On steady state of viscous compressible heat conducting full magnetohydrodynamic equations

Abstract

This paper is concerned with the study of equations of viscous compressible and heat-conducting full magnetohydrodynamic (MHD) steady flows in a horizontal layer under the gravitational force and a large temperature gradient across the layer. We assume as boundary conditions, periodic conditions in the horizontal directions, while in the vertical directions, slip-boundary is assumed for the velocity, vertical conditions for the magnetic field, and fixed temperature or fixed heat flux are prescribed for the temperature. The existence of stationary solution in a small neighborhood of a stationary profile close to hydrostatic state is obtained in Sobolev spaces as a fixed point of some nonlinear operator.

1 Introduction

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids in the presence of a magnetic field or, more precisely, in their macroscopic interaction with a magnetic field. Magnetohydrodynamics is known to be of great importance in several directions. First of all, its applications cover a very wide range of areas in physics from liquid metals to cosmic plasmas. We can mention among others, in astrophysics (with the study of solar structure, the solar wind bathing the earth and other planets), in geophysics (with the planetary magnetism produced by currents deep in the planet), in high-speed aerodynamics, and in plasma physics. Magnetohydrodynamics is also important in connection with many engineering problems such as sustained plasma confinement for controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, magnetohydrodynamic power generation, electro-magnetic casting of metals, and plasma accelerators for ion thrusters for spacecraft propulsion. Due to their practical relevance, magnetohydrodynamic problems have long been the subject of intense cross-disciplinary research, but except for relatively simplified special cases, the rigorous mathematical analysis of such problems still presents many interesting aspects to be studied.

In magnetohydrodynamic flows, magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself, resulting in a complex interaction between the magnetic and fluid dynamic phenomena, and therefore, both hydrodynamic and electrodynamic effects have to be considered. The set of equations that describe compressible viscous magnetohydrodynamics are a combination of the compressible Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. In this paper, we consider the system of partial differential equations for the three-dimensional viscous compressible full magnetohydrodynamic stationary flows (see [23, 25, 26])

$$\begin{aligned} &\varrho (v\cdot \nabla )v -\mu \Delta v - (\mu +\lambda )\nabla \nabla \cdot v = - R\nabla (\varrho T) - g\varrho e_{3}+ (\nabla \times B)\times B , \end{aligned}$$

(1.1)

$$\begin{aligned} & \nabla \cdot (\varrho v) = 0, \end{aligned}$$

(1.2)

$$\begin{aligned} & c_{V}\varrho (v\cdot \nabla ) T - \kappa \Delta T + R\varrho T \nabla \cdot v = 2\mu \bigl\vert D(v) \bigr\vert ^{2} + \lambda ( \nabla \cdot v)^{2} + \nu \vert \nabla \times B \vert ^{2} \end{aligned}$$

(1.3)

$$\begin{aligned} &\nu \nabla \times (\nabla \times B) - \nabla \times (v\times B) = 0, \quad \nabla \cdot B = 0, \end{aligned}$$

(1.4)

where ϱ denotes the density, $v= (v_{1},v_{2},v_{3})$ the velocity, T the absolute temperature, $B = (B_{1},B_{2},B_{3})$ the magnetic field, and $D(v)$ is the deformation tensor defined by

$$ \bigl[D(v)\bigr]_{i,j} = {\frac{1}{2}}(\partial _{x_{i}}v_{j} + \partial _{x_{j}}v_{i}). $$

The constants μ and λ are the first and second viscosity coefficients respectively and satisfy the physical restrictions $\mu >0$ and $2\mu + 3\lambda \geq 0$. The positive constants $c_{V}$, κ, R, and ν are respectively the heat capacity, the ratio of the heat conductivity coefficient over the heat capacity, the universal constant of gases, and the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, while g the gravity acceleration and $e_{3} = (0,0,1)$. In (1.1) and (1.3) we have assumed that the pressure p is given by the law of perfect gases

$$ p = R\varrho T.$$

(1.5)

In magnetohydrodynamics, the displacement current can be neglected (see [23, 26]). As a consequence, equation (1.4) is called the induction equation, and the electric field can be written in terms of the magnetic field B and the velocity v,

$$ E = \nu \nabla \times B- v\times B. $$

Although the electric field E does not appear in the MHD system (1.1)–(1.4), it is indeed induced according to the above identity by the moving conductive flow in the magnetic field.

We recall that, due to their physical importance, complexity, rich phenomena, and mathematical challenges, there have been many studies on magnetohydrodynamics by physicists and mathematicians (see, e.g., [12, 13, 15, 17, 18, 20, 21, 26, 44] and the references cited therein). About the steady MHD equations for incompressible or compressible flows, we refer the interested reader to the articles [1–3, 5, 6, 14, 22, 30, 47] for the existence of strong and weak solutions. In [19], Gerbeau et al. also considered several kinds of unsteady problems and did some numerical analysis (see also [40]). On unsteady MHD, there have been many studies by physicists and mathematicians in the recent years (see, e.g., [13, 16, 17, 27, 44]). In particular, the one-dimensional problem has been studied in many articles (see, e.g., [4, 9, 12, 29]. On the other hand, computational aspects of the system of MHD equations were also studied with considerable interest in developing accurate numerical methods for those systems (see, e.g., [31, 39, 41]).

Almost all the literature mentioned above is concerned with the Cauchy problem or the initial boundary value problem for compressible MHD equations, with the homogenous Dirichlet condition on the magnetic field. In contrast with the extensive research on unsteady MHD flow, we find that there are only few results concerning the steady flow. In [45], Yang et al. established the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary under the assumption that the external force field is small. In [5] the authors improved the work in [45] by considering large external forces and, instead of perfectly conducting boundary conditions on the magnetic field, they considered non-homogeneous Dirichlet boundary conditions that can possibly be large enough.

In this work, we consider the MHD system of equations (1.1)–(1.4) in an infinite horizontal layer, and we investigate the fundamental problem of the existence of a stationary solution in a small neighborhood of a stationary profile close to hydrostatic state. As for boundary conditions, in the horizontal directions, the choice of periodic boundary conditions on velocity, temperature, and magnetic field is made naturally. Such conditions lend themselves better to the study of infinite plane-parallel media than wall-like conditions, which are more characteristic of laboratory experiment simulations. For the vertical direction, slip-boundary conditions are considered on velocity, and the magnetic field has its lines assumed vertical, while fixed temperature or fixed heat flux are prescribed.

More precisely, we consider system (1.1)–(1.4) in the bounded domain

$$ \Omega = \mathbb{T}^{2}\times ]0,h[, $$

where $\mathbb{T}^{2}$ is a two-dimensional torus with the following boundary conditions:

$$\begin{aligned} &v|_{x_{3} =0} = 0, \qquad v_{3}|_{x_{3} =h} = 0, \qquad \partial _{x_{3}}v_{i}|_{x_{3} =h}=0,\quad i=1,2 \end{aligned}$$

(1.6)

$$\begin{aligned} & T|_{x_{3} = 0}= T_{0} + \varepsilon \bigl(x' \bigr), \qquad T|_{x_{3} = h} = T_{0} - {\frac{\gamma -1}{\gamma R}}gh, \end{aligned}$$

(1.7)

$$\begin{aligned} &B_{1}|_{x_{3} =0,h} = B_{2}|_{x_{3} = 0,h} =0, \qquad B_{3}|_{x_{3} =0,h} = B_{0}, \end{aligned}$$

(1.8)

where $T_{0}> 0$ and $B_{0}$ are given large constants, $\varepsilon (x')$ is a small perturbation of the given temperature $T_{0}$ on the lower plane of the horizontal layer, which will be assumed periodic, $x'= (x_{1},x_{2})\in \mathbb{T}^{2}$ so that $x=(x',x_{3})\in \Omega$ and γ is the adiabatic exponent given by

$$ \gamma = {\frac{R}{C_{V}}} + 1. $$

Finally, it is assumed that the total mass of the fluid is prescribed as

$$ \int _{\Omega}\varrho \,dx = M_{0}.$$

(1.9)

From now on, we make the following assumptions:

$$ T_{0}\geq \bar{T}_{0},\qquad M_{0}\geq \bar{M}_{0},\qquad \lVert{ \varepsilon } \rVert _{H^{3}(\mathbb{T}^{2})}\leq \varepsilon _{0}, \qquad \gamma >1,$$

(1.10)

where the positive constants $\bar{T}_{0}$ and $\bar{M}_{0}$ are given large enough, while the constant $\varepsilon _{0}>0$ is sufficiently small.

Remark 1.1

If (see (1.7)) the perturbation $\varepsilon (x')$ is identically zero, then it is clear that the only stationary solution of the system of equations (1.1)–(1.4) with boundary conditions (1.6)–(1.8) and periodic conditions on the horizontal directions is given by the rest state $(0,B_{rs},T_{rs},\varrho _{rs})$, where

$$ B_{rs}:=B_{0}e_{3},\qquad T_{rs}:=T_{0} - {\frac{\gamma -1}{\gamma R}}gx_{3}, \qquad \varrho _{rs}(x_{3})=: \bar{C} T_{rs}^{\frac{1}{\gamma -1}}(x_{3}),$$

(1.11)

with the constant C̄ determined by the condition $\int _{\Omega}\varrho _{rs}(x_{3})\,dx = M_{0}$.

Notice that, if $\varepsilon (x')$ is a nonzero constant $\varepsilon _{0}$, then we can reduce ourselves to case (1.11) with $T_{0}$ replaced by $\bar{T}_{0} = T_{0} + \varepsilon _{0}$.

Remark 1.2

If the perturbation $\varepsilon (x')\not\equiv 0$, i.e., nonidentically zero, and if there exists a stationary solution $(v,B,T,\varrho )$ to our problem, then necessarily v is not identically zero (see Remark 5.2 in Sect. 5). This is obtained from the main result (Theorem 2.1) in the next section. This simply means that the nonhomogeneous repartition of the temperature around $T_{0}$ in the lower plane $x_{3} = 0$ can generate a stationary magnetoconvective motion close to the rest state $(0,B_{rs},T_{rs},\varrho _{rs})$ given by (1.11).

The system of equations (1.1)–(1.4) with boundary conditions (1.6)–(1.8) can be used as a simple model of a three-dimensional plane-parallel atmosphere made up of ideal gas (see [37, 38] and their exhaustive references on subjects related to this issue). More precisely, we consider in a horizontal layer a polytropic atmosphere with a large temperature gradient across the layer and in the presence of a vertical magnetic field. We will then focus on the existence of a magnetoconvective steady flow close to the equilibrium state. This coupling between natural convection (the fluids motions induced by a large temperature gradient) and magnetic field has aroused enormous interest following the observation of astrophysical and geophysical phenomena. The examples of application are numerous: we can cite among others the planetary magnetospheres, the sun, the stars, the solar and stellar winds, the interstellar clouds, the accretion disks, and the galaxies.

