that

$$\begin{aligned}&\displaystyle \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\biggl (|{\widehat{\mathfrak {j}}}_0|^2+\varsigma |{\widehat{\mathfrak {j}}}_1|^2\biggr )+\bigl (\eta _3+\eta _5+\rho ^2(\gamma +{\widetilde{\eta }}_3\overline{\kappa })\bigr )|{\widehat{\mathfrak {j}}}_0|^2 +\varsigma \eta _8|{\widehat{\mathfrak {j}}}_1|^2=\mathfrak {R}({\widehat{k}}\,\overline{{\widehat{\mathfrak {j}}}_0})+\varsigma \mathfrak {R}({\widehat{\ell }}\,\overline{{\widehat{\mathfrak {j}}}_1}). \end{aligned}$$

Hence

$$\begin{aligned}&\sqrt{|{\widehat{\mathfrak {j}}}_0(t)|^2+\varsigma |{\widehat{\mathfrak {j}}}_1(t)|^2}+\min (\eta _3+\eta _5,\eta _8)\int _0^t\sqrt{|{\widehat{\mathfrak {j}}}_0|^2+\varsigma |{\widehat{\mathfrak {j}}}_1|^2}\,\hbox {d}\tau \nonumber \\&\quad \le \sqrt{|{\widehat{\mathfrak {j}}}_0(0)|^2+\varsigma |{\widehat{\mathfrak {j}}}_1(0)|^2}+\int _0^t\sqrt{|{\widehat{k}}|^2+\varsigma |{\widehat{\ell }}|^2}\,\hbox {d}\tau . \end{aligned}$$

(60)

Because ${\widehat{k}}$ and ${\widehat{\ell }}$ are given by the right-hand side of (48), we have

$$\begin{aligned} |({\widehat{k}},{\widehat{\ell }})|\le C\rho ^2|V_1|+C\rho ^3|V|, \end{aligned}$$

and thus, for small enough $\rho ,$

$$\begin{aligned} |V_2(t)|+\int _0^t|V_2|\,\hbox {d}\tau \le CV_2(0)+C\rho ^2\int _0^t|V_1|\,\hbox {d}\tau . \end{aligned}$$

Putting that later inequality together with (59), we conclude that there exists $\rho _0>0$ depending only on the coefficients of System (40) such that if Condition (51) is fulfilled and $\rho \le \rho _0,$ then we have for all $t\ge 0,$

$$\begin{aligned} |V(t)|+\rho ^2\int _0^t|V_1|\,\hbox {d}\tau +\int _0^t|V_2|\,\hbox {d}\tau \le C|V(0)|. \end{aligned}$$

(61)

In particular, this means that (51) is also a necessary and sufficient stability condition for the whole system (40), and that (61) holds provided $\rho \le \rho _0.$

As we are interested in the asymptotics ${{\mathcal {C}}}\rightarrow +\infty$ and as $\varepsilon \approx {{\mathcal {C}}}^{-1},$ it is suitable to check what kind of information is supplied by the above analysis if $\varepsilon$ is small. On the one hand, in that case, the range for which (57) holds true is decreasing with respect to $\varepsilon _1$ and $\varepsilon _2,$ which suggest us to take $\varepsilon _1$ and $\varepsilon _2$ as small as possible. On the other hand, the constant c in (57) is of order $\varepsilon _1,$ hence the decay becomes worse if taking $\varepsilon _1$ smaller. Therefore we need to find some acceptable compromise between having a large range of $\rho$’s in (57) and a good decay.

By looking at Condition (55), we discover that whenever $\varepsilon <{\widetilde{\alpha }}\nu ,$ one can take $\varepsilon _1=\varepsilon /{\widetilde{\alpha }}$ and $\varepsilon _2=0.$ As the error term in (54) is equal to $\varepsilon _1\rho ^3\mathfrak {R}\bigl ((\nu {\widehat{\mathfrak {a}}}-\varepsilon {\widehat{\mathfrak {t}}}){\widehat{\mathfrak {d}}}\bigr ),$ this gives for some constants c and $\rho _0$ independent of $\varepsilon$,

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}{{\mathcal {L}}}_\rho ^2+c\varepsilon \rho ^2{{\mathcal {L}}}_\rho ^2\le 0\quad \hbox {if }\ 0\le \rho \le \rho _0\varepsilon ^{-1/2}, \end{aligned}$$

from which we get in particular,

$$\begin{aligned} \forall t\ge 0,\; \varepsilon \rho ^2\int _0^t|{\widehat{\mathfrak {t}}}|\,\hbox {d}\tau \le C|(\mathfrak {a},\mathfrak {d},\mathfrak {t})(0)|. \end{aligned}$$

(62)

In order to improve the decay information that we have on ${\widehat{\mathfrak {d}}}$ and ${\widehat{\mathfrak {t}}}$ [which will be of fundamental importance in our study of the nonlinear system (25)], let us focus on the following linearized Navier–Stokes–Fourier system:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\hbox {d}}{\hbox {d}t}{\widehat{\mathfrak {a}}}+\rho {\widehat{\mathfrak {d}}} ={\widehat{k}}\,{:=}\,\varepsilon \rho ^2{\widehat{\mathfrak {t}}},\\ \frac{\hbox {d}}{\hbox {d}t}{\widehat{\mathfrak {d}}}+\nu \rho ^2{\widehat{\mathfrak {d}}}-\rho {\widehat{\mathfrak {a}}}-{\widetilde{\alpha }}\rho {\widehat{\mathfrak {t}}}=0,\\ \frac{\hbox {d}}{\hbox {d}t}{\widehat{\mathfrak {t}}}+\kappa \rho ^2{\widehat{\mathfrak {t}}}+\alpha \rho {\widehat{\mathfrak {d}}}=0.\end{array}\right. \end{aligned}$$

Assuming that ${\widehat{k}}=0$ for a while, we have:

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}})|^2+\nu \rho ^2|{\widehat{\mathfrak {d}}}|^2-{\widetilde{\alpha }}\rho \mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {d}}}})=0,\\&\quad \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}|{\widehat{\mathfrak {t}}}|^2+\kappa \rho ^2|{\widehat{\mathfrak {t}}}|^2+\alpha \rho \mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {d}}}})=0. \end{aligned}$$

Therefore

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Bigl (\alpha |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}})|^2+{\widetilde{\alpha }}|{\widehat{\mathfrak {t}}}|^2\Bigr )+ \alpha \nu \rho ^2|{\widehat{\mathfrak {d}}}|^2+{\widetilde{\alpha }}\kappa \rho ^2|{\widehat{\mathfrak {t}}}|^2=0. \end{aligned}$$

(63)

In addition,

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}|\nu \rho {\widehat{a}}|^2+\nu \rho ^2\mathfrak {R}\bigl (\nu \rho {\widehat{a}}\,\overline{{\widehat{\mathfrak {d}}}}\bigr )=0 \end{aligned}$$

and

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\mathfrak {R}(\nu \rho {\widehat{\mathfrak {a}}}\,\overline{{\widehat{\mathfrak {d}}}})-\nu \rho ^2|{\widehat{\mathfrak {a}}}|^2+\nu \rho ^2|{\widehat{\mathfrak {d}}}|^2-{\widetilde{\alpha }}\rho ^2\nu \mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {a}}}}) +\nu \rho ^2\mathfrak {R}(\nu \rho {\widehat{a}}\,\overline{{\widehat{\mathfrak {d}}}})=0. \end{aligned}$$

Therefore

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\bigl (|\nu \rho {\widehat{\mathfrak {a}}}|^2-2\mathfrak {R}(\nu \rho {\widehat{a}}\,\overline{{\widehat{\mathfrak {d}}}})\bigr )+\nu \rho ^2|{\widehat{\mathfrak {a}}}|^2 -\nu \rho ^2|{\widehat{\mathfrak {d}}}|^2+{\widetilde{\alpha }}\rho ^2\nu \mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {a}}}})=0. \end{aligned}$$

Combining the above identities, we conclude that for any $K\in \mathbb {R},$ we have

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}{{\mathcal {L}}}_\rho ^2 +\nu \rho ^2|{\widehat{\mathfrak {a}}}|^2+(K\alpha -1)\nu \rho ^2|{\widehat{\mathfrak {d}}}|^2+K{\widetilde{\alpha }}\kappa \rho ^2|{\widehat{\mathfrak {t}}}|^2 +{\widetilde{\alpha }}\nu \rho ^2\mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {a}}}})=0 \end{aligned}$$

with

$$\begin{aligned} {{\mathcal {L}}}_\rho ^2\,{:=}\,K\alpha |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}})|^2+K{\widetilde{\alpha }}|{\widehat{\mathfrak {t}}}|^2+|\nu \rho {\widehat{\mathfrak {a}}}|^2-2\mathfrak {R}(\nu \rho {\widehat{\mathfrak {a}}}\,\overline{{\widehat{\mathfrak {d}}}}). \end{aligned}$$

Because

$$\begin{aligned} \bigl |{\widetilde{\alpha }}\nu \rho ^2\mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {a}}}})\bigr |\le \frac{\nu \rho ^2}{2}|{\widehat{\mathfrak {a}}}|^2+\frac{{\widetilde{\alpha }}^2\nu \rho ^2}{2}|{\widehat{\mathfrak {t}}}|^2, \end{aligned}$$

we see that if we choose

$$\begin{aligned} K\ge \max \biggl (\frac{2}{\alpha },\frac{\nu {\widetilde{\alpha }}}{\kappa }\biggr ), \end{aligned}$$

then we have ${{\mathcal {L}}}_\rho \approx |\nu \rho {\widehat{\mathfrak {a}}}|+\max (1,\sqrt{\nu /\kappa })|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|$ and

$$\begin{aligned} \nu \rho ^2|{\widehat{\mathfrak {a}}}|^2+(K\alpha -1)\nu \rho ^2|{\widehat{\mathfrak {d}}}|^2+K{\widetilde{\alpha }}\kappa \rho ^2|{\widehat{\mathfrak {t}}}|^2 +{\widetilde{\alpha }}\nu \rho ^2\mathfrak {R}({\widehat{\mathfrak {t}}}\,\overline{{\widehat{\mathfrak {a}}}})\gtrsim \nu \rho ^2|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|^2. \end{aligned}$$

Therefore if $\nu \lesssim \kappa$ then we get

$$\begin{aligned} {{\mathcal {L}}}_\rho (t)\lesssim {{\mathcal {L}}}_\rho (0)e^{-c\min (\frac{1}{\nu },\nu \rho ^2)t}, \end{aligned}$$

whence plugging this information in the equations for ${\widehat{\mathfrak {t}}}$ and for ${\widehat{\mathfrak {d}}}$ and using Duhamel’s formula (to handle nonzero ${\widehat{k}}$), we end up with

$$\begin{aligned} |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+\nu \rho ^2\int _0^t|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K\biggl (|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|+\int _0^t|{\widehat{k}}|\,\hbox {d}\tau \biggr )\quad \hbox {if}\quad \nu \rho \le 1, \end{aligned}$$

and

$$\begin{aligned}&|(\nu \rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\nu \rho ^2\int _0^t|{\widehat{\mathfrak {d}}}|\,\hbox {d}\tau +\kappa \rho ^2\int _0^t|{\widehat{\mathfrak {t}}}|\,\hbox {d}\tau \\&\quad \le K\biggl (|(\nu \rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|+\int _0^t|\rho \nu {\widehat{k}}|\,\hbox {d}\tau \biggr ) \quad \hbox {if}\quad \nu \rho \ge 1. \end{aligned}$$

In the case $\kappa \lesssim \nu ,$ similar computations lead to

$$\begin{aligned} |({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+\kappa \rho ^2\int _0^t|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K\biggl (|({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|+\int _0^t|{\widehat{k}}|\,\hbox {d}\tau \biggr )\quad \hbox {if}\quad \sqrt{\kappa \nu }\rho \le 1, \end{aligned}$$

and to

$$\begin{aligned}&|(\sqrt{\kappa \nu }\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+\sqrt{\frac{\kappa }{\nu }}\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\kappa \rho ^2\int _0^t|{\widehat{\mathfrak {d}}}|\,\hbox {d}\tau +\sqrt{\frac{\kappa }{\nu }}\,\kappa \rho ^2\int _0^t|{\widehat{\mathfrak {t}}}|\,\hbox {d}\tau \\&\quad \le K\biggl (|(\sqrt{\kappa \nu }\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|+\int _0^t|\sqrt{\kappa \nu }\,\rho {\widehat{k}}|\,\hbox {d}\tau \biggr ) \quad \hbox {if}\quad \kappa \rho \ge 1. \end{aligned}$$

Remembering that ${\widehat{k}}=\varepsilon \rho ^2{\widehat{\mathfrak {t}}},$ we easily conclude from the above inequalities that the solution to (50) fulfills

$$\begin{aligned} |(\rho \nu {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+\min (1,\nu \rho )\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\nu \rho ^2\int _0^t|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K|(\rho \nu {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|\quad \hbox {if }\ \nu \le \kappa \end{aligned}$$

(64)

and

$$\begin{aligned}&|(\sqrt{\kappa \nu }\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(t)|+\min (1,\sqrt{\kappa \nu }\rho ) \sqrt{\frac{\kappa }{\nu }}\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau \nonumber \\&\quad +\,\kappa \rho ^2\int _0^t|{\widehat{\mathfrak {t}}}|\,\hbox {d}\tau +\sqrt{\frac{\kappa }{\nu }}\,\kappa \rho ^2\int _0^t|{\widehat{\mathfrak {t}}}|\,\hbox {d}\tau \le K|(\sqrt{\kappa \nu }\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})(0)|\quad \hbox {if }\ \kappa \le \nu \end{aligned}$$

(65)

provided that $\varepsilon \ll \min (\kappa ,\nu )$ and $\rho \ll \varepsilon ^{-1/2}.$

Let us finally resume to the proof of global-in-time decay estimates for the solution to (40). For notational simplicity, we do not track the dependency with respect to $\kappa$ and to $\nu$ any longer. We focus on the case where coefficients $\eta _3$ to $\eta _8$ are of order ${{\mathcal {C}}}$ and $\overline{\kappa },$ $\overline{\nu }$ and $\eta _2$ are of order 1. This implies that $\varepsilon$ is of order ${{\mathcal {C}}}^{-1},$ and we thus get for all $\rho \ll {{\mathcal {C}}},$

$$\begin{aligned} |({\widehat{f}},{\widehat{g}},{\widehat{h}})|\le K\rho ^2|V_2|+K{{\mathcal {C}}}^{-1}\rho ^2|V| \quad \hbox {and}\quad |({\widehat{k}},{\widehat{\ell }})|\le K\rho ^2|V_1|+K{{\mathcal {C}}}^{-1}\rho ^2|V|. \end{aligned}$$

Hence Inequalities (60), (64) and (65) (combined with Duhamel formula) yield for $\rho \lesssim 1,$ c small enough and ${{\mathcal {C}}}$ large enough,

$$\begin{aligned} \begin{aligned}&|V_1(t)|+c\rho ^2\int _0^t|V_1|\,\hbox {d}\tau \le KV_1(0)+K\rho ^2\int _0^t|V_2|\,\hbox {d}\tau \\&\quad \hbox {and}\quad |V_2(t)|+c\,{{\mathcal {C}}}\int _0^t|V_2|\,\hbox {d}\tau \le KV_2(0) +K\rho ^2\int _0^t|V_1|\,\hbox {d}\tau ,\end{aligned} \end{aligned}$$

whence

$$\begin{aligned} |V(t)|+\rho ^2\int _0^t|V_1|\,\hbox {d}\tau +{{\mathcal {C}}}\int _0^t|V_2|\,\hbox {d}\tau \le K|V(0)|. \end{aligned}$$

According to (64) and (65), for $\rho \gtrsim 1,$ working with ${\widetilde{V}}_1\,{:=}\,(\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})$ instead of $V_1=({\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})$ is more appropriate, and we thus have to use the following inequality (that stems from (65) and Duhamel formula):

$$\begin{aligned} |{\widetilde{V}}_1(t)|+\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\rho ^2\int _0^t|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K\biggl (|{\widetilde{V}}_1(0)+\int _0^t|(\rho {\widehat{f}},{\widehat{g}},{\widehat{h}})|\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(66)

A closer look at the structure of $B^{12}_2$ and of $({\widehat{f}},{\widehat{g}},{\widehat{h}})$ defined to be the r.h.s. of (47) reveals that

$$\begin{aligned} |\rho {\widehat{f}}|\lesssim {{\mathcal {C}}}^{-1}\rho ^3|{\widehat{\mathfrak {j}}}_0|+{{\mathcal {C}}}^{-1}\rho ^4|V| \quad \hbox {and}\quad |({\widehat{g}},{\widehat{h}})|\lesssim \rho ^2|V_2|+{{\mathcal {C}}}^{-1}\rho ^3|V|. \end{aligned}$$

Hence inequality (66) implies that for $1\lesssim \rho \ll {{\mathcal {C}}}^{1/3},$ we have

$$\begin{aligned} |{\widetilde{V}}_1(t)|+\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\rho ^2\int _0^t|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K\biggl (|{\widetilde{V}}_1(0)|+\rho ^2\int _0^t|V_2|\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(67)

It is also clear that for $1\lesssim \rho \lesssim {{\mathcal {C}}}^{-1},$

$$\begin{aligned} |({\widehat{k}},{\widehat{\ell }})|\lesssim {{\mathcal {C}}}^{-1}\rho ^2|{\widehat{\mathfrak {a}}}|+\rho ^2|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|+{{\mathcal {C}}}^{-1}\rho ^3|V| \lesssim {{\mathcal {C}}}^{-1}\rho ^2|\rho {\widehat{\mathfrak {a}}}|+\rho ^2|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|+{{\mathcal {C}}}^{-1}\rho ^3|V_2|. \end{aligned}$$

Hence if $\rho \ll {{\mathcal {C}}}^{2/3},$

$$\begin{aligned} |V_2(t)|+{{\mathcal {C}}}\int _0^t|V_2|\,\hbox {d}\tau \le K\biggl (|V_2(0)|+{{\mathcal {C}}}^{-1}\rho ^2\int _0^t|\rho {\widehat{a}}|\,\hbox {d}\tau + \rho ^2\int _0^t|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(68)

Inserting (67) in (68), it is now easy to conclude that for $1\lesssim \rho \ll {{\mathcal {C}}}^{1/3},$ we have

$$\begin{aligned} |V_2(t)|+{{\mathcal {C}}}\int _0^t|V_2|\,\hbox {d}\tau \le K|({\widetilde{V}}_1,V_2)(0)|, \end{aligned}$$

and thus

$$\begin{aligned} |{\widetilde{V}}_1(t)|+\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\rho ^2\int _0^t|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau \le K|({\widetilde{V}}_1,V_2)(0)|. \end{aligned}$$

The case where there is some source term ${\widehat{F}}=({\widehat{A}},{\widehat{D}},{\widehat{\Theta }},{\widehat{J}}_0,{\widehat{J}}_1)$ in (40) may be treated along the same lines, and we end up for all $0\le \rho \ll {{\mathcal {C}}}^{1/3}$ and $t\ge 0$ with

$$\begin{aligned}&|({\widehat{\mathfrak {a}}},\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}},{\widehat{\mathfrak {j}}}_0,{\widehat{\mathfrak {j}}}_1)(t)| +\min (1,\rho )\int _0^t|\rho {\widehat{\mathfrak {a}}}|\,\hbox {d}\tau +\rho ^2\int _0^t|({\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}})|\,\hbox {d}\tau +{{\mathcal {C}}}\int _0^t|({\widehat{\mathfrak {j}}}_0,{\widehat{\mathfrak {j}}}_1)|\,\hbox {d}\tau \nonumber \\&\quad \le K\biggl (|({\widehat{\mathfrak {a}}},\rho {\widehat{\mathfrak {a}}},{\widehat{\mathfrak {d}}},{\widehat{\mathfrak {t}}},{\widehat{\mathfrak {j}}}_0,{\widehat{\mathfrak {j}}}_1)(0)| +\int _0^t|({\widehat{A}},\rho {\widehat{A}},{\widehat{D}},{\widehat{\Theta }},{\widehat{J}}_0,{\widehat{J}}_1)|\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(69)

3.4 Middle-frequency decay estimates

This paragraph is devoted to the proof of global-in-time estimates for the solution to (40) in some suitable frequency range $\rho _\ell \le \rho \le \rho _h$ where $\rho _h$ is prescribed and $\rho _\ell$ will be specified below. Having in mind the study of the nonlinear system (25), it is natural to work at the same level regularity for |D|a,  d,  $|D|^{-1}\Theta ,$ $j_0$ and $j_1$ which, in Fourier variables, corresponds to $(\rho {\widehat{a}},{\widehat{d}},\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1}).$ We thus introduce ${\widehat{\phi }}\,{:=}\,\rho ^{-1}{\widehat{\Theta }}.$ In terms of $({\widehat{a}},{\widehat{d}},{\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1}),$ System (40) rewrites

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left( \begin{array}{c}{\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\phi }}\\ {\widehat{j}}_{0}\\ {\widehat{j}}_{1} \end{array}\right) +\left( \begin{array}{ccccc} 0&{} \quad \rho &{} \quad 0&{} \quad 0&{} \quad 0\\ rho&{} \quad \overline{\nu }\rho ^2&{} \quad -\rho ^2&{} \quad 0&{} \quad -1\\ 0&{} \quad \eta _2&{} \quad \eta _3+\overline{\kappa }\rho ^2&{} \quad -\eta _4\rho ^{-1}&{} \quad 0\\ 0&{} \quad 0&{} \quad -\eta _7\rho &{} \quad \eta _5&{} \quad \eta _6\rho \\ 0&{} \quad 0&{} \quad 0&{} \quad -\eta _6\rho &{} \quad \eta _8\end{array} \right) \left( \begin{array}{c}{\widehat{a}}\\ {\widehat{d}}\\ {\widehat{\phi }}\\ {\widehat{j}}_{0}\\ {\widehat{j}}_{1}\end{array}\right) =\left( \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right) \cdot \end{aligned}$$

(70)

To handle $({\widehat{a}},{\widehat{d}}),$ it is only a matter of using the standard barotropic estimates, considering the coupling with ${\widehat{\phi }}$ and ${\widehat{j}}_{1}$ as source terms. More precisely, taking advantage of the Lyapunov functional

$$\begin{aligned} 2(|{\widehat{a}}|^2+|{\widehat{d}}|^2)+|\nu \rho {\widehat{a}}|^2-2\nu \rho \mathfrak {R}({\widehat{d}} \,\overline{{\widehat{a}}}), \end{aligned}$$

we get for $\rho \nu \ge \rho _\ell$ (for any given $\rho _\ell >0$),

$$\begin{aligned} |(\overline{\nu }\rho {\widehat{a}}, {\widehat{d}})(t)|+\int _0^t|\rho {\widehat{a}}|\,\hbox {d}\tau +\overline{\nu }\rho ^2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau \le C\biggl ( |(\overline{\nu }\rho {\widehat{a}}, {\widehat{d}})(0)|+\int _0^t|{\widehat{j}}_{1}|\,\hbox {d}\tau +\int _0^t|\rho ^2{\widehat{\phi }}|\,\hbox {d}\tau \biggr ) \end{aligned}$$

(71)

for some constant C depending only on $\rho _\ell .$

Let us now concentrate on the system fulfilled by $({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1}),$ namely,

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t{\widehat{\phi }}+(\eta _3+\overline{\kappa }\rho ^2){\widehat{\phi }}-\eta _4\rho ^{-1}{\widehat{j}}_{0}={\widehat{\Phi }}\,{:=}\,-\eta _2{\widehat{d}},\\ \partial _t{\widehat{j}}_{0}+\eta _5{\widehat{j}}_{0}+\eta _6\rho {\widehat{j}}_{1}-\eta _7\rho {\widehat{\phi }}=0,\\ \partial _t{\widehat{j}}_{1}+\eta _8{\widehat{j}}_{1}-\eta _6\rho {\widehat{j}}_{0}=0.\end{array}\right. \end{aligned}$$

(72)

Let us put ${\widehat{\Phi }}$ to 0 for a while. In order to get decay estimates for $({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1}),$ we multiply the equation for ${\widehat{\phi }}$ by $\overline{{\widehat{\phi }}},$ and get:

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}|{\widehat{\phi }}|^2+(\eta _3+\overline{\kappa }\rho ^2)|{\widehat{\phi }}|^2-\eta _4\rho ^{-1}\mathfrak {R}({\widehat{j}}_{0}\,\overline{{\widehat{\phi }}})=0. \end{aligned}$$

(73)

Next, taking the inner product in $\mathbb {C}^2$ of the equations for $({\widehat{j}}_{0},{\widehat{j}}_{1})$ with $({\widehat{j}}_{0},{\widehat{j}}_{1}),$ we find out that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}|({\widehat{j}}_{0},{\widehat{j}}_{1})|^2+\eta _5|{\widehat{j}}_{0}|^2+\eta _8|{\widehat{j}}_{1}|^2-\eta _7\rho \mathfrak {R}({\widehat{j}}_{0}\overline{{\widehat{\phi }}})=0. \end{aligned}$$

(74)

In order to eliminate the last term (which tends to be predominant for large $\rho$), we compute

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\mathfrak {R}({\widehat{\phi }}\,\overline{{\widehat{j}}_{1}})+\bigl ((\eta _3+\eta _8)+\overline{\kappa }\rho ^2\bigr ) \mathfrak {R}({\widehat{\phi }}\,\overline{{\widehat{j}}_{1}}) -\eta _4\rho ^{-1}\mathfrak {R}({\widehat{j}}_{0}\,\overline{{\widehat{j}}_{1}})-\eta _6\rho \mathfrak {R}({\widehat{j}}_{0}\,\overline{{\widehat{\phi }}})=0. \end{aligned}$$

Hence setting for some $A>0$ to be fixed hereafter,

$$\begin{aligned}{{\mathcal {L}}}_\rho ^2\,{:=}\,A|{\widehat{\phi }}|^2+|({\widehat{j}}_{0},{\widehat{j}}_{1})|^2-2B\mathfrak {R}({\widehat{\phi }}\,\overline{{\widehat{j}}_{1}})\quad \hbox {with }\ B\,{:=}\,\frac{1}{\eta _6}\biggl (\eta _7+A\frac{\eta _4}{\rho ^2}\biggr ),\end{aligned}$$

we discover that

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}{{\mathcal {L}}}_\rho ^2+A(\eta _3+\overline{\kappa }\rho ^2)|{\widehat{\phi }}|^2 +\eta _5|{\widehat{j}}_{0}|^2+\eta _8|{\widehat{j}}_{1}|^2\nonumber \\&\quad -\,B(\eta _3+\eta _8+\overline{\kappa }\rho ^2)\mathfrak {R}({\widehat{\phi }}\,\overline{{\widehat{j}}_{1}}) +B\eta _4\rho ^{-1}\mathfrak {R}({\widehat{j}}_{0}\overline{{\widehat{j}}_{1}})=0. \end{aligned}$$

(75)

Note that we have

$$\begin{aligned} B\le \frac{2\eta _7}{\eta _4} \quad \hbox {if}\quad \rho ^2\ge \frac{\eta _4}{\eta _7}A, \end{aligned}$$

(76)

and thus, as may be easily seen by Young inequality,

$$\begin{aligned} {{\mathcal {L}}}_\rho ^2\approx |({\widehat{j}}_{0},{\widehat{j}}_{1},{\widehat{\phi }})|^2\quad \hbox {for }\ \rho \ge \sqrt{ \frac{\eta _4}{\eta _7}A} \end{aligned}$$

if A has been chosen so that

$$\begin{aligned} A>8\biggl (\frac{\eta _7}{\eta _6}\biggr )^2. \end{aligned}$$

(77)

Next, we see that by virtue of (76), we have

$$\begin{aligned} B\eta _4\rho ^{-1}\bigl |\mathfrak {R}({\widehat{j}}_{0}\overline{{\widehat{j}}_{1}})\bigr |\le 2\frac{\eta _4\eta _7}{\eta _6\rho }|{\widehat{j}}_{0}|\,|{\widehat{j}}_{1}|, \end{aligned}$$

whence

$$\begin{aligned} B\eta _4\rho ^{-1}\bigl |\mathfrak {R}({\widehat{j}}_{0}\overline{{\widehat{j}}_{1}})\bigr |\le \frac{\min (\eta _5,\eta _8)}{4}|({\widehat{j}}_{0},{\widehat{j}}_{1})|^2 \quad \hbox {if in addition}\quad \rho \ge \frac{4\eta _4\eta _7}{\eta _6\min (\eta _5,\eta _8)}\cdot \end{aligned}$$

(78)

Finally, still using (76), we have

$$\begin{aligned} B(\eta _3+\eta _8+\overline{\kappa }\rho ^2)\bigl |\mathfrak {R}({\widehat{\phi }}\,\overline{{\widehat{j}}_{1}})\bigr | \le \frac{\min (\eta _5,\eta _8)}{2}|{\widehat{j}}_{1}|^2+\frac{2}{\min (\eta _5,\eta _8)}\biggl (\frac{\eta _7(\eta _3+\eta _8+\overline{\kappa }\rho ^2)}{\eta _6}\biggr )^2|{\widehat{\phi }}|^2. \end{aligned}$$

As we want the last term to be bounded by $\frac{A}{2}(\eta _3+\overline{\kappa }\rho ^2)|{\widehat{\phi }}|^2,$ we eventually require A to be chosen so that

$$\begin{aligned} \frac{2}{\min (\eta _5,\eta _8)}\biggl (\frac{\eta _7(\eta _3+\eta _8+\overline{\kappa }\rho ^2)}{\eta _6}\biggr )^2 \le \frac{A}{2}(\eta _3+\overline{\kappa }\rho ^2)\quad \hbox {for all }\ \quad \rho \le \rho _h. \end{aligned}$$

Easy computations show that a sufficient condition for that is

$$\begin{aligned} A\ge 4\biggl (\frac{(\eta _3+\eta _8)\eta _7}{\eta _3\eta _6}\biggr )^2\biggl (\frac{\eta _3+\overline{\kappa }\rho _h^2}{\min (\eta _5,\eta _8)}\biggr )\cdot \end{aligned}$$

(79)

Let us sum up: our computations show that if A has been chosen so that (77) and (79) are fulfilled, and if we assume that

$$\begin{aligned} \min \biggl (\sqrt{\frac{\eta _4}{\eta _7}A},\frac{4\eta _4\eta _7}{\eta _6\min (\eta _5,\eta _8)}\biggr ) \le \rho \le \rho _h \end{aligned}$$

(80)

where $\rho _h$ is given, then we have ${{\mathcal {L}}}_\rho \approx |({\widehat{j}}_{0},{\widehat{j}}_{1},{\widehat{\phi }})$ and

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}{{\mathcal {L}}}_\rho ^2+A(\eta _3+\overline{\kappa }\rho ^2)|{\widehat{\phi }}|^2+\min (\eta _5,\eta _8)|({\widehat{j}}_{0},{\widehat{j}}_{1})|^2 \le 0. \end{aligned}$$

Let us recall that all the coefficients $\eta _i$ with $3\le i\le 8$ are of the same order ${{\mathcal {C}}}.$ Therefore one can take A of order 1 whenever $\overline{\kappa }\rho _h^2$ is of order ${{\mathcal {C}}},$ and thus for all $\rho$ satisfying (80) and $t\ge 0,$

$$\begin{aligned} |({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)|\le C_0e^{-c_0{{\mathcal {C}}}t} |({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)|. \end{aligned}$$

From that inequality, we conclude thanks to Duhamel’s formula that if we take the r.h.s. $\Phi =-\eta _2{\widehat{d}}$ into account then we have

$$\begin{aligned} |({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)|+{{\mathcal {C}}}\int _0^t|({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \le C_0\biggl (|({\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)|+\rho \int _0^t|{\widehat{d}}|\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(81)

Inserting that inequality in (71), we conclude that for all $\rho _0>0$ there exists a constant K depending only on $\eta _2$ and on the renormalized coefficients $\eta _i'\,{:=}\,\eta _i/{{\mathcal {C}}}$ for $2\le i\le 8$ so that if

$$\begin{aligned} \min \biggl (\frac{1}{\overline{\nu }},\frac{1}{\sqrt{\overline{\kappa }\overline{\nu }}}\biggr )\le \rho \le \rho _0\sqrt{\frac{{{\mathcal {C}}}}{\overline{\kappa }}}, \end{aligned}$$

(82)

then we have for all $t\ge 0$ and large enough ${{\mathcal {C}}},$

$$\begin{aligned}&\overline{\kappa }|(\overline{\nu }\rho {\widehat{a}},{\widehat{d}})(t)|+|(\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)| +\kappa \rho \int _0^t|{\widehat{a}}|\,\hbox {d}\tau +\overline{\nu }\rho ^2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau \nonumber \\&\quad +\,{{\mathcal {C}}}\int _0^t|(\rho ^{-1}\Theta ,{\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \le K\biggl (\overline{\kappa }|(\overline{\nu }\rho {\widehat{a}},{\widehat{d}})(0)|+|(\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)|\biggr )\cdot \end{aligned}$$

(83)

3.5 High-frequency decay estimates

We now want to prove global-in-time estimates for the solution to (40) in the case $\rho \gtrsim \sqrt{{{\mathcal {C}}}/\overline{\kappa }}.$ To this end, as for middle frequencies, it is convenient to work with ${\widehat{\phi }}=\rho ^{-1}{\widehat{\Theta }}$ rather than with ${\widehat{\Theta }}.$ Now from the equation satisfied by ${\widehat{\phi }}$ in (70), we readily get

$$\begin{aligned} |{\widehat{\phi }}(t)|+\overline{\kappa }\rho ^2\int _0^t|{\widehat{\phi }}|\,\hbox {d}\tau \le |{\widehat{\phi }}_0| +\eta _2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau +\eta _4\rho ^{-1}\int _0^t|{\widehat{j}}_{0}|\,\hbox {d}\tau \end{aligned}$$

(84)

while the last two lines of (70) yield

$$\begin{aligned} |({\widehat{j}}_{0},{\widehat{j}}_{1})(t)|+\min (\eta _5,\eta _8)\int _0^t|({\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \le |({\widehat{j}}_{0},{\widehat{j}}_{1})(0)|+\eta _7\rho \int _0^t|{\widehat{\phi }}|\,\hbox {d}\tau . \end{aligned}$$

(85)

Inserting (84) in (85) and omitting from now the dependency with respect to the coefficients $\eta _2$ and $\eta '_i,$ we discover that for $\rho ^2\gg \overline{\kappa }^{-1}{{\mathcal {C}}},$

$$\begin{aligned} |({\widehat{j}}_{0},{\widehat{j}}_{1})(t)|+{{\mathcal {C}}}\int _0^t|({\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \lesssim |({\widehat{j}}_{0},{\widehat{j}}_{1})(0)|+\frac{{{\mathcal {C}}}}{\overline{\kappa }\rho }\biggl (|{\widehat{\phi }}_0|+\int _0^t|{\widehat{d}}|\,\hbox {d}\tau \biggr ),\end{aligned}$$

(86)

and thus, resuming to (84),

$$\begin{aligned} |{\widehat{\phi }}(t)|+\overline{\kappa }\rho ^2\int _0^t|{\widehat{\phi }}|\,\hbox {d}\tau \lesssim |{\widehat{\phi }}_0| +\rho ^{-1}|({\widehat{j}}_{0},{\widehat{j}}_{1})(0)|+ \int _0^t|{\widehat{d}}|\,\hbox {d}\tau . \end{aligned}$$