This work is organized as follows. In Sect. 2, we present some preliminaries of the proof of the main result. In Sect. 3, we introduce a linearized problem, which is crucial for the construction of the nonlinear operator S, for which we establish the existence of a fixed point through the Schauder fixed point theorem. The crucial part of this work is then to show that the operator S satisfies the assumptions of the Schauder fixed point theorem (see Sect. 5). This has required some tedious computations aimed at establishing suitable fine estimates (in Lemmas 4.1–4.10). These estimates rely on some techniques developed in the study of the equations of viscous gazes, in particular those in [8] and also in [35] (in the general case). We can also mention [33, 34, 42, 43].

2 Main result and some preliminaries to the proof

We look for a solution $(v,B,T,\varrho )$ in a small neighborhood of the stationary profile $(0,\hat{B},\hat{T},\hat{\varrho})$ close to the equilibrium state $(0,{B}_{rs},{T}_{rs},{\varrho}_{rs})$ given in (1.11)). This stationary profile is defined by

$$\begin{aligned} & \hat{B}\bigl(x',x_{3}\bigr) := B_{rs},\qquad \hat{T}\bigl(x',x_{3}\bigr) := T_{rs}(x_{3}) + \biggl(1- {\frac{x_{3}}{h}} \biggr)\varepsilon \bigl(x'\bigr), \end{aligned}$$

(2.1)

$$\begin{aligned} & \hat{\varrho}\bigl(x',x_{3}\bigr) :=\hat{M} { \frac{T_{rs}^{\frac{\gamma}{\gamma -1}}(x_{3})}{\hat{T}(x',x_{3})}}\quad \text{with } \hat{M} := M_{0} \biggl[ \int _{\Omega}{\frac{T_{rs}^{ \frac{\gamma}{\gamma -1}}(x_{3})}{\hat{T}(x',x_{3})}}\,dx \biggr]^{-1}. \end{aligned}$$

(2.2)

Notice that T̂ satisfies the same boundary conditions (1.7) and ϱ̂ satisfies (1.9) so that

$$ \int _{\Omega}(\varrho - \hat{\varrho})\,dx = 0.$$

(2.3)

Given (2.1) and (2.2), easy computations show that

$$\begin{aligned} &\partial _{x_{i}}(R\hat{T}\hat{\varrho}) = 0 \quad (i=1,2),\qquad - \partial _{x_{3}}(R\hat{T}\hat{\varrho}) - g\hat{\varrho} = { \frac{g\hat{\varrho}}{T_{rs}}} \biggl(1 - {\frac{x_{3}}{h}} \biggr)\varepsilon \bigl(x'\bigr), \end{aligned}$$

(2.4)

$$\begin{aligned} & C_{V}\hat{\varrho}\nabla \hat{T} - R \hat{T}\nabla \hat{\varrho} = (R + C_{V})\hat{\varrho}\nabla \biggl( \biggl( 1 - { \frac{x_{3}}{h}} \biggr)\varepsilon \bigl(x'\bigr) \biggr) + {\frac{g\hat{\varrho}}{T_{rs}}} \biggl(1 - {\frac{x_{3}}{h}}\biggr) \varepsilon \bigl(x'\bigr){e_{3}}. \end{aligned}$$

(2.5)

This being so, we then set

$$ v = v,\qquad b = B - \hat{B},\qquad \vartheta = T - \hat{T},\qquad \sigma = \varrho - \hat{\varrho},$$

(2.6)

and, since as we seek ϱ close to ϱ̂, we can assume that

$$ \Vert \varrho - \hat{\varrho} \Vert _{L^{\infty}}\leq { \frac{1}{2}} \inf_{\bar{\Omega}} \hat{\varrho}.$$

(2.7)

By considering for a generic vector field M the identities

$$\begin{aligned} &(\nabla \times M)\times M = (M\cdot \nabla )M - {\frac{1}{2}}\nabla \vert M \vert ^{2}, \\ & \nabla \times (v\times M) = v\nabla \cdot M - M\nabla \cdot v + (M \cdot \nabla ) v - (v\cdot \nabla ) M, \\ & \nabla \times (\nabla \times M) = \nabla \nabla \cdot M - \Delta M, \end{aligned}$$

and given (2.6), we then rewrite problem (1.1)–(1.9) with unknowns $(T,v,B,\varrho )$ as a new problem with unknowns $u=(\vartheta ,v,b,\sigma )$ as follows:

$$\begin{aligned} &{-} \kappa \Delta \vartheta = G(u) , \end{aligned}$$

(2.8)

$$\begin{aligned} & {-}\mu \Delta v - (\mu +\lambda )\nabla \nabla \cdot v = - R \nabla ( \hat{T}\sigma + \hat{\varrho}\vartheta ) - g\sigma e_{3} +( \hat{B} \cdot \nabla )b - \hat{B}\cdot \nabla b + F(u), \end{aligned}$$

(2.9)

$$\begin{aligned} & {-} \nu \Delta b = -\hat{B}\nabla \cdot v + (\hat{B}\cdot \nabla )v + H(u), \quad \nabla \cdot b = 0, \end{aligned}$$

(2.10)

$$\mathrm{\nabla }\cdot (\sigma v)=-\mathrm{\nabla }\cdot (\hat{\varrho }v),$$

(2.11)

with the boundary conditions

$$\begin{aligned} &v|_{x_{3} =0} = 0, \qquad v_{3}|_{x_{3} =h}=0 , \qquad \partial _{x_{3}}v_{i}|_{x_{3} = h} = 0\quad i=1,2, \end{aligned}$$

(2.12)

$$\begin{aligned} & \vartheta |_{x_{3} = 0}= \vartheta |_{x_{3} = h} = 0, \end{aligned}$$

(2.13)

$$\begin{aligned} &b|_{x_{3} =0} = b|_{x_{3} =h}=0, \end{aligned}$$

(2.14)

and (see (2.6) and (2.3)) the mean value property

$$ \int _{\Omega}\sigma \,dx = 0.$$

(2.15)

The functions $F(u)$, $G(u)$, and $H(u)$ are given by

$$\begin{aligned} &F(u):= -(\hat{\varrho} + \sigma ) (v\cdot \nabla )v - R\nabla ( \sigma \vartheta ) + g{\frac{\hat{\varrho}}{T_{rs}}} \biggl(1 - {\frac{x_{3}}{h}} \biggr) \varepsilon \bigl(x'\bigr)e_{3} + (\nabla \times b) \times b, \end{aligned}$$

(2.16)

$$\begin{aligned} &G(u):= 2\mu D(v):D(v) + \lambda (\nabla \cdot v)^{2} + Rv\cdot ( \vartheta \nabla \hat{\varrho} + \hat{T}\nabla \sigma + \vartheta \nabla \sigma ) \end{aligned}$$

(2.17)

$$\begin{aligned} & \hphantom{G(u)}\quad + C_{V}(\sigma \nabla \hat{T} + \hat{\varrho}\nabla \vartheta + \sigma \nabla \vartheta )\cdot v - (R + C_{V})\hat{ \varrho} v\cdot \nabla \biggl( \biggl(1 - {\frac{x_{3}}{h}} \biggr) \varepsilon \bigl(x'\bigr) \biggr) \\ &\hphantom{G(u)}\quad-g{\frac{\hat{\varrho}}{T_{rs}}} \biggl(1 - {\frac{x_{3}}{h}} \biggr)\varepsilon \bigl(x'\bigr)v_{3}+ \kappa \Delta \biggl( \biggl(1 - {\frac{x_{3}}{h}} \biggr)\varepsilon \bigl(x'\bigr) \biggr) + \nu \vert \nabla \times b \vert ^{2}, \\ & H(u):= v\nabla \cdot b - b\nabla \cdot v + (b\cdot \nabla )v - (v \cdot \nabla )b . \end{aligned}$$

(2.18)

Thus, problem (1.1)–(1.8) is reduced to finding

$$ u=(\vartheta ,v,b,\sigma ) = (T,v,B,\varrho ) - (\hat{T},0,\hat{B}, \hat{ \varrho}) $$

solution of problem (2.8)–(2.15).

Notice that equation (2.10) is obtained from (1.4), which is written as

$$ - \nu \Delta B - \nabla \times (v \times B) = 0 ,\quad B = b+\hat{B} , $$

because $\nabla \times (\nabla \times B) = \nabla \nabla \cdot B - \Delta B$, $\nabla \cdot B = 0$, $\nabla \cdot \hat{B} = 0$, and so $\nabla \cdot b= 0$.

Our main result is the following theorem.

Theorem 2.1

Under (1.10), the system of equations (1.1)–(1.4) with boundary conditions (1.6)–(1.8) has at least one solution

$$ (T,v,B,\varrho )\in H_{\natural}^{2}(\Omega ) \times H_{\natural}^{3}( \Omega )\times H_{\natural}^{2}( \Omega )\times H_{\natural}^{2}( \Omega ),$$

(2.19)

where ♮ means periodicity in the horizontal directions.

We look for a solution

$$ (T,v,B,\varrho ) = (\vartheta ,v,b,\sigma ) + (\hat{T},0,\hat{B}, \hat{ \varrho}), $$

where $u=(\vartheta ,v,b,\sigma )$ is the appropriate solution of problem (2.8)–(2.15) which will be obtained as a fixed point of some operator constructed from a suitable linearization of the system of equations (2.8)–(2.10).

3 Linearized equations and the nonlinear operator S

We set

$$\begin{aligned} & \widehat{H}^{2}_{\natural}(\Omega ) := H^{2}(\Omega )\cap \widehat{H}^{1}_{ \natural}( \Omega ),\qquad \widehat{H}^{1}_{\natural}(\Omega ):=\bigl\{ b\in H^{1}_{ \natural}(\Omega ): \nabla \cdot b=0\bigr\} , \end{aligned}$$

(3.1)

$$\begin{aligned} & V := \bigl\{ u\in H^{2}_{\natural}(\Omega )\times H^{3}_{\natural}( \Omega )\times \widehat{H}^{2}_{\natural}( \Omega )\times H^{2}_{ \natural}(\Omega ): u \text{ satisfies (2.12)--(2.15)} \bigr\} , \end{aligned}$$

(3.2)

$$\begin{aligned} & \Vert \cdot \Vert _{V}^{2} := \Vert \cdot \Vert _{H^{2}}^{2} + \Vert \cdot \Vert _{H^{3}}^{2} + \Vert \cdot \Vert _{H^{2}}^{2} + \Vert \cdot \Vert _{H^{2}}^{2}, \end{aligned}$$

(3.3)

where $\lVert{\cdot } \rVert _{H^{k}}$ and $\lVert{\cdot } \rVert _{L^{2}}$ denote the usual norms of $H^{k}(\Omega )$ and $L^{2}(\Omega )$. Furthermore, we consider

$$ V_{0} := \bigl(V, \Vert \cdot \Vert _{V_{0}}\bigr)\quad \text{with } \Vert \cdot \Vert _{V_{0}}^{2}:= \Vert \cdot \Vert _{H^{1}}^{2} + \Vert \cdot \Vert _{H^{2}}^{2} + \Vert \cdot \Vert _{H^{1}}^{2} + \Vert \cdot \Vert _{H^{1}}^{2}.$$

(3.4)

Let now $u'=(\vartheta ',v',b',\sigma ')\in V$ and $k>0$ be given. We consider the system of equations

$$\begin{aligned} &{-} \kappa \Delta \vartheta = G', \end{aligned}$$

(3.5)

$$\begin{aligned} & {-}\mu \Delta v - (\mu +\lambda )\nabla \nabla \cdot v = - R \nabla \bigl( \hat{\varrho}\vartheta + \hat{T}\sigma '\bigr) - g\sigma ' e_{3} +( \hat{B}\cdot \nabla )b - \hat{B}\cdot \nabla b + F', \end{aligned}$$

(3.6)

$$\begin{aligned} & {-} \nu \Delta b = -\widehat{B}\nabla \cdot v + (\widehat{B}\cdot \nabla )v + H', \quad \nabla \cdot b = 0, \end{aligned}$$

(3.7)

$$\begin{aligned} & k\bigl(\sigma -\sigma '\bigr) + \nabla \cdot (\sigma v) = - \nabla \cdot ( \hat{\varrho}v), \end{aligned}$$

(3.8)

with boundary conditions (2.12)–(2.14) and (2.15), where (see (2.16)–(2.18)),

$$ G' = G\bigl(u'\bigr),\qquad F'= F\bigl(u'\bigr),\qquad H'= H \bigl(u'\bigr).$$

(3.9)

We obtain in the following lemma an existence result for system (3.5)–(3.8) with (2.12)–(2.14).

Lemma 3.1

Let $u' = (\vartheta ',v',b',\sigma ')\in V$. If $k>0$ is large enough, then system (3.5)–(3.8) with conditions (2.12)–(2.14) has a unique solution $u=(\vartheta ,v,b,\sigma )\in V$ such that

$$\begin{aligned} & \lVert{\vartheta } \rVert _{H^{2}}^{2} \leq c_{\Omega} \bigl\Vert G' \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$

(3.10)

$$\begin{aligned} & \lVert{v} \rVert _{H^{3}}^{2}\leq c_{\Omega} \bigl( \bigl\Vert \sigma ' \bigr\Vert _{H^{2}}^{2} + \bigl\Vert F' \bigr\Vert _{H^{1}}^{2} + \bigl\Vert G' \bigr\Vert _{L^{2}}^{2} + \bigl\Vert H' \bigr\Vert _{L^{2}}^{2} \bigr), \end{aligned}$$

(3.11)

$$\begin{aligned} & \lVert{b} \rVert _{H^{2}}^{2}\leq c_{\Omega} \bigl( \bigl\Vert \sigma ' \bigr\Vert _{H^{1}}^{2} + \bigl\Vert F' \bigr\Vert _{L^{2}}^{2} + \bigl\Vert G' \bigr\Vert _{L^{2}}^{2} + \bigl\Vert H' \bigr\Vert _{L^{2}}^{2} \bigr), \end{aligned}$$

(3.12)

$$\begin{aligned} & k \lVert{\sigma } \rVert _{H^{2}}^{2}\leq c_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2} + k \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2} + c_{\Omega} \lVert{v} \rVert _{H^{3}}^{2}, \end{aligned}$$

(3.13)

where $c_{\Omega}$ is a constant depending only on T̂, B̂, ϱ̂, and Ω.

Proof

Set

$$ W=\bigl\{ (\vartheta ,v,b)\in H^{1}_{\natural}(\Omega )\times H^{1}_{ \natural}(\Omega )\times \hat{H}^{1}_{\natural}( \Omega ): \vartheta |_{x_{3}=0,h}=0,\, v|_{x_{3}=0}= b|_{x_{3}=0,h}=0\bigr\} , $$

and let us start by showing that the system of equations (3.5)–(3.7) with boundary conditions (2.12)–(2.14) has a unique weak solution $(\vartheta ,v,b)\in W$ satisfying

$$ a_{\alpha}\bigl((\vartheta ,v,b),(\chi ,\varphi ,\psi ) \bigr) = L_{\alpha}( \chi ,\varphi ,\psi )\quad \forall (\chi ,\varphi , \psi )\in W, $$

(3.14)

where

$$\begin{aligned} & a_{\alpha}\bigl((\vartheta ,v,b),(\chi ,\varphi ,\psi )\bigr) \\ &\quad := \alpha \kappa \int _{\Omega }\nabla \vartheta \cdot \nabla \chi \,dx + \int _{\Omega }\bigl(\mu \nabla v\cdot \nabla \varphi + ( \mu + \lambda )\nabla \cdot v\nabla \cdot \varphi \bigr)\,dx \\ & \qquad + \nu \int _{\Omega }\nabla b\cdot \nabla \psi \,dx + R \int _{\Omega }\nabla (\hat{\varrho}\vartheta )\cdot \varphi \,dx + \int _{\Omega} \bigl[-(\hat{B}\cdot \nabla )b + \widehat{B}\cdot \nabla b) \bigr]\varphi \,dx \\ & \qquad + \int _{\Omega}\bigl( - (\hat{B}\cdot \nabla )v +\hat{B} \nabla \cdot v\bigr)\psi \,dx, \\ & L_{\alpha}(\chi ,\varphi ,\psi ) := \int _{\Omega } \alpha G'\chi \,dx + \int _{\Omega }F'\varphi \,dx + \int _{\Omega }H'\psi \,dx - \int _{\Omega}\bigl(R \nabla \bigl(\hat{\varrho}^{\gamma -1} \sigma '\bigr) + g\sigma ' e_{3}\bigr) \varphi \,dx, \end{aligned}$$

and $\alpha > 0$ is a real number, which will be appropriately chosen thereafter.

Indeed, by endowing W with the norm $\|(\vartheta ,v,b)\|_{W} = (\|\nabla \vartheta \|_{L^{2}}^{2} + \| \nabla v\|_{L^{2}}^{2} + \|\nabla b\|_{L^{2}}^{2})^{\frac{1}{2}}$, which makes it a Hilbert space, we can easily see that the bilinear form $a_{\alpha}$ is continuous on W. Moreover, since

$$\begin{aligned} & \int _{\Omega} \bigl[(\hat{B}\cdot \nabla )b - \nabla (b \cdot \hat{B}) \bigr]v \,dx + \int _{\Omega }b\cdot \bigl(v \nabla \cdot \hat{B} + (\widehat{B} \cdot \nabla )v +\hat{B}\nabla \cdot v\bigr) \,dx=0, \\ & R \int _{\Omega }\nabla (\hat{\varrho}\vartheta ) \cdot v \,dx = - R \int _{\Omega }\hat{\varrho}\vartheta \nabla \cdot v \,dx\geq - { \frac{\mu +\lambda}{2}} \Vert \nabla \cdot v \Vert _{L^{2}}^{2} - c_{\Omega } \Vert \nabla \vartheta \Vert _{L^{2}}^{2}, \end{aligned}$$

we have

$$\begin{aligned} a_{\alpha}\bigl((\vartheta ,v,b),(\vartheta ,v,b)\bigr) &= \alpha \kappa \Vert \nabla \vartheta \Vert _{L^{2}}^{2} + \mu \Vert \nabla v \Vert _{L^{2}}^{2} + ( \mu + \lambda ) \Vert \nabla \cdot v \Vert _{L^{2}}^{2} \\ & \quad + \nu \Vert \nabla b \Vert _{L^{2}}^{2} + R \int _{\Omega } \nabla (\hat{\varrho}\vartheta )\cdot v \,dx \\ & \geq (\alpha \kappa -c_{\Omega}) \Vert \nabla \vartheta \Vert _{L^{2}}^{2} + \mu \Vert \nabla v \Vert _{L^{2}}^{2} + {\frac{1}{2}}(\mu + \lambda ) \Vert \nabla \cdot v \Vert _{L^{2}}^{2} + \nu \Vert \nabla b \Vert _{L^{2}}^{2} \\ & \geq \inf \biggl(\kappa ,\mu ,{\frac{1}{2}}(\mu + \lambda ), \nu \biggr) \bigl\Vert (\vartheta ,v,b) \bigr\Vert _{W}^{2}, \end{aligned}$$

where we have chosen α such that $\alpha \kappa -c_{\Omega}\geq \kappa$.

So, the bilinear form $a_{\alpha}$ is coercive on W and, since the linear form $L_{\alpha}$ is obviously continuous on W, according to the Lax–Milgram theorem, problem (3.14) has a unique weak solution $(\vartheta ,v,b)\in W$ such that

$$ \Vert \nabla \vartheta \Vert _{L^{2}}^{2}+ \Vert \nabla v \Vert _{L^{2}}^{2} + \Vert \nabla b \Vert _{L^{2}}^{2}\leq c_{\Omega} \bigl( \bigl\Vert \sigma ' \bigr\Vert _{H^{1}}^{2} + \bigl\Vert F' \bigr\Vert _{L^{2}}^{2} + \bigl\Vert G' \bigr\Vert _{L^{2}}^{2} + \bigl\Vert H' \bigr\Vert _{L^{2}}^{2} \bigr).$$

(3.15)

This being, from the classical theory of systems of linear elliptic pdes (see e.g., [24, 32]), according to (3.15), we have (3.10)–(3.12). As for the solution $\sigma \in H^{2}$ of (3.8) and its estimate (3.13), we refer, for instance, the interested reader to [3]. □

Following Lemma 3.1, we then define the nonlinear operator S as follows:

$$ S : V\to V ,\qquad u' = \bigl(\vartheta ',v',b',\sigma ' \bigr), \qquad S\bigl(u'\bigr) = u,$$

(3.16)

where $u= (\vartheta ,v,b,\sigma )\in V$ is the unique solution of system (3.5)–(3.8).