(87)

In order to bound $({\widehat{a}},{\widehat{d}}),$ we combine (71), (86) and (87), so as to get

$$\begin{aligned}&|(\overline{\nu }\rho {\widehat{a}}, {\widehat{d}})(t)|+\int _0^t|\rho {\widehat{a}}|\,\hbox {d}\tau +\overline{\nu }\rho ^2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau \lesssim |(\overline{\nu }\rho {\widehat{a}}, {\widehat{d}})(0)|+\overline{\kappa }^{-1}\rho ^{-1}|(\rho {\widehat{\phi }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)|\\&\quad +\,{{\mathcal {C}}}^{-1}|({\widehat{j}}_{0},{\widehat{j}}_{1})(0)|+({\overline{\kappa }\rho ^2})^{-1}|\rho {\widehat{\phi }}_0| +\overline{\kappa }^{-1}(1+\rho ^{-1})\int _0^t|{\widehat{d}}|\,\hbox {d}\tau . \end{aligned}$$

The last term may be absorbed by the l.h.s. provided that $1+\rho ^{-1}\ll \overline{\kappa }\overline{\nu }\rho ^2.$ Resuming to (86) and (87), and remembering that ${\widehat{\Theta }}=\rho {\widehat{\phi }},$ we conclude that

$$\begin{aligned}&|(\overline{\nu }\rho {\widehat{a}},{\widehat{d}},(\overline{\kappa }\rho )^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)| +\rho \int _0^t|({\widehat{a}},{\widehat{\Theta }})|\,\hbox {d}\tau +\overline{\nu }\rho ^2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau +{{\mathcal {C}}}\int _0^t|({\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \nonumber \\&\quad \lesssim (|(\overline{\nu }\rho {\widehat{a}},{\widehat{d}},(\overline{\kappa }\rho )^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)| \qquad \hbox {for }\ \rho \gg \sqrt{{{\mathcal {C}}}/\overline{\kappa }}\ \hbox { and }\ {{\mathcal {C}}}\gg 1. \end{aligned}$$

(88)

3.6 Final statement of linear estimates

Here we recap the estimates that we obtained so far for (40), if ${{\mathcal {C}}}\gg 1.$

To this end, we first fix ${{\mathcal {C}}}_0$ and $\rho _h$ large enough so that (88) holds true for any ${{\mathcal {C}}}\ge {{\mathcal {C}}}_0$ and $\rho \ge \rho _h\sqrt{{{\mathcal {C}}}/\overline{\kappa }}.$ Then the analysis for the middle frequency ensures that, taking ${{\mathcal {C}}}_0$ larger if needed, Inequality (83) holds true for ${{\mathcal {C}}}\ge {{\mathcal {C}}}_0$ and $\rho _\ell \le \rho \le 2\rho _h\sqrt{{{\mathcal {C}}}/\overline{\kappa }}$ for some $\rho _\ell$ of order 1 depending only on the parameters of the system. Finally, for the low frequencies, one can use (69) for $\rho \le 2\rho _\ell$ if ${{\mathcal {C}}}$ is large enough.

We thus eventually have for ${{\mathcal {C}}}\ge {{\mathcal {C}}}_0$ the following three inequalities:⁵

• Low frequencies $0\le \rho \le 2\rho _\ell$: From (69) and (45), we get

  $$\begin{aligned}&|({\widehat{a}},{\widehat{d}},{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)| +\rho ^2\int _0^t|({\widehat{a}},{\widehat{d}},{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \nonumber \\&\quad +\,{{\mathcal {C}}}\int _0^t|({\widehat{\mathfrak {j}}}_0,{\widehat{j}}_{1})|\,\hbox {d}\tau \, \lesssim |({\widehat{a}},{\widehat{d}},{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)| \end{aligned}$$

  (89)

  with

  $$\begin{aligned} {\widehat{\mathfrak {j}}}_0\,{:=}\,{\widehat{j}}_{0}-\frac{\eta _2\eta _3}{\eta _4(\eta _3+\eta _5)}\rho \biggl ({\widehat{d}}+\frac{1}{\eta _8}{\widehat{j}}_{1}\biggr )-\frac{\eta _3}{\eta _4}{\widehat{\Theta }}. \end{aligned}$$

  (90)

• Middle frequencies $\rho _\ell \le \rho \le 2\rho _h\sqrt{{\mathcal {C}}}$: Inequality (83) gives

  $$\begin{aligned}&|(\rho {\widehat{a}},{\widehat{d}},\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)| +\rho \int _0^t|{\widehat{a}}|\,\hbox {d}\tau +\rho ^2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau +{{\mathcal {C}}}\int _0^t|(\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \nonumber \\&\quad \lesssim |(\rho {\widehat{a}},{\widehat{d}},\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)|. \end{aligned}$$

  (91)

• High frequencies $\rho \ge \rho _h\sqrt{{\mathcal {C}}}$: Inequality (88) implies that

  $$\begin{aligned}&|(\rho {\widehat{a}},{\widehat{d}},\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(t)| +\rho \int _0^t|({\widehat{a}},{\widehat{\Theta }})|\,\hbox {d}\tau +\rho ^2\int _0^t|{\widehat{d}}|\,\hbox {d}\tau +{{\mathcal {C}}}\int _0^t|({\widehat{j}}_{0},{\widehat{j}}_{1})|\,\hbox {d}\tau \nonumber \\&\quad \lesssim (|(\rho {\widehat{a}},{\widehat{d}},\rho ^{-1}{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)|. \end{aligned}$$

  (92)

Remark 3.1

Having a (small) overlap between the three regimes will be important in the sequel as the Fourier splitting device that we will use, namely the Littlewood-Paley decomposition, is not quite orthogonal.

Remark 3.2

Inequality (89) implies that for $0\le \rho \le 2\rho _\ell ,$ we have

$$\begin{aligned} {{\mathcal {C}}}\int _0^t|{\widehat{\zeta }}_0|\,\hbox {d}\tau \lesssim |({\widehat{a}},{\widehat{d}},{\widehat{\Theta }},{\widehat{j}}_{0},{\widehat{j}}_{1})(0)| \quad \hbox {with}\quad {\widehat{\zeta }}_0\,{:=}\,{\widehat{j}}_{0}-\frac{\eta _2\eta _3}{\eta _4(\eta _3+\eta _5)}\rho {\widehat{d}}-\frac{\eta _3}{\eta _4}{\widehat{\Theta }}. \end{aligned}$$

(93)

Let us also emphasize that the inequality for middle frequencies is stronger than that for high frequencies as regards $\Theta .$ Therefore (92) is fulfilled for all $\rho \ge \rho _\ell .$

Finally, regarding the incompressible part of the solution, namely $({\mathcal {P}}{\vec u},{\mathcal {P}}{\vec j}_1),$ we readily have from (33) that for all $\rho \ge 0$ and ${{\mathcal {C}}}\ge 0,$

$$\begin{aligned}&|\widehat{{\mathcal {P}}{\vec u}}(t)|+\rho ^2\int _0^t|\widehat{{\mathcal {P}}{\vec u}}|\,\hbox {d}\tau \le K\bigl (|\widehat{{\mathcal {P}}{\vec u}}(0)|+{{\mathcal {C}}}^{-1}|\widehat{{\mathcal {P}}{\vec j}_1}(0)|\bigr )\nonumber \\&\quad \hbox {and}\quad |\widehat{{\mathcal {P}}{\vec j}_1}(t)|+{{\mathcal {C}}}\int _0^t|\widehat{{\mathcal {P}}{\vec j}_1}|\,\hbox {d}\tau \le K|\widehat{{\mathcal {P}}{\vec j}_1}(0)|. \end{aligned}$$

(94)

4 The paralinearized system

In order to achieve the global existence result of Theorem 2.3, it is tempting to look at (25) as the linear system studied in the previous section. Indeed one expect to be able to handle the r.h.s. of (25) according to Duhamel’s formula. This unfortunately does not work because some of the convection terms will cause a loss of one derivative in the estimates, exactly as for the compressible Euler or Navier–Stokes equations. Paralinearizing (25) (that is including the “principal” part of the convection terms) is a standard way to overcome this difficulty. In the case we are interested in, it turns out that only ${\vec u}\cdot \nabla a$ causes a loss of derivative because the other convection terms may be counterbalanced by the parobolicity of the equations of ${\vec u}$ and of $\Theta .$ However, for symmetry reasons, it is convenient to paralinearize ${\vec u}\cdot \nabla {\vec u},$ too. This eventually leads to the following paralinearized version of System⁶ (25):

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a + T_{\vec v}\cdot \nabla a+\mathrm{div}{\vec u}=F,\\ \partial _t {\vec u}+T_{\vec v}\cdot \nabla {\vec u}- \frac{1}{Re}\ \underline{{{\mathcal {A}}}}{\vec u}+ \frac{1}{Ma^2}\ \underline{\alpha }_1\nabla a+ \frac{1}{Ma^2}\ \underline{\alpha }_2\nabla \Theta -\frac{1}{n}\ {{\mathcal {L}}}(\underline{\sigma }_a+{{\mathcal {L}}}_s\underline{\sigma }_s){\vec j}_1=\vec {G},\\ \partial _t \Theta -\frac{1}{ Pr Re}\,\underline{\kappa }\,\Delta \Theta +\underline{\alpha }_2\, \mathrm{div}{\vec u}-\frac{{{\mathcal {C}}}{{\mathcal {L}}}}{Pr} \ \underline{\sigma }_a\left( j_0-\underline{\alpha }'\Theta \right) =H,\\ \partial _t j_0 + \frac{1}{n}\ {{\mathcal {C}}} \mathrm{div}{\vec j}_1 + {{\mathcal {C}}}{{\mathcal {L}}}\underline{\sigma }_a\left( j_0-\underline{\alpha }'\Theta \right) =J_0,\\ \partial _t {\vec j}_1+ {{\mathcal {C}}} \nabla j_0 +{{\mathcal {C}}}{{\mathcal {L}}}(\underline{\sigma }_a+{{\mathcal {L}}}_s\underline{\sigma }_s){\vec j}_1=\vec {J}_1, \end{array}\right. \end{aligned}$$

(95)

where the velocity field ${\vec v}$ and the source terms F,  ${\vec G},$ H,  $J_0,$ ${\vec J}_1$ are given. The reader may refer to the Appendix for the definition of $T_{\vec v}\cdot \nabla a$ and $T_{\vec v}\cdot \nabla {\vec u}.$

The main result of this part is stated in the following proposition:

Proposition 4.1

There exists a constant ${{\mathcal {C}}}_0>0$ and an integer $k_0$ depending only on the parameters of System (95) such that if

$$\begin{aligned} {{\mathcal {C}}}\ge {{\mathcal {C}}}_0 \end{aligned}$$

(96)

and the threshold between low and high frequencies is at $2^{k_0}$ (see (135)) then the following inequalities hold true for all s and $s'$ in $\mathbb {R}$ $:$

• Low frequencies:

  $$\begin{aligned}&\Vert (a,{\vec u},\Theta ,j_0,{\vec j}_1)(t)\Vert ^\ell _{\dot{B}^s_{2,1}} +\int _0^t\left( \Vert (a,{\vec u},\Theta ,j_0,{\vec j}_1)\Vert _{\dot{B}^{s+2}_{2,1}}^\ell +{{\mathcal {C}}}\Vert (\zeta _0,{\vec j}_1)\Vert _{\dot{B}^s_{2,1}}^\ell \right) \nonumber \\&\quad \lesssim \Vert (a,{\vec u},\Theta ,j_0,{\vec j}_1)(0)\Vert ^\ell _{\dot{B}^s_{2,1}}+\int _0^t\Vert (J_0,{\vec J}_1,H)\Vert _{\dot{B}^{s}_{2,1}}^\ell \,\hbox {d}\tau \nonumber \\&\quad \quad +\,\int _0^t\Vert (F-T_{{\vec v}}\cdot \nabla a,{\vec G}-T_{{\vec v}}\cdot \nabla {\vec u})\Vert _{\dot{B}^s_{2,1}}^\ell \,\hbox {d}\tau , \end{aligned}$$

  (97)

  with

  $$\begin{aligned} \zeta _0\,{:=}\,j_0-\underline{\alpha }'\Theta -\frac{{\underline{\alpha }_2}\underline{\alpha }'}{\underline{\alpha }_1{{\mathcal {C}}}{{\mathcal {L}}}\underline{\sigma }_a\bigl (1+\frac{1}{Pr}\,\underline{\alpha }'\bigr )}\mathrm{div}{\vec u}. \end{aligned}$$

• High frequencies:

  $$\begin{aligned}&\Vert ({\vec u},j_0,{\vec j}_1)(t)\Vert ^h_{\dot{B}^{s'}_{2,1}}+\Vert a(t)\Vert _{\dot{B}^{s'+1}_{2,1}}^h +\Vert \Theta (t)\Vert _{\dot{B}^{s'-1}_{2,1}}^h +\int _0^t\left( \Vert (a,\Theta )\Vert _{\dot{B}^{s'+1}_{2,1}}^h+\Vert {\vec u}\Vert _{\dot{B}^{s'+2}_{2,1}}^h\right) \,\hbox {d}\tau \nonumber \\&\quad +\,{{\mathcal {C}}}\int _0^t\left( \Vert \Theta \Vert _{\dot{B}^{s'-1}_{2,1}}^h+\Vert (j_0,{\vec j}_1)\Vert _{\dot{B}^{s'}_{2,1}}^h\right) \,\hbox {d}\tau \lesssim \Vert ({\vec u},j_0,{\vec j}_1)(0)\Vert ^h_{\dot{B}^{s'}_{2,1}}\nonumber \\&\quad +\,\Vert a(0)\Vert _{\dot{B}^{s'+1}_{2,1}}^h+\Vert \Theta (0)\Vert _{\dot{B}^{s'-1}_{2,1}}^h\nonumber \\&\quad +\,\int _0^t\left( \Vert ({\vec G},J_0,{\vec J}_1)\Vert _{\dot{B}^{s'}_{2,1}}^h+\Vert F\Vert _{\dot{B}^{s'+1}_{2,1}}^h+\Vert H\Vert _{\dot{B}^{s'-1}_{2,1}}^h\right) \,\hbox {d}\tau \nonumber \\&\quad +\,\int _0^t\nabla {\vec v}\Vert _{L^\infty } \left( \Vert (\nabla a,{\vec u})\Vert ^{h}_{\dot{B}^{s'}_{2,1}}+\Vert (a,{\vec u})\Vert ^\ell _{\dot{B}^s_{2,1}}\right) \,\hbox {d}\tau . \end{aligned}$$

  (98)

Proof

As our proof will be essentially based on the results of the previous section, we rescale System (95) as in (36). From the point of view of a priori estimates, this is harmless for the numbers coming into play in the rescaling process are independent of ${{\mathcal {C}}}.$

Now as in [12], we localize that (rescaled) system according to Littlewood-Paley decomposition (shortly introduced in the Appendix). Setting $a_k\,{:=}\,\dot{\Delta }_ka,$ ${\vec u}_k\,{:=}\,\dot{\Delta }_k{\vec u}$ and so on, we get

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t a_k +\mathrm{div}{\vec u}_k=F'_k,\\ \partial _t {\vec u}_k- \overline{\mu }\Delta {\vec u}_k-(\overline{\lambda }+\overline{\mu })\nabla \mathrm{div}{\vec u}_k +\nabla a_k+\nabla \Theta _k-{\vec j}_{1,k}=\vec {G}'_k,\\ \partial _t \Theta _k-\overline{\kappa }\Delta \Theta _k +\eta _2 \mathrm{div}{\vec u}_k+\eta _3\Theta _k- \eta _4j_{0,k}=H_k,\\ \partial _t j_{0,k} + \eta _5 j_{0,k}+\eta _6\mathrm{div}{\vec j}_{1,k} -\eta _7\Theta _k=J_{0,k},\\ \partial _t {\vec j}_{1,k}+\eta _8{\vec j}_{1,k}+ \eta _6 \nabla j_{0,k}=\vec {J}_{1,k}, \end{array}\right. \end{aligned}$$

(99)

with $F'_k\,{:=}\, F_k- \dot{\Delta }_k(T_{{\vec v}}\cdot \nabla a)$ and $\vec {G}'_k\,{:=}\,{\vec G}_k-\dot{\Delta }_k(T_{\vec v}\cdot \nabla {\vec u}).$

Estimates for low frequencies Let

$$\begin{aligned} X_k\,{:=}\,\left( a_k, {\vec u}_k, \Theta _k, j_{0,k}, j_{1,k}\right) ^{T}\quad \hbox {and}\quad Y_k\,{:=}\,\left( F'_k, {\vec G}'_k, H_k, J_{0,k}, {\vec J}_{1,k}\right) ^T\cdot \end{aligned}$$

Denoting by A(D) the infinitesimal generator associated to the semi-group corresponding to the (rescaled) System (99), Duhamel’s formula yields

$$\begin{aligned} X_k(t)=e^{A(D)t}X_k(0)+\int _0^te^{A(D)(t-\tau )}Y_k(\tau )\,\hbox {d}\tau . \end{aligned}$$

(100)

Applying Fourier–Plancherel theorem, following the computations leading to (89), using Remark 3.2 and remembering (94) to handle the incompressible part of ${\vec u}$ and ${\vec j}_1,$ we get some $k_0\in \mathbb {Z}$ such that for all $k\le k_0,$

$$\begin{aligned} \Vert X_k(t)\Vert _{L^2}+2^{2k}\int _0^t\Vert X_k\Vert _{L^2}\,\hbox {d}\tau +{{\mathcal {C}}}\int _0^t\Vert (\zeta _{0,k},{\vec j}_{1,k})\Vert _{L^2}\,\hbox {d}\tau \le C\biggl ( \Vert X_k(0)\Vert _{L^2}+\int _0^t\Vert Y_k\Vert _{L^2}\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(101)

Multiplying both sides by $2^{ks}$ and summing up over $k\le k_0$ yields (97).