Lemma 3.2

The nonlinear operator $S: V_{0}\rightarrow V_{0}$ is continuous. More precisely, for any bounded subset D of V, there exists $c_{D}>0$ such that

$$ \bigl\lVert {S\bigl(u'_{1}\bigr)-S \bigl(u'_{2}\bigr)} \bigr\rVert _{V_{0}} \leq c_{D} \bigl\lVert {u'_{1}-u'_{2}} \bigr\rVert _{V_{0}}\quad \forall u'_{1}, u'_{2}\in D.$$

(3.17)

Proof

Let us first notice that the operator S maps bounded subsets into bounded sets. Indeed, taking into account (3.9), it is easy to see that, for every $u'=(\vartheta ',v',b',\sigma ')\in V$, we have

$$\begin{aligned} & \bigl\lVert {F'} \bigr\rVert _{H^{1}} \leq c_{\Omega}\bigl[ \bigl\lVert {v'} \bigr\rVert _{H^{2}}^{2}\bigl( 1 + \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}\bigr) + \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2}+ \bigl\lVert { \vartheta '} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {b'} \bigr\rVert _{H^{2}}^{2} + \lVert{\varepsilon } \rVert _{H^{1}( \mathbb{T}^{2})}\bigr], \end{aligned}$$

(3.18)

$$\begin{aligned} & \bigl\lVert {G'} \bigr\rVert _{L^{2}} \leq c_{\Omega}\bigl( \bigl\lVert {v'} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {\vartheta '} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {b'} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2}\bigr) \end{aligned}$$

(3.19)

$$\begin{aligned} & \hphantom{\bigl\lVert {G'} \bigr\rVert _{L^{2}}}\quad + c_{\Omega} \bigl\lVert {\vartheta '} \bigr\rVert _{H^{2}} \bigl\lVert {v'} \bigr\rVert _{H^{2}} \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}} + c_{\Omega}\bigl(1+ \bigl\lVert {v'} \bigr\rVert _{H^{2}}\bigr) \lVert{\varepsilon } \rVert _{H^{2}(\mathbb{T}^{2})}, \\ & \bigl\lVert {H'} \bigr\rVert _{L^{2}}\leq c_{\Omega}\bigl( \bigl\lVert {v'} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {b'} \bigr\rVert _{H^{2}}^{2}\bigr). \end{aligned}$$

(3.20)

Hence, by recalling the V-norm (see (3.2)), we obtain

$$\begin{aligned} & \bigl\lVert {F'} \bigr\rVert _{H^{1}} \leq c_{\Omega} \bigl\lVert {u'} \bigr\rVert _{V}^{2}\bigl( 1 + \bigl\lVert {u'} \bigr\rVert _{V}\bigr) + c_{ \Omega} \lVert{\varepsilon } \rVert _{H^{1}(\mathbb{T}^{2})}, \end{aligned}$$

(3.21)

$$\begin{aligned} & \bigl\lVert {G'} \bigr\rVert _{L^{2}}\leq c_{\Omega} \bigl\lVert {u'} \bigr\rVert _{V}^{2}\bigl(1 + \bigl\lVert {u'} \bigr\rVert _{V}\bigr) + c_{\Omega}\bigl(1+ \bigl\lVert {u'} \bigr\rVert _{V}\bigr) \lVert{ \varepsilon } \rVert _{H^{2}(\mathbb{T}^{2})}, \end{aligned}$$

(3.22)

$$\begin{aligned} & \bigl\lVert {H'} \bigr\rVert _{L^{2}}\leq c_{\Omega} \bigl\lVert {u'} \bigr\rVert _{V}^{2}. \end{aligned}$$

(3.23)

Let D be a bounded subset of V. If k is large enough, from (3.10)–(3.13) and the previous inequalities, we can easily show that

$$ \bigl\lVert {S\bigl(u'\bigr)} \bigr\rVert _{V}^{2} = \lVert{v} \rVert _{H^{3}}^{2} + \lVert{\vartheta } \rVert _{H^{2}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} + \lVert{\sigma } \rVert _{H^{2}}^{2} \leq c_{D}\quad \forall u'\in D.$$

(3.24)

This means that $S(D)$ is a bounded set of V. Let now

$$\begin{aligned} & u'_{i} = \bigl(\vartheta '_{i},v'_{i},b'_{i}, \sigma '_{i}\bigr)\in D, \qquad S\bigl(u'_{i} \bigr) = u_{i} = (\vartheta _{i},v_{i},b_{i}, \sigma _{i}), \quad i=1,2, \end{aligned}$$

(3.25)

$$\begin{aligned} & u'_{1} - u'_{2} = \bigl(\vartheta ',v',b',\sigma '\bigr),\qquad S\bigl(u'_{1}\bigr) - S \bigl(u'_{2}\bigr) = u_{1} - u_{2} = (\vartheta ,v,b,\sigma ). \end{aligned}$$

(3.26)

By definition (3.16) of the operator S, we have

$$\begin{aligned} & {-} \kappa \Delta \vartheta = G\bigl(u'_{1}\bigr) - G \bigl(u'_{2}\bigr), \end{aligned}$$

(3.27)

$$\begin{aligned} & {-} \mu \Delta v - (\mu +\lambda )\nabla \nabla \cdot v = - R\nabla \bigl( \hat{T}\sigma '\bigr) - R\nabla (\hat{\varrho}\vartheta ) - g\sigma ' e_{3} \end{aligned}$$

(3.28)

$$\begin{aligned} &\hphantom{- \mu \Delta v - (\mu +\lambda )\nabla \nabla \cdot v}\quad + (\hat{B}\cdot \nabla )b - \hat{B}\cdot \nabla b + F\bigl(u'_{1} \bigr) - F\bigl(u'_{2}\bigr), \\ & {-}\nu \Delta b = v\nabla\cdot\hat{B}- \hat{B}\nabla \cdot v + (\hat{B}\cdot \nabla ) v + H \bigl(u'_{1}\bigr) - H\bigl(u'_{2} \bigr), \end{aligned}$$

(3.29)

$$\begin{aligned} & k\bigl(\sigma -\sigma '\bigr) + \nabla \cdot (\sigma v_{1}) = - \nabla \cdot \bigl(( \hat{\varrho}+ \sigma _{2}) v\bigr)\bigr). \end{aligned}$$

(3.30)

From (3.9) and (2.17)–(2.18), it is easy to see that

$$\begin{aligned} & \bigl\lVert {G\bigl(u'_{1}\bigr)-G \bigl(u'_{2}\bigr)} \bigr\rVert _{H^{-1}} + \bigl\lVert {F\bigl(u'_{1}\bigr)-F\bigl(u'_{2} \bigr)} \bigr\rVert _{L^{2}}+ \bigl\lVert {H\bigl(u'_{1} \bigr)-H\bigl(u'_{2}\bigr)} \bigr\rVert _{H^{-1}} \\ &\quad \leq c_{\Omega}\bigl(1 + \bigl\lVert {u'_{1}} \bigr\rVert _{V}^{2} + \bigl\lVert {u'_{2}} \bigr\rVert _{V}^{2}\bigr) \bigl( \bigl\lVert {v'} \bigr\rVert _{H^{2}} + \bigl\lVert {\vartheta '} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {b'} \bigr\rVert _{H^{1}} + \bigl\lVert {\sigma '} \bigr\rVert _{H^{1}}\bigr). \end{aligned}$$

Now, by applying the regularity results for systems of elliptic PDEs to (3.27)–(3.29), we obtain (see (3.26) and (3.4))

$$\begin{aligned} \lVert{\vartheta } \rVert _{H^{1}} + \lVert{v} \rVert _{H^{2}}+ \lVert{b} \rVert _{H^{1}}&\leq \bigl\lVert {G \bigl(u'_{1}\bigr)-G\bigl(u'_{2} \bigr)} \bigr\rVert _{H^{-1}} + \bigl\lVert {F\bigl(u'_{1} \bigr)-F\bigl(u'_{2}\bigr)} \bigr\rVert _{L^{2}} \\ & \quad + \bigl\lVert {H\bigl(u'_{1}\bigr)-H \bigl(u'_{2}\bigr)} \bigr\rVert _{H^{-1}} \\ & \leq c_{\Omega}\bigl(1 + \bigl\lVert {u'_{1}} \bigr\rVert _{V}^{2} + \bigl\lVert {u'_{2}} \bigr\rVert _{V}^{2}\bigr) \bigl\lVert {u'_{1} - u'_{2}} \bigr\rVert _{V_{0}}. \end{aligned}$$

(3.31)

Now, it remains to estimate in the $H ^{1}$-norm the solution σ of equation (3.30). We first multiply (3.30) by σ, and we integrate over Ω. Applying integration by parts and using the inequality

$$ \int _{\Omega}\nabla \cdot (\sigma v_{1})\sigma = { \frac{1}{2}} \int _{\Omega} \vert \sigma \vert ^{2}\nabla \cdot v_{1}\,dx \leq c_{\Omega} \lVert{v_{1}} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{L^{2}}^{2}, $$

we easily obtain

$$ k \lVert{\sigma } \rVert _{L^{2}}^{2}\leq c_{\Omega} \lVert{v_{1}} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{L^{2}}^{2} + k \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2} + c_{\Omega}\bigl(1 + \lVert{\sigma _{2}} \rVert _{H^{1}}^{2}\bigr) \lVert{v} \rVert _{H^{1}}^{2} .$$

(3.32)

We next differentiate equation (3.30) by applying ∇ to both sides of this equation, then we take the scalar product with ∇σ, and we integrate over Ω. Applying integration by parts and using the inequality

$$\begin{aligned} \int _{\Omega}\nabla \sigma \cdot \nabla \bigl(\nabla \cdot ( \sigma v_{1})\bigr)\,dx &= {\frac{1}{2}} \int _{\Omega} \vert \nabla \sigma \vert ^{2}( \nabla \cdot v_{1}) \,dx + \int _{ \Omega}(\nabla \sigma \cdot \nabla )v_{1}\cdot \nabla \sigma \,dx \\ &\quad + \int _{\Omega}\sigma \bigl(\nabla (\nabla \cdot v_{1})\bigr) \cdot \nabla \sigma \,dx\leq c_{\Omega} \lVert{v_{1}} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{1}}^{2}, \end{aligned}$$

we get

$$ k \lVert{\nabla \sigma } \rVert _{L^{2}}^{2} \leq c_{ \Omega} \lVert{v_{1}} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{H^{1}}^{2} + k \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} + c_{\Omega}\bigl(1 + \lVert{\sigma _{2}} \rVert _{H^{2}}^{2}\bigr) \lVert{v} \rVert _{H^{2}}^{2}.$$

(3.33)