Estimates for high frequencies Here paralinearization is fundamental, as it allows to avoid the loss of one derivative that may be caused by the convection term in the first equation of (99). Even though the final estimate will be the same for any frequency larger than $2^{k_0}$ (where the integer $k_0$ is chosen so that $2\rho _\ell \le 2^{k_0}<4\rho _\ell$), we have to separate our analysis into two sub-cases corresponding to middle frequencies (i.e., $k_0\le k\le 1+\log _2(\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }})$) and $k\ge \log _2(\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }})$ because different Lyapunov functionals have been used to obtain (91) and (92).

Let us first focus on middle frequencies : $k_0\le k\le 1+\log _2(\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }}).$ Following the analysis of the previous section, we introduce the Lyapunov functional

$$\begin{aligned} L_k^2\,{:=}\,2\Vert a_k\Vert _{L^2}^2+2\Vert {\vec u}_k\Vert _{L^2}^2+\overline{\nu }^2\Vert \nabla a_k\Vert _{L^2}^2+2\overline{\nu }(\nabla a_k|u_k), \end{aligned}$$

and find out that

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t} L_k^2+\overline{\mu }\Vert {\mathcal {P}}u_k\Vert _{L^2}^2+\overline{\nu }\Vert (\nabla a_k,\nabla {\mathcal {Q}}{\vec u}_k)\Vert _{L^2}^2 = 2(F'_k|a_k)+2\bigl (({\vec j}_{1,k}-\nabla \Theta _k+{\vec G}'_k)|{\vec u}_k\bigr )\\&\quad +\,\bigl (\overline{\nu }\nabla F'_k|(\overline{\nu }\nabla a_k+{\vec u}_k)\bigr ) +\bigl (\overline{\nu }\nabla a_k|({\vec j}_{1,k}-\nabla \Theta _k+{\vec G}'_k)\bigr ). \end{aligned}$$

The paraconvection terms may be bounded by means of Lemma 4.1 in [12]. More precisely, there exists an integer $N_0$ (depending only on the supports of the functions $\varphi$ and $\chi$ involved in the definition of Littlewood–Paley decomposition) and some constant C so that

$$\begin{aligned}&|(\dot{\Delta }_k(T_{{\vec v}}\cdot \nabla z)|z_k)|\le C\Vert \nabla {\vec v}\Vert _{L^\infty }\Vert z_k\Vert _{L^2}\displaystyle \sum _{|k'-k|\le N_0}\Vert z_{k'}\Vert _{L^2} \quad \hbox {with}\quad z=a,{\vec u},\\&\quad |(\nabla \dot{\Delta }_k(T_{\vec v}\cdot \nabla a)|\nabla a_k)|\le C\Vert \nabla {\vec v}\Vert _{L^\infty }\Vert \nabla a_k\Vert _{L^2}\displaystyle \sum _{|k'-k|\le N_0}\Vert \nabla a_{k'}\Vert _{L^2},\\&\quad \Bigl |\bigl (\overline{\nu }\nabla \dot{\Delta }_k(T_{\vec v}\cdot \nabla a)|{\vec u}_k\bigr )+\bigl (\dot{\Delta }_k(T_{\vec v}\cdot \nabla {\vec u})|\overline{\nu }\nabla a_k\bigr )\Bigr |\\&\quad \le C\Vert (\overline{\nu }\nabla a_k,{\vec u}_k)\Vert _{L^2}\sum _{|k'-k|\le N_0}\Vert (\overline{\nu }\nabla a_{k'},{\vec u}_{k'})\Vert _{L^2}. \end{aligned}$$

Using Bernstein inequality, noticing that

$$\begin{aligned} L_k\approx \Vert \nabla a_k\Vert _{L^2}+\Vert {\vec u}_k\Vert _{L^2}\quad \hbox {if }\ k\ge k_0, \end{aligned}$$

(102)

and integrating with respect to time, we thus get for some constant $C=C(\overline{\lambda },\overline{\mu }),$

$$\begin{aligned}&L_k(t)+\int _0^tL_k\,\hbox {d}\tau \le C\biggl (L_k(0) +\int _0^t\Vert (\nabla F_k,{\vec G}_k)\Vert _{L^2}\,\hbox {d}\tau \nonumber \\&\quad + \sum _{|k'-k|\le N_0}\int _0^t\Vert \nabla {\vec v}\Vert _{L^\infty }L_{k'}\,\hbox {d}\tau +\int _0^t\Vert {\vec j}_{1,k}-\nabla \Theta _k\Vert _{L^2}\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(103)

Of course, from the localized velocity equation, we also gather that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert {\vec u}_k\Vert _{L^2}^2+\overline{\mu }\Vert \nabla {\mathcal {P}}{\vec u}_k\Vert _{L^2}^2+\overline{\nu }\Vert {\mathcal {Q}}{\vec u}_k\Vert _{L^2}^2 =\bigl (({\vec j}_{1,k}-\nabla \Theta _k+{\vec G}'_k)|{\vec u}_k\bigr )-(\nabla a_k|{\vec u}_k), \end{aligned}$$

which implies, for some $C=C(\overline{\lambda },\overline{\mu }),$

$$\begin{aligned}&\Vert {\vec u}_k(t)\Vert _{L^2}+2^{2k}\int _0^t\Vert {\vec u}_k\Vert _{L^2}\,\hbox {d}\tau \le C\biggl (\int _0^t\Vert {\vec G}_k+{\vec j}_{1,k}-\nabla \Theta _k\Vert _{L^2}\,\hbox {d}\tau \\&\quad +\,\sum _{|k'-k|\le N_0}\int _0^t\Vert \nabla {\vec v}\Vert _{L^\infty }\Vert {\vec u}_{k'}\Vert _{L^2}\,\hbox {d}\tau +\int _0^t\Vert \nabla a_k\Vert _{L^2}\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

Therefore combining with (103) and using (102) allows to exhibit the parabolic behavior of ${\vec u}$: there exists some constant $C=C(\overline{\lambda },\overline{\mu })$ such that for all $k\ge k_0,$

$$\begin{aligned}&L_k(t)+\int _0^t\bigl (\Vert \nabla a_k\Vert _{L^2}+2^{2k}\Vert {\vec u}_k\Vert _{L^2}\bigr )\,\hbox {d}\tau \le C\biggl (L_k(0) +\int _0^t\Vert (\nabla F_k,{\vec G}_k)\Vert _{L^2}\,\hbox {d}\tau \nonumber \\&\quad +\, \sum _{|k'-k|\le N_0}\int _0^t\Vert \nabla {\vec v}\Vert _{L^\infty }L_{k'}\,\hbox {d}\tau +\int _0^t\Vert {\vec j}_{1,k}-\nabla \Theta _k\Vert _{L^2}\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(104)

In order to bound $\Theta _k,$ $j_{0,k}$ and ${\vec j}_{1,k},$ we consider the sub-system corresponding to the last three lines of (99), looking at $\eta _2\mathrm{div}{\vec u}_k$ as a source term. Then we introduce the pseudo-differential operator $B(D)\,{:=}\,\eta _6^{-1}(\eta _7\mathrm{Id}\,+A\eta _4(-\Delta )^{-1})$ (where the positive number A is chosen as in (79) with $\rho _h=\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }}$) and the Lyapunov functional

$$\begin{aligned} M_k^2\,{:=}\,A\Vert \Lambda ^{-1}\Theta _k\Vert _{L^2}^2+\Vert (j_{0,k},{\vec j}_{1,k})\Vert _{L^2}^2 -2(B(D)\Lambda ^{-1}\Theta _k|\Lambda ^{-1}\mathrm{div}{\vec j}_{1,k}). \end{aligned}$$

Following the computations leading to (75), we discover that

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}M_k^2+A\eta _3\Vert \Lambda ^{-1}\Theta _k\Vert _{L^2}^2+A\overline{\kappa }\Vert \Theta _k\Vert _{L^2}^2 +\eta _5\Vert j_{0,k}\Vert _{L^2}^2+\eta _8\Vert {\vec j}_{1,k}\Vert _{L^2}^2\\&\quad -\,\Bigl (B(D)((\eta _3+\eta _8)\mathrm{Id}\,+\overline{\kappa }(-\Delta )^{-1})\Lambda ^{-1}\Theta _k|{\vec j}_{1,k}\Bigr ) +\eta _4\Bigl (B(D)(-\Delta )^{-1}j_0|\mathrm{div}{\vec j}_{1,k}\Bigr )\\&\quad =A(\Lambda ^{-1}(H_k-\eta _2\mathrm{div}{\vec u}_k)|\Lambda ^{-1}\Theta _k)+(j_{0,k}|J_{0,k}) +({\vec j}_{1,k}|{\vec J}_{1,k}) \\&\qquad - \bigl ((\Lambda ^{-1}(H_k-\eta _2\mathrm{div}{\vec u}_k)|B(D){\vec j}_{1,k}\bigr ) -({\vec J}_{1,k}|B(D)\Lambda ^{-1}\Theta _k). \end{aligned}$$

Then mimicking the arguments leading to (81) and using Fourier–Plancherel theorem, we eventually get

$$\begin{aligned}&\Vert (\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})(t)\Vert _{L^2} +{{\mathcal {C}}}\int _0^t \Vert (\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})\Vert _{L^2}\,\hbox {d}\tau \\&\quad \lesssim \Vert (\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})(0)\Vert _{L^2} +\int _0^t \Vert (\Lambda ^{-1}H_k,J_{0,k},{\vec J}_{1,k})\Vert _{L^2}\,\hbox {d}\tau +\int _0^t\Vert {\vec u}_k\Vert _{L^2}\,\hbox {d}\tau . \end{aligned}$$

Combining with (104) thus yields for $k_0\le k\le 1+\log _2(\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }}),$

$$\begin{aligned}&\Vert (\nabla a_k,{\vec u}_k,\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})(t)\Vert _{L^2} +\int _0^t\Vert \nabla a_k\Vert _{L^2}\,\hbox {d}\tau +2^{2k}\int _0^t\Vert {\vec u}_k\Vert _{L^2}\,\hbox {d}\tau \nonumber \\&\quad +\,{{\mathcal {C}}}\int _0^t \Vert (\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})\Vert _{L^2}\,\hbox {d}\tau \lesssim \Vert (\nabla a_k,{\vec u}_k,\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})(0)\Vert _{L^2}\nonumber \\&\quad +\,\int _0^t \Vert (\nabla F_k,{\vec G}_k,\Lambda ^{-1}H_k,J_{0,k},{\vec J}_{1,k})\Vert _{L^2}\,\hbox {d}\tau +\sum _{|k'-k|\le N_0} \int _0^t\Vert \nabla {\vec v}\Vert _{L^\infty }\Vert (\nabla a_k,{\vec u}_k)\Vert _{L^2}\,\hbox {d}\tau .\nonumber \\ \end{aligned}$$

(105)

Let us finally go the case $k>\log _2(\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }}).$ Then applying an energy method to the equation of $\Lambda ^{-1}\Theta _k$ yields

$$\begin{aligned}&\Vert \Lambda ^{-1}\Theta _k(t)\Vert _{L^2}+\overline{\kappa }\int _0^t\Vert \Theta _k\Vert _{L^2}\,\hbox {d}\tau \le \Vert \Lambda ^{-1}\Theta _k(0)\Vert _{L^2}\nonumber \\&\quad +\,\int _0^t\bigl (\Vert \Lambda ^{-1}H_k\Vert _{L^2}+\eta _4\Vert \Lambda ^{-1} j_{0,k}\Vert _{L^2} +\eta _2\Vert {\vec u}_k\Vert _{L^2}\bigr )\,\hbox {d}\tau , \end{aligned}$$

(106)

and for the radiative modes, we readily have

$$\begin{aligned}&\Vert (j_{0,k},{\vec j}_{1,k})(t)\Vert _{L^2}+\min (\eta _5,\eta _8)\int _0^t\Vert (j_{0,k},{\vec j}_{1,k})\Vert _{L^2}\,\hbox {d}\tau \le \Vert (j_{0,k},{\vec j}_{1,k})(0)\Vert _{L^2}\nonumber \\&\quad +\,\eta _7\int _0^t\Vert \Theta _k\Vert _{L^2}\,\hbox {d}\tau +\int _0^t\Vert (J_{0,k},{\vec J}_{1,k})\Vert _{L^2}\,\hbox {d}\tau . \end{aligned}$$

(107)

Hence combining with (104) and taking $\rho _0$ large enough, we get for $k>\log _2(\rho _0\sqrt{{{\mathcal {C}}}/\overline{\kappa }}),$

$$\begin{aligned}&\Vert (\nabla a_k,{\vec u}_k,\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})(t)\Vert _{L^2}\\&\qquad +\int _0^t\bigl (\Vert \nabla a_k\Vert _{L^2}+2^{2k}\Vert {\vec u}_k\Vert _{L^2} +2^k\Vert \Theta _k\Vert _{L^2}+{{\mathcal {C}}}\Vert (j_{0,k},{\vec j}_{1,k})\Vert _{L^2}\bigr )\hbox {d}\tau \\&\quad \lesssim \Vert (\nabla a_k,{\vec u}_k,\Lambda ^{-1}\Theta _k,j_{0,k},{\vec j}_{1,k})(0)\Vert _{L^2} +\int _0^t\bigl (\Vert (\nabla F_k,{\vec G}_k,\Lambda ^{-1}H_k, J_{0,k},{\vec J}_{1,k})\Vert _{L^2}\,\hbox {d}\tau \\&\quad \quad +\,\sum _{|k'-k|\le N_0} \int _0^t\Vert \nabla v\Vert _{L^\infty }\Vert (\nabla a_{k'},{\vec u}_{k'})\Vert _{L^2}\,\hbox {d}\tau . \end{aligned}$$

Putting together with Inequality (105), multiplying both sides by $2^{ks'}$ and summing up over $k\ge k_0$ completes the proof of (98). Note that owing to the sum over $k',$ there is a small overlap with low frequencies, which explains the presence of the last term of (98).

5 The well-posedness issue in the critical regularity framework

This section is mainly devoted to the proof of Theorem 2.3. In passing, we sketch the proof of our local-in-time statement in the critical framework (Theorem 2.2) and justify the nonrelativistic limit pointed out in Corollary 2.1.

Let us first say a few words on the uniqueness issue, which is the consequence of stability estimates in a suitable space. As usual, as a part of System (24) [or (25)] is quasilinear hyperbolic, proving (directly) stability estimates in the solution space $E^{\frac{n}{2}-1}$ is hopeless. The loss of one derivative coming from the density equation induces us to use the larger space $E^{\frac{n}{2}-2}$ (or rather, its local-in-time version).

In high dimension $n\ge 4$ indeed, one can prove stability estimates in $E^{\frac{n}{2}-2},$ just by combining standard hyperbolic and parabolic estimates, and product laws. The proof goes along the lines of that for the nonradiative polytropic compressible Navier–Stokes equations in [7] and does not present any new difficulty (apart from wearisomeness). The case $n=3$ turns out to be critical and one cannot achieve stability estimates in $E^{-\frac{1}{2}}$ by a direct application of hyperbolic and parabolic estimates because some nonlinear terms are not under control. For example, the product of two functions in $\dot{B}^{\frac{1}{2}}_{2,1}(\mathbb {R}^3)$ and $\dot{B}^{-\frac{1}{2}}_{2,1}(\mathbb {R}^3),$ respectively, does not belong to $\dot{B}^{-\frac{3}{2}}_{2,1}(\mathbb {R}^3)$ but to the slightly larger Besov space $\dot{B}^{-\frac{3}{2}}_{2,\infty }(\mathbb {R}^3).$ This obstacle may be overcome by proving stability estimates in a wider space (roughly, Besov spaces $\dot{B}^s_{2,1}$ have been changed to $\dot{B}^s_{2,\infty }$ in the definition of $E^{-\frac{1}{2}}$), and using a logarithmic interpolation inequality. This is just an adaptation of the corresponding proof for nonradiative flows (see [8]).

As for the existence issue, it is very similar to that of the barotropic case (see Subsection 5.1. in [12]). It is only a matter of combining a priori estimates for transport equations (to handle a), hyperbolic symmetrizable systems with constant coefficients (radiative unknowns) and parabolic equations or systems (for the temperature and velocity). The main difficulty is that the velocity and temperature equations have nonconstant coefficients, depending on a in their leading order, and that a has critical regularity. Exactly as in [12], this may be overcome by splitting a into some (smooth) low-frequency part $\dot{S}_ma,$ and small high-frequency part $(\mathrm{Id}\,-\dot{S}_m)a,$ treated as a remainder source term. The parameter $m\in \mathbb {Z}$ has to be adjusted conveniently according to the decay of the high frequencies of the initial data $a^0.$ As presenting the whole proof would be a bit lengthy, and does not require any new idea compared to the barotropic case, we skip the details.

The rest of this section is devoted to the global existence statement of Theorem 2.3. The key is the proof of the global a priori estimates (29) and (30) for smooth solutions to (25). As those estimates are uniquely based on energy arguments, Friedrichs method (used in, e.g., Chap. 10 of [1] in the nonradiative case) allows to construct a sequence of approximate smooth solutions satisfying exactly the same estimates.

The proof of global a priori estimates for a smooth enough solution $(a,{\vec u},\Theta ,j_0,{\vec j}_1)$ to System (25) relies on Proposition 4.1 with $s=s'=\frac{n}{2}-1,$ ${\vec v}={\vec u},$ and source terms⁷

$$\begin{aligned} F:= & {} T_{\vec u}\cdot \nabla a-{\vec u}\cdot \nabla a-a\mathrm{div}{\vec u},\\ {\vec G}:= & {} T_{\vec u}\cdot \nabla {\vec u}-{\vec u}\cdot \nabla {\vec u}+\frac{1}{1+a}\bigl (\mathrm{div}(2\mu D{\vec u})+\nabla (\lambda \mathrm{div}{\vec u})- \underline{{{\mathcal {A}}}}{\vec u}\bigr )\\&+\,\bigl (\pi '_0(1)-\frac{\pi '_0(1+a)}{1+a}+\pi '_1(1)-(1+\Theta )\frac{\pi '_1(1+a)}{1+a}\bigr )\nabla a\\&+\,\bigl (\pi _1(1)-\frac{\pi _1(1+a)}{1+a}\bigr )\nabla \Theta -\bigl (\frac{\sigma _a+\sigma _s}{n}\bigr )\bigl (\frac{a}{1+a}\Bigr ){\vec j}_1,\\ H:= & {} -{\vec u}\cdot \nabla \Theta +\frac{1}{1+a}\mathrm{div}\bigl ((\kappa -\underline{\kappa })\nabla \Theta \bigr ) +\bigl (\pi _1(1)-(1+\Theta )\frac{\pi _1(1+a)}{1+a}\bigr )\mathrm{div}{\vec u}\\&+\,\frac{1}{n}\bigl (\frac{\sigma _a+\sigma _s}{1+a}\bigr ){\vec j}_1\cdot {\vec u}-{{\mathcal {C}}}\,\sigma _a\,\frac{a}{1+a}\,\bigl (\underline{\alpha }'\Theta -j_0\bigr )\\&+\,\frac{1}{1+a}\bigl (2\underline{\mu }D{\vec u}:D{\vec u}+\underline{\lambda }(\mathrm{div}{\vec u})^2\bigr ),\\ J_0:= & {} 0 \quad \hbox {and}\quad {\vec J}_1\,{:=}\,{{\mathcal {C}}}(\sigma _s-\underline{\sigma }_s){\vec j}_1. \end{aligned}$$

Denoting

$$\begin{aligned} X(t)&{:=}&\Vert (a,\Theta )(t)\Vert ^\ell _{\dot{B}^{\frac{n}{2}-1}_{2,1}}+\Vert a(t)\Vert ^h_{\dot{B}^{\frac{n}{2}}_{2,1}}+ \Vert \Theta (t)\Vert ^h_{\dot{B}^{\frac{n}{2}-2}_{2,1}}+\Vert ({\vec u},j_0,{\vec j}_1)(t)\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\\&+\,\int _0^t\Bigl (\Vert (a,\Theta ,j_0,{\vec j}_1)\Vert _{\dot{B}^{\frac{n}{2}+1}_{2,1}}^\ell +\Vert {\vec u}\Vert _{\dot{B}^{\frac{n}{2}+1}_{2,1}} +\Vert (a,\Theta )\Vert ^h_{\dot{B}^{\frac{n}{2}}_{2,1}}\Bigr )\hbox {d}\tau \\&+\,{{\mathcal {C}}}\int _0^t\Bigl (\Vert {\vec j}_1\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}+\Vert \zeta _0\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^\ell +\Vert j_0\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^h+\Vert \Theta \Vert _{\dot{B}^{\frac{n}{2}-2}_{2,1}}^h\Bigr )\,\hbox {d}\tau , \end{aligned}$$

we get

$$\begin{aligned}&X(t)\lesssim X(0)+\int _0^t\Vert \nabla {\vec u}\Vert _{L^\infty } X\,\hbox {d}\tau +\int _0^t\bigl ( \Vert J_0\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}+\Vert {\vec J}_1\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\bigr )\,\hbox {d}\tau \nonumber \\&\quad +\,\int _0^t\Vert (F-T_{\vec u}\cdot \nabla a,{\vec G}-T_{\vec u}\cdot \nabla {\vec u},H)\Vert ^\ell _{\dot{B}^{\frac{n}{2}-1}_{2,1}} \nonumber \\&\quad +\, \int _0^t\bigl (\Vert F\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}^h+\Vert {\vec G}\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^h +\Vert H\Vert _{\dot{B}^{\frac{n}{2}-2}_{2,1}}^h\bigr )\,\hbox {d}\tau . \end{aligned}$$

(108)

Note that $\Vert \nabla {\vec u}\Vert _{L^\infty }\lesssim \Vert {\vec u}\Vert _{\dot{B}^{\frac{n}{2}+1}_{2,1}}.$ Therefore in order to close the estimates globally for small X(0) (that is small data), it suffices to bound the last three integrals in (108) by $CX^2(t).$ For that, we shall use repeatedly the fact that

$$\begin{aligned} \Vert a\Vert ^\ell _{\dot{B}^{\frac{n}{2}-1}_{2,1}} +\Vert a\Vert ^h_{\dot{B}^{\frac{n}{2}}_{2,1}}\approx \Vert a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}\cap \dot{B}^{\frac{n}{2}}_{2,1}} \ \hbox { and }\ \Vert \Theta \Vert ^\ell _{\dot{B}^{\frac{n}{2}-1}_{2,1}} +\Vert \Theta \Vert ^h_{\dot{B}^{\frac{n}{2}-2}_{2,1}}\approx \Vert \Theta \Vert _{\dot{B}^{\frac{n}{2}-2}_{2,1}+\dot{B}^{\frac{n}{2}-1}_{2,1}}. \end{aligned}$$

(109)

Another useful property is that, owing to interpolation and Hölder inequality,

$$\begin{aligned} \Vert a\Vert _{L^2_t(\dot{B}^{\frac{n}{2}}_{2,1})}\lesssim X(t). \end{aligned}$$

(110)

We shall finally assume that

$$\begin{aligned} \Vert a\Vert _{L^\infty (\mathbb {R}^+\times \mathbb {R}^n)}\quad \hbox {is small enough}, \end{aligned}$$

(111)

a property that will be used implicitly whenever composition estimates are applied. Of course, as we will get eventually that $\Vert a\Vert _{L^\infty \big (\dot{B}^{\frac{n}{2}}_{2,1}\big )}$ is small, and as $\dot{B}^{\frac{n}{2}}_{2,1}$ is embedded in $L^\infty ,$ Assumption (111) may be justified a posteriori.

Step 1. Estimates for F and $F-T_{{\vec u}}\cdot \nabla a$ Because $F-T_{\vec u}\cdot \nabla a=-{\vec u}\cdot \nabla a -a\mathrm{div}{\vec u}$ and the product maps $\dot{B}^{\frac{n}{2}}_{2,1}\times \dot{B}^{\frac{n}{2}-1}_{2,1}$ in $\dot{B}^{\frac{n}{2}-1}_{2,1},$ we readily have

$$\begin{aligned} \Vert F-T_{\vec u}\cdot \nabla a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\lesssim \Vert {\vec u}\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}\Vert \nabla a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}. \end{aligned}$$

Similarly, Lemma 7.2 implies that

$$\begin{aligned} \Vert F\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}\lesssim \Vert {\vec u}\Vert _{\dot{B}^{\frac{n}{2}+1}_{2,1}}\Vert a\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}. \end{aligned}$$

Therefore for all $t>0,$

$$\begin{aligned} \int _0^t\bigl (\Vert F-T_{\vec u}\cdot \nabla a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^\ell +\Vert F\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}^h\bigr )\,\hbox {d}\tau \lesssim X^2(t). \end{aligned}$$

(112)

Step 2. Estimates for ${\vec G}$ and ${\vec G}-T_{{\vec u}}\cdot \nabla {\vec u}$ Arguing as for F,  we get

$$\begin{aligned} \Vert {\vec u}\cdot \nabla {\vec u}-T_{{\vec u}}\cdot \nabla {\vec u}\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}} +\Vert {\vec u}\cdot \nabla {\vec u}\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\lesssim \Vert {\vec u}\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}^2. \end{aligned}$$

Next, we observe that

$$\begin{aligned} \frac{1}{1+a}\,\mathrm{div}\left( \mu D{\vec u}\right) -\underline{\mu }\,\mathrm{div}D{\vec u}=\frac{1}{1+a}\mathrm{div}\left( (\mu -\underline{\mu })D{\vec u}\right) -\frac{a}{1+a}\,\underline{\mu }\,\mathrm{div}D{\vec u}\end{aligned}$$

and a similar equality for $\frac{1}{1+a}\,\nabla (\lambda \mathrm{div}{\vec u})-\underline{\lambda }\nabla \mathrm{div}{\vec u}.$ Hence combining Lemmas 7.3 and 7.4 yields

$$\begin{aligned} \textstyle \Vert \frac{1}{1+a}\bigl (\mathrm{div}(2\mu D{\vec u})+\nabla (\lambda \mathrm{div}{\vec u})- \underline{{{\mathcal {A}}}}{\vec u}\bigr )\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\lesssim \left( 1+\Vert a\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}\right) \Vert a\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}\Vert \nabla {\vec u}\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}. \end{aligned}$$

The terms of ${\vec G}$ involving the pressure may be written

$$\begin{aligned} \pi _2(a)\nabla a+ \pi _3(a)\Theta \nabla a+\pi _4(a)\nabla \Theta \quad \hbox {with }\ \pi _2(0)=\pi _4(0)=0. \end{aligned}$$

Combining Lemma 7.4 and product estimates, we get

$$\begin{aligned} \Vert \pi _2(a)\nabla a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\lesssim \Vert a\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}\Vert \nabla a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}. \end{aligned}$$

For the other two terms, we decompose $\Theta$ into $\Theta ^\ell +\Theta ^h$ so as to write that

$$\begin{aligned} \pi _3(a)\Theta \nabla a=\pi _3(a)\Theta ^\ell \nabla a+\pi _3(a)\Theta ^h\nabla a \quad \hbox {and}\quad \pi _4(a)\nabla \Theta =\pi _4(a)\nabla \Theta ^\ell +\pi _4(a)\nabla \Theta ^h. \end{aligned}$$

We thus get

$$\begin{aligned} \begin{aligned} \Vert \pi _3(a)\Theta \nabla a\Vert _{L^1_t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )}&\lesssim \left( 1+\Vert a\Vert _{L^\infty _t\big (\dot{B}^{\frac{n}{2}}_{2,1}\big )}\right) \left( \Vert \Theta ^\ell \Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}}_{2,1}\big )}\Vert \nabla a\Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )}\right. \\&\quad \left. +\,\Vert \Theta ^h\Vert _{L^1_t\big (\dot{B}^{\frac{n}{2}}_{2,1}\big )}\Vert \nabla a\Vert _{L^\infty _t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )}\right) ,\\ \Vert \pi _4(a)\nabla \Theta \Vert _{L^1_t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )}&\lesssim \Vert a\Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}}_{2,1}\big )}\Vert \nabla \Theta ^\ell \Vert _{L^2\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )} +\Vert a\Vert _{L^\infty _t\big (\dot{B}^{\frac{n}{2}}_{2,1}\big )}\Vert \nabla \Theta ^h\Vert _{L^1_t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )}. \end{aligned} \end{aligned}$$

The last term of ${\vec G}$ reads $\pi _5(a){\vec j}_1$ for some smooth function $\pi _5$ vanishing at 0. Hence we just have

$$\begin{aligned} \Vert \pi _5(a){\vec j}_1\Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) }\lesssim \Vert a\Vert _{L^\infty _t\left( \dot{B}^{\frac{n}{2}}_{2,1}\right) } \Vert {\vec j}_1\Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) }. \end{aligned}$$

So finally, putting together all the inequalities of this step, we conclude that for all $t>0,$

$$\begin{aligned} \int _0^t\left( \Vert {\vec G}-T_{\vec u}\cdot \nabla a\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}} +\Vert {\vec G}\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\right) \,\hbox {d}\tau \lesssim X^2(t). \end{aligned}$$

(113)

Step 3. Estimates for H According to (109), it suffices to bound H in $L^1\big (\mathbb {R}^+;\dot{B}^{\frac{n}{2}-1}_{2,1}+\dot{B}^{\frac{n}{2}-2}_{2,1}\big ).$ Now decomposing $\Theta$ in $\Theta ^\ell +\Theta ^h,$ and using product laws in Besov spaces, we get if $n\ge 3,$

$$\begin{aligned} \Vert {\vec u}\cdot \nabla \Theta \Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}+\dot{B}^{\frac{n}{2}-2}_{2,1}\right) }\lesssim \Vert {\vec u}\Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}}_{2,1}\big )} \bigl (\Vert \nabla \Theta ^\ell \Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )}+\Vert \nabla \Theta ^h\Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}-2}_{2,1}\big )}\bigr ). \end{aligned}$$

Next, using Lemma 7.4,

$$\begin{aligned}\textstyle \Vert \frac{1}{1+a}\mathrm{div}\bigl ((\kappa -\underline{\kappa })\nabla \Theta \bigr ) \Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}+\dot{B}^{\frac{n}{2}-2}_{2,1}\right) }\!\lesssim \!\Vert a\Vert _{L^\infty _t\left( \dot{B}^{\frac{n}{2}}_{2,1}\right) } \left( \!\Vert \nabla \Theta ^\ell \Vert _{L^1_t(\dot{B}^{\frac{n}{2}}_{2,1})}+\Vert \nabla \Theta ^h\Vert _{L^1_t(\dot{B}^{\frac{n}{2}-1}_{2,1})}\!\right) . \end{aligned}$$

The term involving the pressure may be written

$$\begin{aligned} \Bigl (\pi _1(1)-(1+\Theta )\frac{\pi _1(1+a)}{1+a}\Bigr )\mathrm{div}{\vec u}= \bigl (\pi _6(a)+\pi _7(a)\,\Theta \bigr )\mathrm{div}{\vec u}\quad \hbox {with }\ \pi _6(0)=0. \end{aligned}$$

Now we have

$$\begin{aligned} \begin{array}{rcl} \Vert \pi _6(a)\mathrm{div}{\vec u}\Vert _{L^1_t(\dot{B}^{\frac{n}{2}-1}_{2,1})}&{}\lesssim &{} \Vert a\Vert _{L^2_t(\dot{B}^{\frac{n}{2}}_{2,1})}\Vert \mathrm{div}{\vec u}\Vert _{L^2_t\big (\dot{B}^{\frac{n}{2}-1}_{2,1}\big )},\\ \Vert \pi _7(a)\Theta \mathrm{div}{\vec u}\Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}+\dot{B}^{\frac{n}{2}-2}_{2,1}\right) }&{}\lesssim &{}\bigl (1+\Vert a\Vert _{L^\infty _t\bigr (\dot{B}^{\frac{n}{2}}_{2,1}\bigr )}\bigr ) \Vert \mathrm{div}{\vec u}\Vert _{L^2_t\bigr (\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr )}\bigl (\Vert \Theta ^\ell \Vert _{L^2_t\bigr (\dot{B}^{\frac{n}{2}}_{2,1}\bigr )}+ \Vert \Theta ^h\Vert _{L^2_t\bigr (\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr )}\bigr ). \end{array} \end{aligned}$$

The next term reads $\pi _8(a){\vec j}_1\cdot {\vec u}$ with $\pi _8$ a smooth function of a. We thus get

$$\begin{aligned} \Vert \pi _8(a){\vec j}_1\cdot {\vec u}\Vert _{L^1_t\bigr (\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr )} \lesssim \bigl (1+\Vert a\Vert _{L^\infty _t\bigr (\dot{B}^{\frac{n}{2}}_{2,1}\bigr )}\bigr )\Vert {\vec u}\Vert _{L^2_t\bigr (\dot{B}^{\frac{n}{2}}_{2,1}\bigr )} \Vert {\vec j}_1\Vert _{L^2_t\bigr (\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr )}. \end{aligned}$$

The last but one term reads ${{\mathcal {C}}}\pi _9(a)\bigl (j_0-\underline{\alpha }'\Theta \bigr )$ for some function $\pi _9$ vanishing at 0. To bound it, it is crucial to exploit the fact that $\zeta _0^\ell$ and $\Theta ^h$ satisfy fast decay properties. More precisely, remembering the definition of $\zeta _0,$ we have the decomposition

$$\begin{aligned} {{\mathcal {C}}}\pi _9(a)\bigl (j_0-\underline{\alpha }'\Theta \bigr )=\pi _9(a)\biggl ({{\mathcal {C}}}\bigl (\zeta _0^\ell +j_0^h-\underline{\alpha }'\Theta ^h\bigr )+\frac{\underline{\alpha }_2\underline{\alpha }'}{\underline{\alpha }_1\underline{\sigma }_a(1+\underline{\alpha }')}\mathrm{div}{\vec u}^\ell \biggr )\cdot \end{aligned}$$

Now we have

$$\begin{aligned} \begin{aligned}&{{\mathcal {C}}}\Vert \pi _9(a)\bigl (\zeta _0^\ell + j_0^h\bigr )\Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) }\lesssim {{\mathcal {C}}}\bigl (\Vert \zeta _0\Vert ^\ell _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) } +\Vert j_0\Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) }^h\bigr )\Vert a\Vert _{L^\infty _t\left( \dot{B}^{\frac{n}{2}}_{2,1}\right) }\lesssim X^2(t),\\&{{\mathcal {C}}}\Vert \pi _9(a)\Theta ^h\Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-2}_{2,1}\right) }\lesssim {{\mathcal {C}}}\Vert \Theta \Vert ^h_{L^1_t\left( \dot{B}^{\frac{n}{2}-2}_{2,1}\right) } \Vert a\Vert _{L^\infty _t\left( \dot{B}^{\frac{n}{2}}_{2,1}\right) }\lesssim X^2(t),\\&\Vert \pi _9(a)\mathrm{div}{\vec u}^\ell \Vert _{L^1_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) }\lesssim \Vert a\Vert _{L^2_t\left( \dot{B}^{\frac{n}{2}}_{2,1}\right) } \Vert \mathrm{div}{\vec u}\Vert ^\ell _{L^2_t\left( \dot{B}^{\frac{n}{2}-1}_{2,1}\right) }\lesssim X^2(t). \end{aligned} \end{aligned}$$

Finally,

$$\begin{aligned} \textstyle \Vert \frac{1}{1+a}\bigl (2\mu D{\vec u}:D{\vec u}+\lambda (\mathrm{div}{\vec u})^2\bigr )\Vert _{\dot{B}^{\frac{n}{2}-2}_{2,1}} \lesssim \bigr (1+\Vert a\Vert _{\dot{B}^{\frac{n}{2}}_{2,1}}\bigr )\Vert \nabla u\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}^2. \end{aligned}$$

Combining all the above inequalities, we conclude that for all $t\in \mathbb {R}^+,$

$$\begin{aligned} \int _0^t\Vert H\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}+\dot{B}^{\frac{n}{2}-2}_{2,1}}\,\hbox {d}\tau \lesssim X^2(t). \end{aligned}$$

(114)

Step 4. Estimates for ${\vec J}_1$ Because ${\vec J}_1$ has the same form as the last term of ${\vec G},$ it may be bounded exactly as in the second step:

$$\begin{aligned} \int _0^t\Vert {\vec J}_1\Vert _{\dot{B}^{\frac{n}{2}-1}_{2,1}}\,\hbox {d}\tau \lesssim {{\mathcal {C}}}\Vert {\vec j}_1\Vert _{L^1_t\bigr (\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr )}\,\Vert a\Vert _{L^\infty _t\bigr (\dot{B}^{\frac{n}{2}}_{2,1}\bigr )}\lesssim X^2(t). \end{aligned}$$

(115)

Step 5. Conclusion Plugging inequalities (112), (113), (114) and (115) in (108), we get for a constant C depending only on the coefficients of the system, and for all $t\in \mathbb {R}^+,$

$$\begin{aligned} X(t)\le C\bigl (X(0)+X^2(t)\bigr ), \end{aligned}$$

which allows to close the estimates for all time, if X(0) is small enough.