Adding (3.32) and (3.33), we obtain, using (3.31) (see also (3.25) and (3.26)), that

$$\begin{aligned} k \lVert{\sigma } \rVert _{H^{1}}^{2} &\leq c_{\Omega} \bigl\lVert {S\bigl(u'_{1}\bigr)} \bigr\rVert _{V} \lVert{\sigma } \rVert _{H^{1}}^{2} + k \bigl\lVert {\sigma '} \bigr\rVert _{H^{1}}^{2} + c_{\Omega}\bigl(1 + \bigl\lVert {S\bigl(u'_{2} \bigr)} \bigr\rVert _{V}^{2}\bigr) \lVert{v} \rVert _{H^{2}}^{2} \\ & \leq c_{\Omega} \bigl\lVert {S\bigl(u'_{1} \bigr)} \bigr\rVert _{V} \lVert{\sigma } \rVert _{H^{1}}^{2} + c_{\Omega} \bigl(1 + \bigl\lVert {S\bigl(u'_{2} \bigr)} \bigr\rVert _{V}^{2} \bigr) \bigl(1 + \bigl\lVert {u'_{1}} \bigr\rVert _{V}^{2} + \bigl\lVert {u'_{2}} \bigr\rVert _{V}^{2} \bigr) \bigl\lVert {u'_{1}-u'_{2}} \bigr\rVert _{V_{0}}^{2}. \end{aligned}$$

Hence, if k is large enough so that (see (3.24)) $k - c_{\Omega} \lVert{S(u'_{1})} \rVert _{V}\geq k - c_{ \Omega }c_{D}\geq 1$, we obtain

$$ \lVert{\sigma } \rVert _{H^{1}}\leq c_{\Omega} \bigl(1 + \bigl\lVert {S\bigl(u'_{2}\bigr)} \bigr\rVert _{V} \bigr) \bigl(1 + \bigl\lVert {u'_{1}} \bigr\rVert _{V} + \bigl\lVert {u'_{2}} \bigr\rVert _{V} \bigr) \bigl\lVert {u'_{1}-u'_{2}} \bigr\rVert _{V_{0}}.$$

(3.34)

Now, adding up inequalities (3.34) and (3.31), we obtain

$$\begin{aligned} & \lVert{v} \rVert _{H^{2}} + \lVert{\vartheta } \rVert _{H^{1}} + \lVert{b} \rVert _{H^{1}} + \lVert{\sigma } \rVert _{H^{1}} \\ &\quad \leq c_{\Omega}\bigl(1 + \bigl\lVert {S\bigl(u'_{2} \bigr)} \bigr\rVert _{V}^{2}\bigr) \bigl(1 + \bigl\lVert {u'_{1}} \bigr\rVert _{V}^{2} + \bigl\lVert {u'_{2}} \bigr\rVert _{V}^{2} \bigr) \bigl\lVert {u'_{1}-u'_{2}} \bigr\rVert _{V_{0}}, \end{aligned}$$

from which (see (3.26)) follows (3.17), and this completes the proof of Lemma 3.2. □

Remark 3.3

The $V_{0}$-continuous operator S (i.e., continuous in the norm of $V_{0}$) will be shown to satisfy the assumptions of the Schauder fixed point theorem. Following the work done in [7, 8], we will show that

$$ S\bigl(B^{a}_{V}\bigr)\subseteq B^{a}_{V},$$

(3.35)

where $B^{a}_{V} = \overline{B(0,a)}$ is the closed ball of V with radius a sufficiently small. Moreover, since the embedding $H^{k}(\Omega )\hookrightarrow H^{k-1}(\Omega )$ is compact, the closed ball $B^{a}_{V}$ is (see (3.4)) $V_{0}$-compact, and hence, the operator S has a fixed point $u=(\vartheta ,v,b,\sigma )\in B^{a}_{V}$. We will establish the crucial inclusion (3.35) in Sect. 5.

Lemma 3.4

Let $a > 0$ be sufficiently small. If the operator S has a fixed point in $B^{a}_{V}$ through (3.35), then the boundary value problem (3.5)–(2.14) has at least one solution u in V.

Proof

Indeed, if (3.35) is satisfied, then, according to Remark 3.3, the operator S has a fixed point $u\in B^{a}_{V}$. By the definition (see (3.16)) of the operator S, we get that $u\in V$ and satisfies the system of equations (2.8)–(2.11) with boundary conditions (2.12)–(2.14). □

Remark 3.5

If we set

$$ T: = \hat{T} + \vartheta ,\qquad B: = b + \hat{B},\qquad \varrho : = \hat{\varrho} + \sigma , $$

(see (2.6)), following Lemma 3.4, $(T,v,B,\varrho )\in H^{2}_{\natural}(\Omega )\times H^{3}_{\natural}( \Omega )\times H^{2}_{\natural}(\Omega )\times H^{2}_{\natural}( \Omega )$ and satisfies equations (1.1)–(1.4) with boundary conditions (1.6)–(1.8). Thus, our main result Theorem 2.1 is proved.

From Remark 3.3 and Lemma 3.4, we can then turn our focus to the crucial point (3.35) to be established. To this aim, we need adequate estimates of the solution $u = S(u')$ of the system of equations (3.5)–(3.8) with conditions (2.12)–(2.15) from which we will obtain (3.35) in Remark 3.3. These estimates will require a more elaborate treatment and hence will be discussed in the next technical section.

We will need some estimates of the nonlinear terms

$$ F'=F\bigl(u'\bigr),\qquad G'=G \bigl(u'\bigr)),\qquad H'=H\bigl(u' \bigr),\qquad v\sigma = v\bigl(u'\bigr)\sigma \bigl(u' \bigr), $$

which appear in equations (3.5)–(3.8).

Indeed, let $u' = (\vartheta ',v',b',\sigma ')\in B_{V}^{a}$. If a is small enough then

$$ \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {G'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2} \leq c_{\Omega }a^{3}.$$

(3.36)

In fact, considering (3.13) and (3.11), we have

$$\begin{aligned} & \lVert{v} \rVert _{H^{3}}^{2}\leq c_{\Omega} \bigl( \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {G'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} \bigr), \end{aligned}$$

(3.37)

$$\begin{aligned} & k \lVert{\sigma } \rVert _{H^{2}}^{2}\leq c_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2} + k \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2} + c_{\Omega} \lVert{v} \rVert _{H^{3}}^{2}. \end{aligned}$$

(3.38)

Since (see (3.21)–(3.23))

$$ \bigl\lVert {F'} \bigr\rVert _{H^{1}} + \bigl\lVert {G'} \bigr\rVert _{L^{2}} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}\leq c_{ \Omega} \bigl\lVert {u'} \bigr\rVert _{V}^{2} \bigl(1 + \bigl\lVert {u'} \bigr\rVert _{V}\bigr) + c_{\Omega} \bigl\lVert {u'} \bigr\rVert _{V} \lVert{\varepsilon } \rVert _{H^{2}}, $$

(3.39)

then for all $u' = (\vartheta ',v',b',\sigma ')\in B_{V}^{a}$, we have

$$ \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}} + \bigl\lVert {F'} \bigr\rVert _{H^{1}} + \bigl\lVert {G'} \bigr\rVert _{L^{2}} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}\leq c_{\Omega} \bigl[a + a^{2}(1 + a) + a \Vert \varepsilon \Vert _{H^{2}(\mathbb{T}^{2})}\bigr]\leq c_{a}.$$

(3.40)

If k is large enough so that

$$ k- c_{\Omega}\bigl( \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {G'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2}\bigr) \geq k- c_{a}>0, $$

then from (3.37) and (3.38) it follows that

$$ \lVert{\sigma } \rVert _{H^{2}}^{2}\leq c_{\Omega} \bigl( \bigl\lVert {\sigma '} \bigr\rVert _{H^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {G'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} \bigr),$$

(3.41)

whence, given (3.37), we get

$$ \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2}\leq c_{\Omega} \bigl( \bigl\lVert { \sigma '} \bigr\rVert _{H^{2}} + \bigl\lVert {F'} \bigr\rVert _{H^{1}} + \bigl\lVert {G'} \bigr\rVert _{L^{2}} + \bigl\lVert {H'} \bigr\rVert _{L^{2}} \bigr)^{3},$$

(3.42)

where (see (3.39))

$$ \bigl\lVert {F'} \bigr\rVert _{H^{1}}+ \bigl\lVert {G'} \bigr\rVert _{L^{2}} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}\leq c_{ \Omega} \bigl(a^{2}(1 + a) + a \Vert \varepsilon \Vert _{H^{2}(\mathbb{T}^{2})} \bigr).$$

(3.43)

From (3.43), (3.42), and (3.40) it follows

$$\begin{aligned} & \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {G'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2} \\ & \quad \leq c_{\Omega}\bigl(a^{4}\bigl(1 + a^{2} \bigr) + a^{2} \Vert \varepsilon \Vert _{H^{2}( \mathbb{T}^{2})}^{2} \bigr) + c_{\Omega}\bigl[a + a^{2}(1 + a) + a \Vert \varepsilon \Vert _{H^{2}(\mathbb{T}^{2})}\bigr]^{3}. \end{aligned}$$

By choosing (see (1.10)) $\varepsilon _{0}\leq a$, it is easy to see that for all $a\in \,]0,1[$, we have

$$ c_{\Omega}\bigl(a^{4}\bigl(1 + a^{2}\bigr) + a^{2} \Vert \varepsilon \Vert _{H^{2}(\mathbb{T}^{2})}^{2} \bigr) + c_{\Omega}\bigl[a + a^{2}(1 + a) + a \Vert \varepsilon \Vert _{H^{2}(\mathbb{T}^{2})}\bigr]^{3} \leq c_{\Omega }a^{3}, $$

and hence we obtain (3.36). □

4 Estimates of the solutions $(v,\vartheta ,b,\sigma )$

This section is devoted to the estimates of the solutions $u=(\vartheta ,v,b,\sigma )$ of equations (3.5)–(3.8) with boundary conditions (2.12)–(2.14). While the estimate of ϑ is obtained in (3.10), the estimates of v, b, and σ will be obtained in the following Lemmas 4.1–4.10 and are based largely on the ideas developed in the works [7, 8]. We recall here that these lemmas will be proven under assumptions (1.10). Regarding the positive constant k, which appears in equation (3.8), it can be chosen arbitrarily large as in Lemma 3.1. We set

$$ k := {\frac{\bar{\kappa}k_{1}}{2}}\quad \text{with } k_{1} = {\frac{R}{2\mu + \lambda}}\hat{M} \biggl(T_{0} - { \frac{\gamma -1}{\gamma R}}gh \biggr)^{\frac{\gamma}{\gamma -1}},$$

(4.1)

and κ̄ being a large positive constant satisfying in particular

$$ \bar{\kappa}\geq 16 \biggl(1 - {\frac{\gamma -1}{\gamma RT_{0}}}gh \biggr)^{-{ \frac{\gamma}{\gamma -1}}}.$$

(4.2)

In the following lemmas, we denote by $C'_{k}$ $(k = 1, \ldots 10)$ the constants that depend on Ω and B̂ but neither on $T_{0}$ nor on $M_{0}$ (provided that they are larger than some constant) and by $\tilde{C}_{k}$ $(k = 1, \ldots 9)$ the constants that depend on Ω, B̂, $T_{0}$, and $M_{0}$. In addition, in the proof of each lemma, if it is not necessary to specify them, one will indicate by $C_{\Omega}$ the constants that depend neither on $T_{0}$ nor on $M_{0}$ and by Ĉ those that depend on $T_{0}$ and/or $M_{0}$.