For the sake of completeness, we here sketch the proof of Corollary 2.1 (for more details, the reader may refer to our recent papers [13, 14] where very similar arguments are used to justify weak convergence in different asymptotics). One can argue as follows:

• 1. The uniform estimate provided by (29) ensures that, up to extraction, we have

  $$\begin{aligned}(a^\varepsilon ,{\vec u}^\varepsilon ,\Theta ^\varepsilon ,j_0^\varepsilon ,{\vec j}_1^\varepsilon )\rightharpoonup (a,{\vec u},\Theta ,j_0,{\vec j}_1)\end{aligned}$$

  for the weak $*$ topology associated to the space $E^{\frac{n}{2}-1}.$ Standard (omitted) arguments ensure that the limit solution $(a,{\vec u},\Theta ,j_0,{\vec j}_1)$ belongs to the superspace $E^{\frac{n}{2}-1}_w$ of $E^{\frac{n}{2}-1},$ where strong time continuity has been replaced by weak continuity.

• 2. Inequality (30) ensures that ${\vec j}_1^\varepsilon \rightarrow \vec {0}$ in $L^1\left( \mathbb {R}_+;\dot{B}^{\frac{n}{2}-1}_{2,1}\right) .$ From the last equation of (25), we thus gather that $\nabla j_0\equiv 0.$ Because $j_0$ is in $L^\infty \left( \mathbb {R}_+;\dot{B}^{\frac{n}{2}-1}_{2,1}\right)$ and thus tends weakly to 0 at infinity, we conclude that $j_0\equiv 0.$

• 3. Inequality (30) also implies that $\Theta _0^h\equiv 0$ and that $\zeta _{0,\varepsilon }^\ell \rightarrow 0.$ As $j_0^\ell \equiv 0$ and

  $$\begin{aligned} \zeta _{0,\varepsilon }=j_{0,\varepsilon }-\underline{\alpha }'\Theta _\varepsilon -\varepsilon \frac{\underline{\alpha }_2\underline{\alpha }'}{{{\mathcal {L}}}\underline{\alpha }_1\underline{\sigma _a}(1+Pr^{-1}\underline{\alpha }')}\mathrm{div}{\vec u}^\varepsilon \rightharpoonup j_0-\underline{\alpha }'\Theta , \end{aligned}$$

  this implies that $\Theta ^\ell \equiv 0.$ We thus have $\Theta \equiv 0.$

• 4. What we proved hitherto already ensures that the last term of the velocity equation of (25) goes (strongly) to 0 in $L^1\bigr (\mathbb {R}_+;\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr ).$ To pass to the limit in the other terms of the first two equations of (25), we need some compactness. The easiest way to achieve it, is to bound $\partial _ta^\varepsilon$ and $\partial _t{\vec u}^\varepsilon$ in some suitable space. Taking advantage of the equations for $a^\varepsilon$ and ${\vec u}^\varepsilon ,$ of the uniform bounds provided by (29) and of product laws in Besov spaces, one can show that $(\partial _t a^\varepsilon )$ and $(\partial _t {\vec u}^\varepsilon )$ are bounded in $L^2_{loc}\bigr (\mathbb {R}_+;\dot{B}^{\frac{n}{2}-1}_{2,1}\bigr )$ and $L^2_{loc}\bigr (\mathbb {R}_+;\dot{B}^{\frac{n}{2}-2}_{2,1}\bigr ),$ respectively. Using compact embedding in Besov spaces, one can conclude to strong convergence results in good enough norms so as to pass to the limit in all the nonlinear terms. Consequently, $(a,{\vec u})$ satisfies (32).

• 5. Because (32) admits a unique solution in the space given by Corollary 2.1 (up to replacing strong continuity by weak continuity), $(a,{\vec u})$ is uniquely determined, and thus strongly continuous in time. We conclude that all limits of subsequences of $(a^\varepsilon ,{\vec u}^\varepsilon ,\Theta ^\varepsilon ,j_0^\varepsilon ,{\vec j}_1^\varepsilon )$ are the same. Hence the whole family converges to $(a,{\vec u},0,0,\vec {0}).$

6 Local existence for general large data

Given some reference constant positive temperature ${\bar{\vartheta }},$ we want to establish the local well posedness of the following system governing the evolution of $\varrho ,$ ${\vec u},$ $\Theta \,{:=}\,\vartheta -{\bar{\vartheta }},$ $j_0\,{:=}\,I_0-b({\bar{\vartheta }})$ and ${\vec j}_1\,{:=}\,{\vec I}_1$:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t\varrho +\mathrm{div}(\varrho {\vec u})=0,\\ \varrho (\partial _t{\vec u}+{\vec u}\cdot \nabla {\vec u})+\nabla p-\mathrm{div}(2\mu D{\vec u}+\lambda \mathrm{div}{\vec u}\,\mathrm{Id}\,) =\Bigl (\displaystyle \frac{\sigma _a+\sigma _s}{n}\Bigr ){\vec j}_1,\\ C_v\varrho (\partial _t\Theta +{\vec u}\cdot \nabla \Theta )-\mathrm{div}(\kappa \nabla \Theta )=2\mu D{\vec u}:D{\vec u}+\lambda (\mathrm{div}{\vec u})^2\\ \quad -\vartheta \partial _\vartheta p\,\mathrm{div}{\vec u}+\sigma _a(j_0+b({\bar{\vartheta }})-b(\vartheta ))+\Bigl (\displaystyle \frac{\sigma _a+\sigma _s}{n}\Bigr ){\vec j}_1\cdot {\vec u},\\ \partial _tj_0+\displaystyle \frac{1}{n}\mathrm{div}{\vec j}_1=\sigma _a(b(\vartheta )-b({\bar{\vartheta }})-j_0),\\ \partial _t{\vec j}_1+\nabla j_0+(\sigma _a+\sigma _s){\vec j}_1=\vec {0}. \end{array}\right. \end{aligned}$$

(116)

Recall that the pressure function is given by (9). Here in contrast with the previous section, all the coefficients are allowed to depend (smoothly) on both $\varrho$ and $\vartheta$ provided the positivity condition (22) is fulfilled, and the distribution function b may be anything (if smooth of course).

The rest of this section is devoted to the proof of Theorem 2.1. Given that a lot of regularity is available, proving uniqueness is not a big issue, and is thus skipped. As regards the proof of local existence, it may be worked out from the a priori estimates for (116) that we shall derive below (use, e.g., a Friedrichs scheme, or an iterative method based on a linearization of the system). Those estimates will be obtained by combining results on linear equations (see the three propositions just below) and nonlinear inequalities stated in the Appendix.

More precisely, estimating the density will rely on the following proposition (see, e.g., Th. 3.14 in [1]):

Proposition 6.1

Let a satisfy the transport equation

$$\begin{aligned} \partial _ta+{\vec v}\cdot \nabla a=f. \end{aligned}$$

Then for any $s\in (-\frac{n}{2},\frac{n}{2}+1]$ there exists a constant C so that for all $t\ge 0,$

$$\begin{aligned} \Vert a(t)\Vert _{B^s_{2,1}}\le e^{C\int _0^t\Vert \nabla {\vec v}\Vert _{B^{\frac{n}{2}}_{2,1}}\,\mathrm{d}\tau } \biggl (\Vert a(0)\Vert _{B^s_{2,1}}+\int _0^te^{-C\int _0^\tau \Vert \nabla {\vec v}\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau '} \Vert f(\tau )\Vert _{B^s_{2,1}}\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

The estimates for the velocity and temperature equations will be based on the following statement that has been proved in [11] for homogeneous Besov norms⁸.

Proposition 6.2

Let $-\frac{n}{2}<s\le \frac{n}{2}$ and $u:[0,T]\times \mathbb {R}^n\rightarrow \mathbb {R}$ satisfying

$$\begin{aligned} \partial _tu-a\mathrm{div}(b\nabla u)=f,\qquad u|_{t=0}=u^0 \end{aligned}$$

for some functions a and b such that

$$\begin{aligned} \alpha \,{:=}\,\inf _{(t,x)\in [0,T]\times \mathbb {R}^n} (ab)(t,x)\,>0. \end{aligned}$$

Then there exist $\kappa =\kappa (s,n)$ and $C=C(s,n)$ such that for all $t\in [0,T],$

$$\begin{aligned}&\Vert u(t)\Vert _{B^s_{2,1}}+\kappa \alpha \int _0^t\Vert u\Vert _{B^{s+2}_{2,1}}\,\hbox {d}\tau \le \Vert u^0\Vert _{B^s_{2,1}}+\int _0^t\Vert f\Vert _{B^s_{2,1}}\,\mathrm{d}\tau \\&\quad +\,C\int _0^t\bigl (\alpha +\alpha ^{-1}\Vert b\nabla a,a\nabla b\Vert _{B^{\frac{n}{2}}_{2,1}}^2\bigr )\Vert u\Vert _{B^s_{2,1}}\,\hbox {d}\tau . \end{aligned}$$

Similarly, if ${\vec v}:[0,T]\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ satisfies

$$\begin{aligned} \partial _t{\vec v}-2a\mathrm{div}(\mu D{\vec v})-b\nabla (\lambda \mathrm{div}{\vec v})={\vec g},\qquad {\vec v}|_{t=0}={\vec v}^0 \end{aligned}$$

with

$$\begin{aligned} \alpha \,{:=}\,\min \left( \inf _{(t,x)\in [0,T]\times \mathbb {R}^n} (a\mu )(t,x), \inf _{t,x}(2a\mu +b\lambda )(t,x)\right) >0 \end{aligned}$$

then there exist $\kappa =\kappa (s,n)$ and $C=C(s,n)$ such that

$$\begin{aligned}&\Vert {\vec v}(t)\Vert _{B^s_{2,1}}+\kappa \alpha \int _0^t\Vert {\vec v}\Vert _{B^{s+2}_{2,1}}\,\hbox {d}\tau \le \Vert {\vec v}^0\Vert _{B^s_{2,1}}+\int _0^t\Vert {\vec g}\Vert _{B^s_{2,1}}\,\hbox {d}\tau \\&\quad +\,C\int _0^t\bigl (\alpha +\alpha ^{-1}\Vert \mu \nabla a,a\nabla \mu ,\lambda \nabla b,b\nabla \lambda \Vert _{B^{\frac{n}{2}}_{2,1}}^2\bigr )\Vert {\vec v}\Vert _{B^s_{2,1}}\,\hbox {d}\tau . \end{aligned}$$

Finally, the radiative modes $j_0$ and ${\vec j}_1$ will be handled thanks to the obvious following result, in the spirit of [12], page 189:

Proposition 6.3

Let $(j_0,{\vec j}_1):[0,T]\times \mathbb {R}^n\rightarrow \mathbb {R}\times \mathbb {R}^n$ satisfy

$$\begin{aligned} \left\{ \begin{array}{l}\partial _tj_0+\frac{1}{n}\mathrm{div}{\vec j}_1=J_0,\\ \partial _t{\vec j}_1+\nabla j_0={\vec J}_1.\end{array}\right. \end{aligned}$$

Then for any $s\in \mathbb {R}$ and $t\in [0,T],$ we have

$$\begin{aligned} \bigl \Vert (j_0,{\textstyle \frac{1}{\sqrt{n}}}{\vec j}_1)(t)\bigr \Vert _{B^s_{2,1}}\le \bigl \Vert (j_0,{\textstyle \frac{1}{\sqrt{n}}}{\vec j}_1)(0)\bigr \Vert _{B^s_{2,1}} +\int _0^t\bigl \Vert (J_0,{\textstyle \frac{1}{\sqrt{n}}}{\vec J}_1)(\tau )\bigr \Vert _{B^s_{2,1}}\,\hbox {d}\tau . \end{aligned}$$

In the rest of this section, we concentrate on the proof of a priori estimates for a smooth solution $(\varrho ,{\vec u},\Theta ,j_0,{\vec j}_1)$ to (116), on $[0,T]\times \mathbb {R}^n.$ We assume in addition that $\varrho (t,x)$ and $\vartheta (t,x)$ are bounded by above and from below for all (t, x) in $[0,T]\times \mathbb {R}^n$:

$$\begin{aligned}&0<\textstyle \frac{1}{2}\,\varrho _*\le \varrho (t,x)\le 2\varrho ^*<\infty \end{aligned}$$

(117)

$$\begin{aligned}&\hbox { and }\quad 0<\textstyle \frac{1}{2}\,\vartheta _*\le \vartheta (t,x)\le 2\vartheta ^*<\infty . \end{aligned}$$

(118)

Step 1. Estimates for the density Because

$$\begin{aligned} (\partial _t+{\vec u}\cdot \nabla )\varrho ^{\pm 1}\pm \varrho ^{\pm 1}\mathrm{div}{\vec u}=0, \end{aligned}$$

we readily have

$$\begin{aligned} \Vert \varrho ^{\pm 1}(t)\Vert _{L^\infty }\le e^{\int _0^t\Vert \mathrm{div}{\vec u}\Vert _{L^\infty }\,\hbox {d}\tau }\Vert \varrho ^{\pm 1}(0)\Vert _{L^\infty }. \end{aligned}$$

(119)

This implies that (117) if fulfilled if

$$\begin{aligned} \varrho _*\le \varrho _0\le \varrho ^* \end{aligned}$$

(120)

and we have some control on $\mathrm{div}{\vec u}$ in $L^1(0,T;L^\infty ).$

Next, differentiating the above equation with respect to the space variable yields

$$\begin{aligned} (\partial _t+{\vec u}\cdot \nabla )\nabla \varrho ^{\pm 1}+\nabla {\vec u}\cdot \nabla \varrho ^{\pm 1}\pm \nabla \varrho ^{\pm 1}\mathrm{div}{\vec u}\pm \varrho ^{\pm 1}\nabla \mathrm{div}{\vec u}=0. \end{aligned}$$

Hence using the fact that $B^{\frac{n}{2}}_{2,1}$ is an algebra, that (see Lemma 7.3)

$$\begin{aligned} \Vert \varrho ^{\pm 1}\nabla \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}} \lesssim \Vert \varrho ^{\pm 1}\Vert _{L^\infty } \Vert \nabla \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}+\Vert \mathrm{div}{\vec u}\Vert _{L^\infty }\Vert \nabla \varrho ^{\pm 1}\Vert _{B^{\frac{n}{2}}_{2,1}}, \end{aligned}$$

and Proposition 6.1, we obtain

$$\begin{aligned} \Vert \nabla \varrho ^{\pm 1}(t)\Vert _{B^{\frac{n}{2}}_{2,1}}\le e^{CU(t)}\biggl (\Vert \nabla \varrho ^{\pm 1}(0)\Vert _{B^{\frac{n}{2}}_{2,1}}+C\Vert \varrho ^{\pm 1}\Vert _{L^\infty } \int _0^t e^{-CU(\tau )}\Vert \nabla \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau \biggr ), \end{aligned}$$

with $U(t)\,{:=}\,\int _0^t\Vert \nabla {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau .$

So assuming from now on that

$$\begin{aligned} CU(T)\le \log 2\quad \hbox {and}\quad C\int _0^T\Vert \nabla \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\,dt\le 2, \end{aligned}$$

we get for all $t\in [0,T],$

$$\begin{aligned} \Vert \nabla \varrho ^{\pm 1}(t)\Vert _{B^{\frac{n}{2}}_{2,1}}\le A_0^{\pm }\,{:=}\,2\bigl (\Vert \nabla \varrho ^{\pm 1}(0)\Vert _{B^{\frac{n}{2}}_{2,1}} +2\Vert \varrho ^{\pm 1}\Vert _{L^\infty }\bigr ). \end{aligned}$$

(121)