Lemma 4.1

Let $u'=(\vartheta ',v',b',\sigma ')\in V$ and $u=S(u') = (\vartheta ,v,b,\sigma )\in V$ be the solution of system (3.5)–(3.7) whose existence is guaranteed by Lemma 3.1. Under hypothesis (1.10) and the assumption that the constant k given in (4.1)₁ is large, we have

$$\begin{aligned} & {\frac{\mu}{R}} \lVert{\nabla v} \rVert _{L^{2}}^{2} + {\frac{\mu +\lambda}{R}} \lVert{\nabla \cdot v} \rVert _{L^{2}}^{2} + {\frac{\nu}{R}} \lVert{ \nabla b} \rVert _{L^{2}}^{2} + C'_{1}T_{0} \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2} + k \bigl\Vert \bigl(\hat{T} \hat{\varrho}^{-1} \bigr)^{\frac{1}{2}}\sigma \bigr\Vert _{L^{2}}^{2} \\ & \quad \leq k \bigl\Vert \bigl(\hat{T} \hat{\varrho}^{-1} \bigr)^{\frac{1}{2}}\sigma ' \bigr\Vert _{L^{2}}^{2} + C'_{1} \bigl( \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2}+ \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2}+ \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{L^{2}}^{2} \bigr) + \tilde{C}_{1} \lVert{\vartheta } \rVert _{H^{1}}^{2}. \end{aligned}$$

(4.3)

Proof

Let us first observe that

$$\begin{array}{rl}& -{\int }_{\mathrm{\Omega }}[v\mathrm{\nabla }\left(\hat{T}{\sigma }^{\prime }\right)+\frac{\hat{T}}{\hat{\varrho }}\sigma \mathrm{\nabla }\cdot (\hat{\varrho }v)]dx\\ & =-{\int }_{\mathrm{\Omega }}\frac{\hat{T}}{\hat{\varrho }}(\sigma -{\sigma }^{\prime })\mathrm{\nabla }\cdot (\hat{\varrho }v)dx-{\int }_{\mathrm{\Omega }}\hat{T}(v\cdot \mathrm{\nabla }log\hat{\varrho }){\sigma }^{\prime }dx,& \\ & {\int }_{\mathrm{\Omega }}[(\hat{B}\cdot \mathrm{\nabla })b-\mathrm{\nabla }(b\cdot \hat{B})]vdx+{\int }_{\mathrm{\Omega }}b\cdot ((\hat{B}\cdot \mathrm{\nabla })v-\hat{B}\mathrm{\nabla }\cdot v)dx=0.\end{array}$$

We multiply equations (3.6)–(3.8) by $R^{-1}v$, b, and $\hat{T} \hat{\varrho}^{-1}\sigma$ respectively and integrate the resulting equations over Ω. By using the above identities, we find after integration by parts that

$$\begin{array}{rll}& \frac{\mu }{R}{\parallel \mathrm{\nabla }v\parallel }_{L^2}^2+\frac{\mu +\lambda }{R}{\parallel \mathrm{\nabla }\cdot v\parallel }_{L^2}^2+\frac{\nu }{R}{\parallel \mathrm{\nabla }b\parallel }_{L^2}^2+\frac{k}{2}{\parallel {\left(\hat{T}{\hat{\varrho }}^{-1}\right)}^{\frac{1}{2}}(\sigma -{\sigma }^{\prime })\parallel }_{L^2}^2& \\ & +\frac{k}{2}{\parallel {\left(\hat{T}{\hat{\varrho }}^{-1}\right)}^{\frac{1}{2}}\sigma \parallel }_{L^2}^2=\frac{k}{2}{\parallel {\left(\hat{T}{\hat{\varrho }}^{-1}\right)}^{\frac{1}{2}}{\sigma }^{\prime }\parallel }_{L^2}^2+\sum _{i=1}^4{I}_i,& \end{array}$$

(4.4)

where

$$\begin{aligned} &I_{1}: = - \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}}\bigl( \sigma - \sigma '\bigr)\nabla \cdot (\hat{\varrho}v)\,dx,\qquad I_{2}; = {\frac{1}{R}} \int _{\Omega }\bigl(H'\cdot b+ F' \cdot v + R \hat{\varrho}\vartheta \nabla \cdot v\bigr)\,dx, \\ & I_{3} := - {\frac{g}{R}} \int _{\Omega }(e_{3}\cdot v) \sigma ' \,dx - \int _{\Omega}\bar{T}(v\cdot \nabla \log \hat{\varrho})\sigma ' \,dx, \qquad I_{4} := - \int _{ \Omega}{\frac{\hat{T}}{\hat{\varrho}}}\sigma \nabla (v\sigma ) \,dx. \end{aligned}$$

Recalling the expressions of T̂ and ϱ̂ in (2.1) and using (1.10), one can easily see that $\lVert{\nabla \log \hat{\varrho}} \rVert _{L^{\infty}}$ is small enough so that

$$ \lVert{v\cdot \nabla \log \hat{\varrho}} \rVert _{L^{2}}^{2} \leq {\frac{\mu}{2\mu +\lambda}} \lVert{\nabla v} \rVert _{L^{2}}^{2}.$$

(4.5)

Thus, by using (4.1) and (4.2), we obtain

$$\begin{aligned} &I_{1}\leq {\frac{k}{2}} \bigl\lVert { \bigl({\hat{T} \hat{\varrho}^{-1}} \bigr)^{\frac{1}{2}}\bigl(\sigma - \sigma '\bigr)} \bigr\rVert _{L^{2}}^{2} + { \frac{2}{\bar{\kappa}k_{1}}} \lVert{\hat{T}\hat{\varrho}} \rVert _{L^{\infty}}\bigl( \lVert{ \nabla \cdot v} \rVert _{L^{2}}^{2} + \lVert{v\cdot \nabla \log \hat{\varrho}\bigr)} \rVert _{L^{2}}^{2}) \\ &\hphantom{I_{1}} \leq {\frac{k}{2}} \bigl\lVert { \bigl({\hat{T}\hat{\varrho}^{-1}} \bigr)^{\frac{1}{2}}\bigl(\sigma - \sigma ' \bigr)} \bigr\rVert _{L^{2}}^{2} + {\frac{\mu +\lambda}{16R}} \lVert{\nabla \cdot v} \rVert _{L^{2}}^{2} + { \frac{\mu}{8R}} \lVert{ \nabla v} \rVert _{L^{2}}^{2}, \\ &I_{2} \leq {\frac{\nu}{4R}} \lVert{\nabla b} \rVert _{L^{2}}^{2} + {\frac{3}{16R}}(\mu +\lambda ) \lVert{\nabla \cdot v} \rVert _{L^{2}}^{2} + { \frac{\mu}{16R}} \lVert{\nabla v} \rVert _{L^{2}}^{2} \\ &\hphantom{I_{2}}\quad + C_{\Omega}\bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2}\bigr) + \hat{C} \lVert{ \vartheta } \rVert _{H^{1}}^{2}, \\ &I_{3} \leq {\frac{\mu}{16R}} \lVert{\nabla v} \rVert _{L^{2}}^{2} + C_{\Omega} \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2} . \end{aligned}$$

As for the last term $I_{4}$, we have

$$ I_{4} = -{\frac{1}{2}} \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}} \vert \sigma \vert ^{2}( \nabla \cdot v)\,dx + {\frac{1}{2}} \int _{\Omega } \vert \sigma \vert ^{2} v\cdot \nabla \biggl({\frac{\hat{T}}{\hat{\varrho}}} \biggr)\,dx\leq C_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{L^{2}}^{2}. $$

Combining these estimates with (4.4), we obtain

$$\begin{aligned} &{\frac{3\mu}{4R}} \lVert{\nabla v} \rVert _{L^{2}}^{2} + {\frac{3}{4R}}(\mu +\lambda ) \lVert{\nabla \cdot v} \rVert _{L^{2}}^{2} + { \frac{3\nu}{4R}} \lVert{\nabla b} \rVert _{L^{2}}^{2} + {\frac{k}{2}} \bigl\Vert \bigl({\hat{T}\hat{\varrho}^{-1}} \bigr)^{\frac{1}{2}}\bigl(\sigma - \sigma '\bigr) \bigr\Vert _{L^{2}}^{2} \\ &\quad + {\frac{k}{2}} \bigl\Vert \bigl({\hat{T}\hat{\varrho}^{-1}} \bigr)^{\frac{1}{2}}\sigma \bigr\Vert _{L^{2}}^{2} \leq {\frac{k}{2}} \bigl\Vert \bigl({\hat{T}\hat{\varrho}^{-1}} \bigr)^{\frac{1}{2}}\sigma ' \bigr\Vert _{L^{2}}^{2} + C_{\Omega} \bigl\lVert { \sigma '} \bigr\rVert _{L^{2}}^{2} \\ &\quad + \hat{C} \lVert{\vartheta } \rVert _{H^{1}}^{2} + C_{\Omega} \bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{L^{2}}^{2} \bigr). \end{aligned}$$

(4.6)

Next, let us establish the $L^{2}$-estimate of $\sigma '$, which appears in the right-hand side of (4.6).

To this aim, we introduce the following auxiliary problem:

$$ \left \{ \begin{gathered} \nabla \cdot \varphi = \sigma '\text{ in } \Omega =\mathbb{T}^{2} \times ]0,h[, \\ \varphi \bigl(x',0\bigr) = \varphi \bigl(x',h \bigr) = 0.\end{gathered} \right .$$

(4.7)

There exists (see [10]) at least one solution $\varphi \in H^{1}(\Omega )$ to problem (4.7) such that

$$ \lVert{ \varphi } \rVert _{H^{1}}\leq c_{\Omega} \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}.$$

(4.8)

Now, we rewrite equation (3.6) as

$$ - R\nabla \bigl(\hat{T}\sigma '\bigr) = - \mu \Delta v- (\mu + \lambda )\nabla \nabla \cdot v + R\nabla (\hat{\varrho}\vartheta ) + g\sigma 'e_{3} -( \hat{B}\cdot \nabla )b + \hat{B}\cdot \nabla b - F', $$

multiply it by $R^{-1}\varphi$, and integrate over Ω. Integrating by parts, taking into account (4.7), we obtain that

$$\begin{aligned} \int _{\Omega}\hat{T} \bigl\vert \sigma ' \bigr\vert ^{2}\,dx& = {\frac{g}{R}} \int _{\Omega}\sigma ' e_{3}\cdot \varphi \,dx \\ &\quad + \frac{1}{R} \int _{\Omega} \bigl[\mu \nabla v\cdot \nabla \varphi + (\mu + \lambda ) (\nabla \cdot v) ( \nabla \cdot \varphi ) + R\nabla (\hat{\varrho}\vartheta )\cdot \varphi - F'\cdot \varphi \bigr]\,dx \\ & \quad + \frac{1}{R} \int _{\Omega} \bigl[-( \hat{B}\cdot \nabla )b + \hat{B}\cdot \nabla b \bigr]\cdot \varphi \,dx . \end{aligned}$$

(4.9)

Considering (4.8), we obtain that

$$ T_{0} \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2} \leq C_{\Omega} \bigl[\mu \lVert{\nabla v} \rVert _{L^{2}}^{2} + (\mu + \lambda ) \lVert{\nabla \cdot v} \rVert _{L^{2}}^{2} + \nu \lVert{\nabla \cdot b} \rVert _{L^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} \bigr] + \hat{C} \lVert{ \vartheta } \rVert _{H^{1}}^{2}. $$

If we multiply now the inequality above by $(4RC_{\Omega})^{-1}$ and add it to (4.6), we obtain estimate (4.3), and this completes the proof the lemma. □

Lemma 4.2

Under the same hypotheses of Lemma 4.1, we have

$$\begin{aligned} & \sum_{i=1} ^{2} \biggl[{ \frac{\mu}{R}} \lVert{ \nabla \partial _{x_{i}} v} \rVert _{L^{2}}^{2} + {\frac{\mu + \lambda}{R}} \lVert{\nabla \cdot \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + { \frac{\nu}{R}} \lVert{\nabla \cdot \partial _{x_{i}}b} \rVert _{L^{2}}^{2} + k\big\| \bigl(\hat{T} \hat{\varrho}^{-1} \bigr)^{\frac{1}{2}}\partial _{x_{i}}\sigma \big\| _{L^{2}}^{2} \biggr] \\ &\quad \leq \sum_{i=1} ^{2}k\big\| \bigl( \hat{T} \hat{\varrho}^{-1}\bigr)^{\frac{1}{2}}\partial _{x_{i}}\sigma '\big\| _{L^{2}}^{2} + C'_{2} \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2} + \tilde{C}_{2} \lVert{ \vartheta } \rVert _{H^{1}}^{2} \\ & \qquad + C'_{2} \bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{1}}^{2} \bigr) . \end{aligned}$$

(4.10)

Proof

Let us notice that, since $\partial _{x_{i}}v$ and $\partial _{x_{i}}b$ ($i=1,2$) satisfy the same boundary conditions (2.12) and (2.14), we have

$$\begin{aligned} &- \int _{\Omega} \biggl[(\partial _{x_{i}} v)\nabla \cdot \bigl(\partial _{x_{i}}\bigl(\hat{T}\sigma '\bigr)\bigr) + {\frac{\hat{T}}{\hat{\varrho}}} \partial _{x_{i}}\nabla \cdot (\hat{\varrho}v) \partial _{x_{i}}\sigma \biggr]\,dx \\ &\quad = - \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}}\bigl(\partial _{x_{i}} \nabla \cdot (\hat{\varrho}v)\bigr) \bigl(\partial _{x_{i}}\sigma - \partial _{x_{i}} \sigma ' \bigr)\,dx + \int _{\Omega}(\partial _{x_{i}}\hat{T}) \sigma '\partial _{x_{i}}\nabla \cdot v\,dx \\ &\qquad - \int _{\Omega}\hat{T} \biggl[(\partial _{x_{i}} \log \hat{\varrho})\nabla \cdot v + {\frac{1}{\hat{\varrho}}}\partial _{x_{i}}(v \cdot \nabla \hat{\varrho}) \biggr]\partial _{x_{i}}\sigma ' \,dx \\ & \int _{\Omega} \bigl[(\hat{B}\cdot \nabla )\partial _{x_{i}}b - \nabla (\partial _{x_{i}}b\cdot \hat{B}) \bigr] \partial _{x_{i}}v \,dx + \int _{\Omega }\partial _{x_{i}}b\cdot \bigl((\hat{B} \cdot \nabla )\partial _{x_{i}}v - \hat{B}\nabla \cdot \partial _{x_{i}}v\bigr) \,dx=0. \end{aligned}$$

Now, we differentiate equations (3.6)–(3.8) by applying on both sides $\partial _{x_{i}}$ $(i=1,2)$ and multiply them by $R^{-1}\partial _{x_{i}}v$, $R^{-1}\partial _{x_{i}}b$, and $\hat{T}\hat{\varrho}^{-1}\partial _{x_{i}}\sigma$ respectively. We integrate over Ω and apply the integration by parts to obtain

$$\begin{array}{rl}& \frac{\mu }{R}{\parallel \mathrm{\nabla }{\partial }_{x_i}v\parallel }_{L^2}^2+\frac{\mu +\lambda }{R}{\parallel \mathrm{\nabla }\cdot {\partial }_{x_i}v\parallel }_{L^2}^2+\frac{\nu }{R}{\parallel \mathrm{\nabla }{\partial }_{x_i}b\parallel }_{L^2}^2+\frac{k}{2}{\parallel {\left(\hat{T}{\hat{\varrho }}^{-1}\right)}^{\frac{1}{2}}{\partial }_{x_i}\sigma \parallel }_{L^2}^2\\ & +\frac{k}{2}{\parallel {\left(\hat{T}{\hat{\varrho }}^{-1}\right)}^{\frac{1}{2}}({\partial }_{x_i}\sigma -{\partial }_{x_i}{\sigma }^{\prime })\parallel }_{L^2}^2=\frac{k}{2}{\parallel {\left(\hat{T}{\hat{\varrho }}^{-1}\right)}^{\frac{1}{2}}{\partial }_{x_i}{\sigma }^{\prime }\parallel }_{L^2}^2+\sum _{i=1}^4{I}_i,& \end{array}$$

(4.11)

where

$$\begin{aligned} &I_{1}: = - \int _{\Omega}{\frac{\bar{T}}{\hat{\varrho}}} \bigl(\partial _{x_{i}}\sigma - \partial _{x_{i}}\sigma ' \bigr) \partial _{x_{i}}\nabla \cdot (\hat{\varrho}v)\,dx, \\ &I_{2} := \int _{\Omega} \bigl[(\partial _{x_{i}}\nabla \cdot v) \partial _{x_{i}}(\hat{\varrho}\vartheta ) - R^{-1} \bigl(F'\cdot \partial _{x_{i}}\partial _{x_{i}}v - H'\cdot \partial _{x_{i}} \partial _{x_{i}}b \bigr) \bigr]\,dx, \\ &I_{3}: = - \int _{\Omega}\sigma ' \bigl[\partial _{x_{i}} \bigl(\hat{T}(\partial _{x_{i}}\log \hat{\varrho}) \nabla \cdot v + \hat{\varrho}^{-1}\partial _{x_{i}}(v\cdot \nabla \hat{\varrho}) \bigr) + (\partial _{x_{i}}\hat{T})\partial _{x_{i}}\nabla \cdot v \bigr]\,dx, \\ &I_{4} = : {\frac{g}{R}} \int _{\Omega} \sigma ' \partial _{x_{i}} \partial _{x_{i}}v_{3}\,dx,\qquad I_{5}: = - \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}}(\partial _{x_{i}} \sigma )\partial _{x_{i}}\nabla \cdot (\sigma v)\,dx. \end{aligned}$$

Now, from the arguments similar to the ones used in the proof of Lemma 4.1 (particularly for the term $I_{1}$), we get

$$\begin{aligned} &I_{1} \leq {\frac{k}{2}} \bigl\lVert { \bigl({\hat{T} \hat{\varrho}^{-1}} \bigr)^{\frac{1}{2}}(\partial _{x_{i}} \sigma - \partial _{x_{i}}\sigma '} \bigr\rVert _{L^{2}}^{2} + { \frac{\mu +\lambda}{2R}} \bigl\lVert {\nabla \cdot ( \partial _{x_{i}}v)} \bigr\rVert _{L^{2}}^{2} + {\frac{\mu}{6R}} \lVert{ \nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2}, \\ &I_{2}\leq {\frac{\mu}{6R}} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + { \frac{\nu}{2R}} \lVert{ \nabla \partial _{x_{i}}b} \rVert _{L^{2}}^{2} + C_{\Omega}\bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2}\bigr) + \hat{C} \lVert{\vartheta } \rVert _{H^{1}}^{2}, \\ &I_{3} + I_{4} \leq {\frac{\mu}{6R}} \lVert{ \nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + C_{\Omega} \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2}. \end{aligned}$$

As for the term $I_{5}$, we have

$$\begin{aligned} I_{5} &= - {\frac{1}{2}} \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}} \vert \partial _{x_{i}}\sigma \vert ^{2}(\nabla \cdot v)\,dx + { \frac{1}{2}} \int _{\Omega } \vert \partial _{x_{i}}\sigma \vert ^{2} \bigl(v\cdot \nabla \bigl(\hat{T}\hat{\varrho}^{-1} \bigr) \bigr)\,dx \\ &\quad - \int _{\Omega }\bar{T}\hat{\varrho}^{-1}( \partial _{x_{i}}\sigma )\nabla \cdot (\sigma \partial _{x_{i}} v)\,dx \leq C_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{H^{1}}^{2}. \end{aligned}$$

(4.12)

By adding these estimates to (4.11) and summing on $i=1,2$, we obtain (4.10). □

Lemma 4.3

Under the same hypotheses of Lemma 4.1, we have

$$\begin{aligned} (\bar{\kappa}+ 1)C_{0}T_{0}^{2} \lVert{ \partial _{x_{3}}\sigma } \rVert _{L^{2}}^{2}& \leq ( \bar{\kappa}- 1)C_{0}T_{0}^{2} \bigl\lVert { \partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} \\ &\quad + C'_{3} \Biggl(\sum _{i=1}^{2} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + \lVert{\nabla v} \rVert _{L^{2}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}+ \lVert{\nabla b} \rVert _{L^{2}}^{2} \Biggr) \\ &\quad + \tilde{C}_{3}\bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \lVert{\nabla \vartheta } \rVert _{H^{1}}^{2}\bigr) + C'_{3} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{1}}^{2}, \end{aligned}$$