In what follows, we shall denote $A_0\,{:=}\,\max (A_0^-,A_0^+).$

Step 2. Estimates for the radiative functions From Proposition 6.3, we have (omitting $1/\sqrt{n}$ for notational simplicity):

$$\begin{aligned}&\Vert (j_0,{\vec j}_1)(t)\Vert _{B^{\frac{n}{2}}_{2,1}}\le \Vert (j_0,{\vec j}_1)(0)\Vert _{B^{\frac{n}{2}}_{2,1}}\nonumber \\&\quad +\, \int _0^t\Vert \sigma _a(b(\vartheta )-b({\bar{\vartheta }})-j_0)\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau +\int _0^t\Vert (\sigma _a+\sigma _s){\vec j}_1\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau . \end{aligned}$$

(122)

The first term of the r.h.s. may be bounded according to Lemmas 7.3 and 7.4:

$$\begin{aligned} \begin{array}{ll} &{}\Vert \sigma _a(b(\vartheta )-b({\bar{\vartheta }})-j_0)\Vert _{B^{\frac{n}{2}}_{2,1}}\\ &{}\quad \lesssim \Vert \sigma _a\Vert _{L^\infty }\Vert b(\vartheta )-b({\bar{\vartheta }})-j_0\Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert b(\vartheta )-b({\bar{\vartheta }})-j_0\Vert _{B^{\frac{n}{2}}_{2,1}}\Vert \nabla \sigma _a\Vert _{B^{\frac{n}{2}-1}_{2,1}}\\ &{}\quad \lesssim \left( \Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}+\Vert j_0\Vert _{B^{\frac{n}{2}}_{2,1}}\right) \left( 1+\Vert \nabla \varrho \Vert _{B^{\frac{n}{2}-1}_{2,1}}+\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}-1}_{2,1}}\right) . \end{array} \end{aligned}$$

A similar inequality may be proved for the last term of the r.h.s. of (122). Using (121), we thus end up with

$$\begin{aligned} \Vert (j_0,{\vec j}_1)(t)\Vert _{B^{\frac{n}{2}}_{2,1}}\le \Vert (j_0,{\vec j}_1)(0)\Vert _{B^{\frac{n}{2}}_{2,1}}+ C\int _0^t\left( A_0+\Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) \Vert (\Theta ,j_0,{\vec j}_1)\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau , \end{aligned}$$

and thus, denoting $J(t)\,{:=}\,\Vert (j_0,{\vec j}_1)(t)\Vert _{B^{\frac{n}{2}}_{2,1}}$ and using Gronwall Lemma,

$$\begin{aligned} J(t)\le e^{C\int _0^t(A_0+\Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}})\,\hbox {d}\tau } \biggl (J(0)+ C\int _0^t\left( A_0+\Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) \Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau \biggr )\cdot \end{aligned}$$

(123)

Step 3. Estimates for the velocity field Fix some positive real number ${\bar{\varrho }},$ and set ${\bar{\mu }}\,{:=}\,\mu ({\bar{\varrho }},{\bar{\vartheta }})$ and ${\bar{\lambda }}\,{:=}\,\lambda ({\bar{\varrho }},{\bar{\vartheta }}).$ Let ${\vec u}_L$ be the solution to the following constant coefficients Lamé system:

$$\begin{aligned} \partial _t{\vec u}_L-\overline{{{\mathcal {A}}}}{\vec u}_L=0,\qquad {\vec u}_L|_{t=0}={\vec u}_0\qquad \hbox {with }\ \overline{{{\mathcal {A}}}}\,{:=}\,{\bar{\mu }}\Delta +({\bar{\lambda }}+{\bar{\mu }})\nabla \mathrm{div}. \end{aligned}$$

The fluctuation ${\vec u}_{F}\,{:=}\,{\vec u}-{\vec u}_L$ fulfills

$$\begin{aligned} \partial _t{\vec u}_F-\frac{1}{\varrho }\mathrm{div}\bigl (2\mu D{\vec u}_F+\lambda \mathrm{div}{\vec u}_F\mathrm{Id}\,\bigr )={\vec g}_F,\qquad {\vec u}_F|_{t=0}=0, \end{aligned}$$

with

$$\begin{aligned} {\vec g}_F\,{:=}\,-{\vec u}\cdot \nabla {\vec u}-\frac{1}{\varrho }\nabla p + \biggl (\displaystyle \frac{\sigma _a+\sigma _s}{n\varrho }\biggr ){\vec j}_1 +\frac{1}{\varrho }\mathrm{div}(2\mu D{\vec u}_L+\lambda \mathrm{div}{\vec u}_L\,\mathrm{Id}\,)-\overline{{{\mathcal {A}}}}{\vec u}_L. \end{aligned}$$

Applying the second part of Proposition 6.2 yields for some $C=C(\varrho _*,\varrho ^*,\lambda ,\mu ,n),$

$$\begin{aligned} X_{{\vec u}_F}(t)\le C\exp \biggl (C\int _0^t\bigl (1+\Vert \mu \nabla \varrho ^{-1},\lambda \nabla \varrho ^{-1},\varrho ^{-1}\nabla \mu ,\varrho ^{-1}\nabla \lambda \Vert ^2_{B^{\frac{n}{2}}_{2,1}}\bigr )\,\hbox {d}\tau \biggr )\int _0^t\Vert {\vec g}_F\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau , \end{aligned}$$

(124)

where we have used the notation

$$\begin{aligned} X_z(t)\,{:=}\,\Vert z(t)\Vert _{B^{\frac{n}{2}}_{2,1}}+\int _0^t\Vert z\Vert _{B^{\frac{n}{2}+2}_{2,1}}\,\hbox {d}\tau . \end{aligned}$$

From Lemmas 7.3 and 7.4, we gather that

$$\begin{aligned} \begin{array}{lll} \Vert \mu \nabla \varrho ^{-1}\Vert _{B^{\frac{n}{2}}_{2,1}}&{}\lesssim &{} \Vert \mu \Vert _{L^\infty }\Vert \nabla \varrho ^{-1}\Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert \varrho ^{-1}\Vert _{L^\infty }\Vert \nabla \mu \Vert _{B^{\frac{n}{2}}_{2,1}}\\ &{}\lesssim &{} A_0\left( 1+\Vert \nabla \varrho ,\nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) \lesssim A_0\left( A_0+\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) .\end{array} \end{aligned}$$

A similar bound holds for the other terms of the exponential in (124). Let us now bound the terms of ${\vec g}_F.$ First, because $B^{\frac{n}{2}}_{2,1}$ is an algebra, we have

$$\begin{aligned} \Vert {\vec u}\cdot \nabla {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\lesssim \Vert {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\Vert \nabla {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}. \end{aligned}$$

Next, using Lemmas 7.3 and 7.4, Inequality (121) and the expression of p in (9),

$$\begin{aligned} \Vert \varrho ^{-1}\nabla p\Vert _{B^{\frac{n}{2}}_{2,1}} \lesssim A_0\left( 1+\Vert \Theta \Vert _{B^{\frac{n}{2}+1}_{2,1}}\right) . \end{aligned}$$

Likewise, we get

$$\begin{aligned} \textstyle \Vert \bigl (\frac{\sigma _a+\sigma _s}{n\varrho }\bigr ){\vec j}_1\Vert _{B^{\frac{n}{2}}_{2,1}}\lesssim \left( 1+A_0+\Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) \Vert {\vec j}_1\Vert _{B^{\frac{n}{2}}_{2,1}} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{lll} \Vert \varrho ^{-1}\mathrm{div}(\mu D{\vec u}_L)\Vert _{B^{\frac{n}{2}}_{2,1}}&{}\lesssim &{} \Vert \varrho ^{-1}\Vert _{L^\infty }\Vert \mu D{\vec u}_L\Vert _{B^{\frac{n}{2}+1}_{2,1}} +\Vert \mu D{\vec u}_L\Vert _{L^\infty }\Vert \nabla \varrho ^{-1}\Vert _{B^{\frac{n}{2}}_{2,1}}\\ &{}\lesssim &{} \Vert \mu \Vert _{L^\infty }\Vert \nabla {\vec u}_L\Vert _{B^{\frac{n}{2}+1}_{2,1}}+\Vert \nabla {\vec u}_L\Vert _{L^\infty }\left( \Vert \nabla \mu \Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert \mu \Vert _{L^\infty }\Vert \nabla \varrho ^{-1}\Vert _{B^{\frac{n}{2}}_{2,1}}\right) \\ &{}\lesssim &{} \Vert \nabla {\vec u}_L\Vert _{B^{\frac{n}{2}+1}_{2,1}}+\Vert \nabla {\vec u}_L\Vert _{B^{\frac{n}{2}}_{2,1}}\Vert (\nabla \varrho ,\nabla \varrho ^{-1},\nabla \Theta )\Vert _{B^{\frac{n}{2}}_{2,1}}.\end{array} \end{aligned}$$

A similar inequality may be proved for $\varrho ^{-1}\nabla (\lambda \mathrm{div}{\vec u}_L)$ and we end up, by virtue of (121), with

$$\begin{aligned} \Vert \varrho ^{-1}\mathrm{div}(2\mu D{\vec u}_L+\lambda \mathrm{div}{\vec u}_L\,\mathrm{Id}\,)\Vert _{B^{\frac{n}{2}}_{2,1}} \lesssim \Vert \nabla {\vec u}_L\Vert _{B^{\frac{n}{2}+1}_{2,1}}+\Vert \nabla {\vec u}_L\Vert _{B^{\frac{n}{2}}_{2,1}}\left( A_0+\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) . \end{aligned}$$

Putting all the above inequalities together, we conclude that

$$\begin{aligned}&X_{{\vec u}_F}(t)\le Ce^{C(t(1+A_0^2)+\int _0^t\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}^2\,\hbox {d}\tau )} \biggl (\sqrt{t}X^2_{{\vec u}}(t)+(A_0+1)tJ(t)\nonumber \\&\quad +\bigl (X_\Theta (t)+A_0\sqrt{t}\bigr )\bigl (\sqrt{t} (1+A_0+J(t)) +\Vert \nabla {\vec u}_L\Vert _{L^2_t(B^{\frac{n}{2}}_{2,1})}\bigr ) +\Vert \nabla {\vec u}_L\Vert _{L^1_t(B^{\frac{n}{2}+1}_{2,1})}\biggr )\cdot \nonumber \\ \end{aligned}$$

(125)

Step 4. Estimates for the temperature Let ${\bar{\kappa }}\,{:=}\,\frac{\kappa ({\bar{\varrho }},{\bar{\vartheta }})}{C_v{\bar{\varrho }}}$ and $\Theta _L$ be the solution to the following constant coefficient heat equation:

$$\begin{aligned} \partial _t\Theta _L-{\bar{\kappa }}\Delta \Theta _L=0,\qquad \Theta _L|_{t=0}=u_0. \end{aligned}$$

The fluctuation $\Theta _{F}\,{:=}\,\Theta -\Theta _L$ fulfills

$$\begin{aligned} \partial _t\Theta _F-\frac{1}{C_v\varrho }\mathrm{div}(\kappa \nabla \Theta _F)=h_F,\qquad \Theta _F|_{t=0}=0, \end{aligned}$$

with

$$\begin{aligned} h_F:= & {} -{\vec u}\cdot \nabla \Theta +\frac{1}{C_v\varrho }\bigl (2\mu D{\vec u}:D{\vec u}+\lambda (\mathrm{div}{\vec u})^2\bigr ) -\frac{1}{C_v}\frac{\pi _1(\varrho )}{\varrho }\,\vartheta \,\mathrm{div}{\vec u}\\&+\,\frac{\sigma _a}{C_v\varrho }(j_0+b({\bar{\vartheta }})-b(\vartheta ))+\frac{(\sigma _a+\sigma _s)}{C_v\varrho }{\vec j}_1\cdot {\vec u}+\frac{1}{C_v\varrho }\mathrm{div}(\kappa \nabla \Theta _L)-{\bar{\kappa }}\Delta \Theta _L. \end{aligned}$$

The maximum principle guarantees that $\vartheta _*\le {\bar{\vartheta }}+\Theta _L\le \vartheta ^*$ if

$$\begin{aligned} \vartheta _*\le \vartheta _0\le \vartheta ^*. \end{aligned}$$

(126)

Therefore remembering that $B^{\frac{n}{2}}_{2,1}\hookrightarrow L^\infty ,$ to ensure that (118) is fulfilled, it suffices to establish that $\sup _{t\in [0,T]}\Vert \Theta _F(t)\Vert _{B^{\frac{n}{2}}_{2,1}}$ is small enough.

Now applying the first part of Proposition 6.2 yields

$$\begin{aligned} X_{\Theta _F}(t)\le C\exp \biggl (C\int _0^t\bigl (1+\Vert (\kappa \nabla \varrho ^{-1},\varrho ^{-1}\nabla \kappa )\Vert ^2_{B^{\frac{n}{2}}_{2,1}}\bigr )\,\hbox {d}\tau \biggr ) \int _0^t\Vert h_F\Vert _{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau . \end{aligned}$$

(127)

As in the previous step, we have

$$\begin{aligned} \Vert (\kappa \nabla \varrho ^{-1},\varrho ^{-1}\nabla \kappa )\Vert _{B^{\frac{n}{2}}_{2,1}}\lesssim \left( A_0+\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) A_0. \end{aligned}$$

So we now have to bound $h_F$ in $B^{\frac{n}{2}}_{2,1}.$ Because $B^{\frac{n}{2}}_{2,1}$ is an algebra, we have

$$\begin{aligned} \Vert {\vec u}\cdot \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\lesssim \Vert {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}. \end{aligned}$$

Lemmas 7.3 and 7.4 ensure that

$$\begin{aligned} \begin{array}{rcl} \Vert \varrho ^{-1}\mu D{\vec u}:D{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}&{}\lesssim &{} \Vert \varrho ^{-1}\mu \Vert _{L^\infty }\Vert \nabla {\vec u}\otimes \nabla {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert \nabla {\vec u}\otimes \nabla {\vec u}\Vert _{B^{\frac{n}{2}-1}_{2,1}}\Vert \nabla (\varrho ^{-1}\mu )\Vert _{B^{\frac{n}{2}}_{2,1}}\\ &{}\lesssim &{} \Vert \nabla {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}\left( \Vert \nabla {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert (\nabla \varrho ,\nabla \Theta )\Vert _{B^{\frac{n}{2}}_{2,1}}\Vert \nabla {\vec u}\Vert _{B^{\frac{n}{2}-1}_{2,1}}\right) . \end{array} \end{aligned}$$

Similarly, we have

$$\begin{aligned} \begin{array}{rcl} \Vert \varrho ^{-1}\pi _1(\varrho )\,\vartheta \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}&{}\lesssim &{}\Vert \varrho ^{-1}\pi _1(\varrho )\,\vartheta \Vert _{L^\infty } \Vert \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}-1}_{2,1}}\Vert \nabla \left( \varrho ^{-1}\pi _1(\varrho )\,\vartheta \right) \Vert _{B^{\frac{n}{2}}_{2,1}}\\ &{}\lesssim &{}\Vert \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}+(A_0+\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}})\Vert \mathrm{div}{\vec u}\Vert _{B^{\frac{n}{2}-1}_{2,1}},\\ \Vert \varrho ^{-1}\sigma _a j_0\Vert _{B^{\frac{n}{2}}_{2,1}}&{}\lesssim &{}(1+\Vert (\nabla \varrho ,\nabla \Theta )\Vert _{B^{\frac{n}{2}-1}_{2,1}})\Vert j_0\Vert _{B^{\frac{n}{2}}_{2,1}},\\ \Vert \varrho ^{-1}\sigma _a(b(\vartheta )-b({\bar{\vartheta }}))\Vert _{B^{\frac{n}{2}}_{2,1}} &{}\lesssim &{} \Vert \varrho ^{-1}\sigma _a\Vert _{L^\infty }\Vert b(\vartheta )-b({\bar{\vartheta }})\Vert _{B^{\frac{n}{2}}_{2,1}} +\Vert b(\vartheta )-b({\bar{\vartheta }})\Vert _{L^\infty }\Vert \nabla (\varrho ^{-1}\sigma _a)\Vert _{B^{\frac{n}{2}-1}_{2,1}}\\ &{}\lesssim &{} \Vert \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}(1+\Vert (\nabla \varrho ,\nabla \Theta )\Vert _{B^{\frac{n}{2}-1}_{2,1}}),\\ \Vert \varrho ^{-1}(\sigma _a+\sigma _s){\vec j}_1\cdot {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}} &{}\lesssim &{}(1+\Vert (\nabla \varrho ,\nabla \Theta )\Vert _{B^{\frac{n}{2}-1}_{2,1}})\Vert {\vec j}_1\Vert _{B^{\frac{n}{2}}_{2,1}}\Vert {\vec u}\Vert _{B^{\frac{n}{2}}_{2,1}}. \end{array} \end{aligned}$$

Finally, arguing exactly as for the corresponding term in ${\vec g}_F,$ we get

$$\begin{aligned} \Vert \varrho ^{-1}\mathrm{div}(\kappa \nabla \Theta _L)\Vert _{B^{\frac{n}{2}}_{2,1}}\lesssim \Vert \nabla \Theta _L\Vert _{B^{\frac{n}{2}+1}_{2,1}}+\Vert \nabla \Theta _L\Vert _{B^{\frac{n}{2}}_{2,1}}\left( A_0+\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}\right) . \end{aligned}$$

Putting together all the previous inequalities, we conclude that

$$\begin{aligned}&X_{\Theta _F}(t)\le Ce^{C((A_0^2+1)t+\int _0^t\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}^2\,\hbox {d}\tau )} \biggl (\Vert \nabla \Theta _L\Vert _{L_t^2\big (B^{\frac{n}{2}}_{2,1}\big )}^2+\Vert \nabla \Theta _L\Vert _{L_t^1\big (B^{\frac{n}{2}+1}_{2,1}\big )}A_0 \nonumber \\&\quad +\,X_{{\vec u}_F}^2+X_{\Theta _F}^2+A_0\sqrt{t}X^2_{{\vec u}} +X_{\vec u}\left( X_{{\vec u}_F}+\Vert \nabla {\vec u}_L\Vert _{L^2_t\big (B^{\frac{n}{2}}_{2,1}\big )}\right) \left( X_{\Theta _F}+\Vert \nabla \Theta _L\Vert _{L^2\big (B^{\frac{n}{2}}_{2,1}\big )}\right) \nonumber \\&\quad +\,(\sqrt{t}+\sqrt{t}X_\Theta +A_0t)X_{\vec u}+t(1+A_0+X_\Theta )(X_\Theta +J(1+X_{\vec u}))\nonumber \\&\quad +\,\sqrt{t}A_0\Vert \nabla \Theta _L\Vert _{L^2_t(B^{\frac{n}{2}}_{2,1})}\biggr )\cdot \end{aligned}$$

(128)

Step 5. Closure of the estimates Let $T^*\le T$ be the largest time for which

$$\begin{aligned} J(t)\le 2\Vert (j_0^0,{\vec j}_1^0)\Vert _{B^{\frac{n}{2}}_{2,1}},\quad X_{{\vec u}}(t)\le 2\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}},\quad \ X_\Theta (t)\le 2 \Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}\quad \hbox {on }\ [0,T^*] \end{aligned}$$

(129)

and, in addition,

$$\begin{aligned} \begin{array}{ll} X_{{\vec u}_F}(T^*)\le \eta _{{\vec u}},&{}\qquad X_{{\vec u}_L}(T^*)\le \eta _\Theta ,\\ \Vert \nabla {\vec u}_L\Vert _{L^1_{T^*}(B^{\frac{n}{2}+1}_{2,1})}\le \varepsilon _1,&{}\qquad \Vert \nabla {\vec u}_L\Vert _{L^2_{T^*}(B^{\frac{n}{2}}_{2,1})}\le \varepsilon _2,\\ \Vert \nabla \Theta _L\Vert _{L^1_{T^*}(B^{\frac{n}{2}+1}_{2,1})}\le {\widetilde{\varepsilon }}_1,&{}\qquad \Vert \nabla \Theta _L\Vert _{L^2_{T^*}(B^{\frac{n}{2}}_{2,1})}\le {\widetilde{\varepsilon }}_2,\end{array} \end{aligned}$$

(130)

for suitably small $\eta _{{\vec u}},$ $\eta _\Theta ,$ $\varepsilon _1,$ $\varepsilon _2,$ ${\widetilde{\varepsilon }}_1$ and ${\widetilde{\varepsilon }}_2$ (that will be fixed hereafter). Note that the time continuity properties of the solution and the fact that $\nabla {\vec u}_L,\nabla \Theta _L\in L^1_{loc}(\mathbb {R}^+;B^{\frac{n}{2}+1}_{2,1})$ and that $\nabla {\vec u}_L,\nabla \Theta _L\in L^2_{loc}(\mathbb {R}^+;B^{\frac{n}{2}}_{2,1})$ ensures that $T^*>0,$ as well as (117) and (118) if the data fulfill (120) and (126).

Therefore from Inequality (123) we deduce that for all $t\in [0,T^*],$

$$\begin{aligned} J(t)\le e^{C(A_0t+2\sqrt{t}\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}})} \Bigl (\Vert (j_0^0,{\vec j}_1^0)\Vert _{B^{\frac{n}{2}}_{2,1}}+2Ct(A_0+2\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}})\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}\Bigr )\cdot \end{aligned}$$

Hence $J(t)\le 2 \Vert (j_0^0,{\vec j}_1^0)\Vert _{B^{\frac{n}{2}}_{2,1}}$ on $[0,T^*]$ whenever (say)

$$\begin{aligned} C\left( A_0T^*+2\sqrt{T^*}\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}\right)\le & {} \textstyle {\log \frac{3}{2}}\quad \hbox {and}\quad 6CT^*\left( A_0+2\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}\right) \Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}\nonumber \\\le & {} \Vert \big (j_0^0,{\vec j}_1^0\big )\Vert _{B^{\frac{n}{2}}_{2,1}}. \end{aligned}$$

(131)

Next, if we assume in addition that

$$\begin{aligned} \biggl (\bigl (A_0^2+1\bigr )T^*+\int _0^{T^*}\Vert \nabla \Theta \Vert _{B^{\frac{n}{2}}_{2,1}}^2\,dt\biggr )\quad \hbox {is small enough} \end{aligned}$$

(132)

then Inequality (125) implies that on $[0,T^*],$ we have

$$\begin{aligned} X_{{\vec u}_F}(t)\le 4C\Bigl (\sqrt{t}\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}^2 +(\Vert \Theta _0\Vert _{B^{\frac{n}{2}}_{2,1}}+\sqrt{t} A_0)(\sqrt{t}(\Vert (j_0^0,{\vec j}_1^0)\Vert _{B^{\frac{n}{2}}_{2,1}}+A_0+1)+\varepsilon _2)+\varepsilon _1\Bigr )\cdot \end{aligned}$$

(133)

And finally, from (128), we infer that

$$\begin{aligned} X_{\Theta _F}(t)\le & {} 4C\Bigl (X_{{\vec u}_F}^2+A_0\sqrt{t}\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}^2 +\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}(X_{{\vec u}_F}+\varepsilon _2)(X_{\Theta _F}+{\widetilde{\varepsilon }}_2)\nonumber \\&+\,\sqrt{t}\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}(1+\Vert \Theta _0\Vert _{B^{\frac{n}{2}}_{2,1}}+\sqrt{t}A_0) +t(1+A_0+\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}})(\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}\nonumber \\&+\,\Vert (j_0^0,{\vec j}_1^0)\Vert _{B^{\frac{n}{2}}_{2,1}}(1+\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}))\nonumber \\&+\,A_0{\widetilde{\varepsilon }}_1+{\widetilde{\varepsilon }}_2^2+X^2_{\Theta _F}+\sqrt{t}A_0{\widetilde{\varepsilon }}_2\Bigr )\cdot \end{aligned}$$

(134)

To conclude, it suffices to notice that for given $\eta _{\vec u}>0,$ then after taking $\varepsilon _1$ and $\varepsilon _2$ small enough, Inequality (133) will guarantee that $X_{{\vec u}_F}\le \eta _{\vec u}$ on a small time interval the length of which may be bounded by below in terms of $\eta _{\vec u}$ and of the norms of the data. Let us underscore that imposing the values of $\varepsilon _1$ and $\varepsilon _2$ may be converted into a (not so explicit) smallness condition $T^*,$ through the whole function ${\vec u}^0.$

Similarly, taking ${\widetilde{\varepsilon }}_1$ and ${\widetilde{\varepsilon }}_2$ small enough, (134) will imply that $X_{\Theta _F}\le \eta _\Theta$ on some time interval depending only on the data.

To complete the proof of a priori estimates, we still have to check that (129) and (132) are indeed fulfilled if t,  $\eta _{\vec u}$ and $\eta _\Theta$ have been chosen sufficiently small. Given that $X_{{\vec u}}\le X_{{\vec u}_L}+X_{{\vec u}_F},$ $X_\Theta \le X_{\Theta _L}+X_{\Theta _F},$ $X_{{\vec u}_L}(t)\le (1+t)\Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}$ and $X_{\Theta _L}(t)\le (1+t)\Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}},$ Inequality (129) is fulfilled if t is small enough and

$$\begin{aligned} \eta _{\vec u}\le \frac{1}{2} \Vert {\vec u}^0\Vert _{B^{\frac{n}{2}}_{2,1}}\quad \hbox {and}\quad \eta _\Theta \le \frac{1}{2} \Vert \Theta ^0\Vert _{B^{\frac{n}{2}}_{2,1}}. \end{aligned}$$

Finally, we have

$$\begin{aligned} \int _0^t\Vert \nabla \Theta \Vert ^2_{B^{\frac{n}{2}}_{2,1}}\,\hbox {d}\tau \le 2{\widetilde{\varepsilon }}_2^2+2\eta _\Theta ^2, \end{aligned}$$

hence (132) is fulfilled if t,  ${\widetilde{\varepsilon }}_2$ and $\eta _\Theta$ have been chosen small enough. Therefore there exists a positive time $T^*$ that may be computed in terms of the norms of the data, and of the free solutions ${\vec u}_L$ and $\Theta _L$ so that (129) is fulfilled on $[0,T^*].$ This completes the proof of a priori estimates in the case of large data and general coefficients.

Besov spaces and Littlewood–Paley decomposition

Here we shortly recall the definition of Besov spaces, paraproduct, and a few nonlinear estimates that have been used extensively in the paper. The reader will find more material in the textbooks [1] and [27].

Let $\chi :\mathbb {R}^n\rightarrow [0,1]$ be a smooth nonincreasing radial function supported in B(0, 1) with, additionally, $\chi \equiv 1$ on B(0, 1 / 2). Let $\varphi (\xi )\,{:=}\,\chi (\xi /2)-\chi (\xi ).$ We define the nonhomogeneous Littlewood-Paley decomposition by setting

$$\begin{aligned} \Delta _{-1}u\,{:=}\,\chi (D)u={\mathcal {F}}^{-1}(\chi {\mathcal {F}}u)\quad \hbox {and}\quad \Delta _ku\,{:=}\,\varphi (2^{-k}D)u={\mathcal {F}}^{-1}(\varphi (2^{-k}\cdot ){\mathcal {F}}u)\ \hbox { if }\ k\ge 0. \end{aligned}$$

Note that by construction,

$$\begin{aligned} \chi +\sum _{k\ge 0} \varphi (2^{-k}\cdot )\equiv 1\quad \hbox {in }\ \mathbb {R}^n, \end{aligned}$$

and thus

$$\begin{aligned} \sum _{k\ge -1}\Delta _k=\mathrm{Id}\,\hbox { holds in }\ {\mathcal {S}}'(\mathbb {R}^n). \end{aligned}$$

Likewise, we define the homogeneous Littlewood-Paley decomposition by

$$\begin{aligned} \dot{\Delta }_ku\,{:=}\,\varphi (2^{-k}D)u={\mathcal {F}}^{-1}(\varphi (2^{-k}\cdot ){\mathcal {F}}u)\ \hbox {for all }\ k\in \mathbb {Z}. \end{aligned}$$

Because

$$\begin{aligned} \sum _{k\in \mathbb {Z}} \varphi (2^{-k}\xi )=1\quad \hbox {if and only if }\ \xi \not =0, \end{aligned}$$

in contrast with its nonhomogeneous counterpart, we only have

$$\begin{aligned} \sum _{k\in \mathbb {Z}}\dot{\Delta }_k=\mathrm{Id}\,\hbox { in }\ {\mathcal {S}}'(\mathbb {R}^n) \ \ \hbox { modulo polynomials.} \end{aligned}$$

Definition 7.1

For $s\in \mathbb {R}$ we set

$$\begin{aligned} \Vert u\Vert _{B^s_{2,1}}\,{:=}\,\sum _{k\ge -1} 2^{sk}\Vert \Delta _ku\Vert _{L^2} \quad \hbox {and}\quad \Vert u\Vert _{\dot{B}^s_{2,1}}\,{:=}\,\sum _{k\in \mathbb {Z}} 2^{sk}\Vert \dot{\Delta }_ku\Vert _{L^2}. \end{aligned}$$

The nonhomogeneous Besov space $B^s_{2,1}=B^s_{2,1}(\mathbb {R}^n)$ is the set of tempered distributions u so that $\Vert u\Vert _{B^s_{2,1}}<\infty ,$ while the homogeneous Besov space $\dot{B}^s_{2,1}=\dot{B}^s_{2,1}(\mathbb {R}^n)$ is the set of tempered distributions u so that $\Vert u\Vert _{\dot{B}^s_{2,1}}<\infty$ and, in addition⁹

$$\begin{aligned} \lim _{\lambda \rightarrow +\infty }\theta (\lambda D)u=0\quad \hbox {in }\ L^\infty \end{aligned}$$

for some test function $\theta$ with $\theta (0)\not =0.$

As in [5, 7, 12], working with different Besov norms for low and high frequencies is the key to the proof of global results. This motivates our introducing the notation

$$\begin{aligned} \Vert u\Vert ^\ell _{\dot{B}^s_{2,1}}\,{:=}\,\sum _{k\le k_0}2^{ks}\Vert \dot{\Delta }_ku\Vert _{L^2}\quad \hbox {and}\quad \Vert u\Vert ^h_{\dot{B}^s_{2,1}}\,{:=}\,\sum _{k>k_0}2^{ks}\Vert \dot{\Delta }_ku\Vert _{L^2}, \end{aligned}$$

(135)

where $k_0\in \mathbb {Z}$ is a fixed integer depending only on the coefficients of System (25).

Likewise, we shall use the notation

$$\begin{aligned} u^\ell \,{:=}\,\sum _{k\le k_0}\dot{\Delta }_ku\quad \hbox {and}\quad u^h\,{:=}\,\sum _{k>k_0}\dot{\Delta }_ku. \end{aligned}$$

(136)

The paraproduct between two tempered distributions u and v is defined by

$$\begin{aligned} T_uv\,{:=}\,\sum _k \dot{S}_{k-1} u\dot{\Delta }_kv\quad \hbox {with}\quad S_{k-1}\,{:=}\,\chi (2^{k-1}D). \end{aligned}$$

For sufficiently smooth functions (or distributions) the following Bony decomposition holds true:

$$\begin{aligned} uv=T_uv+R(u,v)+T_vu, \end{aligned}$$

where the remainder operator is defined by

$$\begin{aligned} R(u,v)\,{:=}\,\sum _k \dot{\Delta }_ku\, (\dot{\Delta }_{k-1}v+\dot{\Delta }_kv+\dot{\Delta }_{k+1}v\bigr ). \end{aligned}$$

The paraproduct and remainder operators are continuous in a number of classical functional spaces. In the present paper, we need the following result:

Lemma 7.2

Let s and t be two real numbers.

• If $t\le n/2$ then $T:\dot{B}^t_{2,1}\times \dot{B}^s_{2,1}\rightarrow \dot{B}^{s+t-\frac{n}{2}}_{2,1}.$

• If $0<s+t\le n$ then $R:\dot{B}^t_{2,1}\times \dot{B}^s_{2,1}\rightarrow \dot{B}^{s+t-\frac{n}{2}}_{2,1}.$

The following product laws in Besov spaces (or their homogeneous analogue) have been used a number of times in the paper.

Lemma 7.3

If $s,t>0$ then

$$\begin{aligned} \Vert au\Vert _{B^s_{2,1}}\lesssim \Vert a\Vert _{L^\infty }\Vert u\Vert _{B^s_{2,1}}+\Vert u\Vert _{B^{-t}_{\infty ,\infty }}\Vert \nabla a\Vert _{B^{s-1+t}_{2,1}} \end{aligned}$$

where $\Vert u\Vert _{B^{-t}_{\infty ,\infty }}\,{:=}\,\sup _{k\ge -1}2^{-k}\Vert \Delta _ku\Vert _{L^\infty }.$

Moreover

$$\begin{aligned} \Vert au\Vert _{B^s_{2,1}}\lesssim \Vert a\Vert _{L^\infty }\Vert u\Vert _{B^s_{2,1}}+\Vert u\Vert _{L^\infty }\Vert \nabla a\Vert _{B^{s-1}_{2,1}} \end{aligned}$$

and, for $j=1,\ldots ,n,$

$$\begin{aligned} \Vert a\partial _jv\Vert _{B^s_{2,1}}\lesssim \Vert a\Vert _{L^\infty }\Vert \partial _jv\Vert _{B^s_{2,1}}+\Vert v\Vert _{L^\infty }\Vert \nabla a\Vert _{B^{s}_{2,1}}. \end{aligned}$$

Proof

Use Bony’s decomposition

$$\begin{aligned} au=T_au+R(a,u)+T_ua, \end{aligned}$$

and the fact that, as in the paraproduct $T_ua$ the low frequencies of a are not involved, one may replace $\Vert a\Vert _{B^{s+t}_{2,1}}$ with $\Vert \nabla a\Vert _{B^{s-1+t}_{2,1}}$ (see Remark 2.83 in [1]). The last inequality stems from the first one with $t=1,$ once noticed that $\Vert \partial _jv\Vert _{B^{-1}_{\infty ,\infty }}\lesssim \Vert v\Vert _{L^\infty }$ (a consequence of Bernstein inequality).

Lemma 7.4

Let $s>0.$ Let $F:\mathbb {R}\rightarrow \mathbb {R}$ be a smooth function vanishing at 0. Then we have

$$\begin{aligned} \Vert F(a)\Vert _{B^s_{2,1}}\le C(\Vert a\Vert _{L^\infty }) \Vert a\Vert _{B^s_{2,1}} \quad \hbox {and}\quad \Vert \nabla (F(a))\Vert _{B^{s-1}_{2,1}}\le C(\Vert a\Vert _{L^\infty }) \Vert \nabla a\Vert _{B^{s-1}_{2,1}}. \end{aligned}$$

More generally, if $G:\mathbb {R}^2\rightarrow \mathbb {R}$ is a smooth function vanishing at 0 then

$$\begin{aligned} \Vert G(a,\theta )\Vert _{B^s_{2,1}}\le C(\Vert a\Vert _{L^\infty },\Vert \theta \Vert _{L^\infty }) \Vert (a,\theta )\Vert _{B^s_{2,1}} \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla (G(a,\theta ))\Vert _{B^{s-1}_{2,1}}\le C(\Vert a\Vert _{L^\infty },\Vert \theta \Vert _{L^\infty }) \Vert (\nabla a,\nabla \theta )\Vert _{B^{s-1}_{2,1}}. \end{aligned}$$

The statements with no gradient are completely standard (see, e.g., [27], Chap. 5). The first refined inequality with a gradient may be found in [10], Prop. 4. Its extension to two variables is not a big deal.