(4.13)

where κ̄ is given by (4.2) and

$$ C_{0} := {\frac{R}{2\mu + \lambda}} \biggl(1 - { \frac{\gamma -1}{\gamma RT_{0}}}gh \biggr)^{\frac{\gamma}{\gamma -1}}.$$

(4.14)

Proof

Using the identity

$$ \Delta v_{3} = \partial _{x_{3}}\nabla \cdot v + \partial _{x_{1}}( \partial _{x_{1}}v_{3} - \partial _{x_{3}}v_{1}) + \partial _{x_{2}}( \partial _{x_{2}}v_{3} - \partial _{x_{3}}v_{2}), $$

it follows from equation (3.7) that

$$\begin{aligned} \partial _{x_{3}}\nabla \cdot v &= -{\frac{\mu}{2\mu + \lambda}} \bigl[ \partial _{x_{1}}(\partial _{x_{1}}v_{3} - \partial _{x_{3}}v_{1}) + \partial _{x_{2}}(\partial _{x_{2}}v_{3} - \partial _{x_{3}}v_{2}) \bigr] \\ &\quad + {\frac{1}{2\mu +\lambda}} \bigl[R\partial _{x_{3}}( \bar{\varrho}\vartheta ) + R\hat{T}\partial _{x_{3}}\sigma ' + ( R \partial _{x_{3}}\hat{T} + g)\sigma ' + \partial _{x_{3}}(b\cdot \hat{B}) - (\hat{B}\cdot \nabla )b_{3} - F'_{3} \bigr]. \end{aligned}$$

(4.15)

Next we differentiate equation (3.8) by applying $\partial _{x_{3}}$, multiply the equation by $\partial _{x_{3}}\sigma$, and integrate over Ω. Integrating by parts, taking into account (4.15), we get

$$ \int _{\Omega }k \biggl[\bigl(\partial _{x_{3}}\sigma - \partial _{x_{3}}\sigma '\bigr)\partial _{x_{3}} \sigma + {\frac{R}{2\mu + \lambda}}\hat{\varrho}\hat{T}(\partial _{x_{3}} \sigma ) \bigl(\partial _{x_{3}} \sigma '\bigr) \biggr] \,dx = \sum_{i=1}^{6}I_{i},$$

(4.16)

where

$$\begin{aligned} &I_{1}: = {\frac{-1}{2\mu +\lambda}} \int _{\Omega} \hat{\varrho}\bigl(R\sigma ' \partial _{x_{3}}\hat{T} + g\sigma '\bigr)\partial _{x_{3}} \sigma \,dx, \\ &I_{2} := {\frac{1}{2\mu +\lambda}} \int _{\Omega} \hat{\varrho}\bigl(F'_{3} - R\partial _{x_{3}}(\hat{\varrho}\vartheta )\bigr) \partial _{x_{3}}\sigma \,dx, \\ &I_{3} := {\frac{\mu}{2\mu +\lambda}} \int _{\Omega} \hat{\varrho}(\partial _{x_{3}}\sigma ) \bigl(\partial _{x_{1}} (\partial _{x_{1}}v_{3} - \partial _{x_{3}}v_{1}) + \partial _{x_{2}}(\partial _{x_{2}}v_{3} - \partial _{x_{3}}v_{2}) \bigr)\,dx, \\ &I_{4} := - \int _{\Omega}\bigl(\partial _{x_{3}}(v\cdot \nabla \hat{\varrho}) + (\partial _{x_{3}}\hat{\varrho})\nabla \cdot v\bigr) \partial _{x_{3}}\sigma \,dx, \\ &I_{5} := \int _{\Omega}\hat{\varrho}\bigl((\hat{B}\cdot \nabla )b_{3} - \hat{B}\cdot \partial _{x_{3}}b\bigr)\partial _{x_{3}} \sigma \,dx, \\ &I_{6} :=- \int _{\Omega}(\partial _{x_{3}}\sigma ) \nabla \cdot \bigl(\partial _{x_{3}}(v\sigma )\bigr) \,dx. \end{aligned}$$

Using the following identity in $\mathbb{R}$

$$ (X-Y)X + a XY = {\frac{1 + a}{2}} X^{2} + { \frac{1 - a}{2}}(X - Y)^{2} - {\frac{1 - a}{2}} Y^{2} $$

and taking into account expressions (2.1) of ϱ̂ and T̂, one obtains for $T_{0}$ large enough that

$$\begin{array}{rll}& {\int }_{\mathrm{\Omega }}k[({\partial }_{x_3}\sigma -{\partial }_{x_3}{\sigma }^{\prime }){\partial }_{x_3}\sigma +\frac{R}{2\mu +\lambda }\hat{\varrho }\hat{T}({\partial }_{x_3}\sigma )\left({\partial }_{x_3}{\sigma }^{\prime }\right)]dx& \\ & \ge \frac{k+k_1}{2}{\parallel {\partial }_{x_3}\sigma \parallel }_{L^2}^2-\frac{k-k_1}{2}{\parallel {\partial }_{x_3}{\sigma }^{\prime }\parallel }_{L^2}^2+\frac{k-k_1^{\prime }}{2}{\parallel {\partial }_{x_3}\sigma -{\partial }_{x_3}{\sigma }^{\prime }\parallel }_{L^2}^2,\end{array}$$

where $k_{1}$ is the constant given in (4.1) and

$$ k'_{1} := {\frac{M_{0} R}{2\mu + \lambda}}T_{0}^{ \frac{\gamma}{\gamma -1}}. $$

Moreover, from expressions (2.1) (see also (1.11)) of ϱ̂ and T̂, it follows that

$$\begin{aligned} &I_{1}\leq {\frac{k - k'_{1}}{6}} \bigl\lVert {\partial _{x_{3}} \sigma - \partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + {\frac{k_{1}}{12}} \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + C_{\Omega }M_{0}T_{0}^{{\frac{2-\gamma}{\gamma -1}}} \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}, \\ &I_{2} \leq \hat{C}\bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \lVert{\vartheta } \rVert _{H^{1}}^{2}\bigr) + {\frac{k_{1}}{8}} \lVert{\partial _{x_{3}}\sigma } \rVert _{L^{2}}^{2}, \\ &I_{3} \leq {\frac{k - k'_{1}}{6}} \bigl\lVert {\partial _{x_{3}} \sigma - \partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + {\frac{k_{1}}{12}} \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + C_{\Omega }M_{0}T_{0}^{{\frac{2-\gamma}{\gamma -1}}} \sum_{i=1}^{2} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2}, \\ &I_{4} \leq {\frac{k - k'_{1}}{6}} \bigl\lVert {\partial _{x_{3}} \sigma - \partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + {\frac{k_{1}}{12}} \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + C_{\Omega }M_{0}T_{0}^{{\frac{4-3\gamma}{\gamma -1}}} \lVert{v} \rVert _{L^{2}}^{2}, \\ &I_{5} \leq C_{\Omega }M_{0}T_{0}^{{\frac{\gamma -2}{\gamma -1}}} \lVert{\nabla b} \rVert _{L^{2}}^{2} + { \frac{k_{1}}{8}} \lVert{\partial _{x_{3}}\sigma } \rVert _{L^{2}}^{2}. \end{aligned}$$

As for the last $I_{6}$ term, one has

$$\begin{aligned} I_{6} &= -{\frac{1}{2}} \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}} \vert \partial _{x_{3}}\sigma \vert ^{2}(\nabla \cdot v)\,dx - \int _{\Omega }(\partial _{x_{3}}\sigma ) (\nabla \sigma \cdot v)\,dx + \int _{\Omega }(\partial _{x_{3}} \sigma ) \bigl(\nabla \cdot (\partial _{x_{3}}v)\bigr)\sigma\,dx \\ & \leq C_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{H^{1}}^{2} . \end{aligned}$$

Combining these estimates with (4.16), we obtain

$$\begin{aligned} \biggl(k +{\frac{k_{1}}{2}} \biggr) \lVert{\partial _{x_{3}} \sigma } \rVert _{L^{2}}^{2}&\leq \biggl(k - { \frac{k_{1}}{2}} \biggr) \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + C_{\Omega }M_{0} T_{0}^{{\frac{2-\gamma}{\gamma -1}}} \bigl( \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}+ \lVert{ \nabla b} \rVert _{L^{2}}^{2} \bigr) \\ &\quad + M_{0}C_{\Omega }T_{0}^{{\frac{2-\gamma}{\gamma -1}}} \sum _{i=1}^{2} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + C_{ \Omega }M_{0}T_{0}^{{\frac{4-3\gamma}{\gamma -1}}} \lVert{v} \rVert _{L^{2}}^{2} \\ &\quad + \hat{C}\bigl( \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \lVert{\vartheta } \rVert _{H^{1}}^{2}\bigr) + C_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{1}}^{2}. \end{aligned}$$

By multiplying now both sides of this inequality by

$$ \bigl(M_{0}T_{0}^{\frac{2-\gamma}{\gamma -1}} \bigr)^{- 1} = T_{0}^{2} \bigl(M_{0} T_{0}^{\frac{\gamma}{\gamma -1}} \bigr)^{- 1} $$

and taking into account (4.1), we obtain (4.13), which completes the proof of the lemma. □

Lemma 4.4

Under the assumptions of Lemma 4.1, we have

$$\begin{aligned} & \lVert{v} \rVert _{H^{2}}^{2} + T_{0}^{2} \bigl\lVert { \nabla \sigma '} \bigr\rVert _{L^{2}}^{2} \\ & \quad \leq C'_{4}\sum_{i=1}^{2} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + C'_{4} \bigl(T_{0}^{2} \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + \lVert{\nabla b} \rVert _{l^{2}}^{2} + \lVert{ \nabla v} \rVert _{L^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} \bigr). \end{aligned}$$

(4.17)

Proof

We first rewrite equation (3.6) as the following Stokes problem in Ω:

$$\begin{aligned} &{-}\mu \Delta v + R\nabla \bigl(\hat{T}\sigma '\bigr) = - (\mu + \lambda )\nabla \nabla \cdot v - R\nabla (\hat{\varrho}\vartheta ) - g\sigma ' e_{3} + ( \hat{B}\cdot \nabla )b - \nabla (b\cdot \hat{B}) +F', \\ &\nabla \cdot v = \nabla \cdot v, \end{aligned}$$

with boundary conditions (2.12). From the classical estimates (see [11] and [29]) of the solutions of the Stokes problem, one has

$$ \lVert{v} \rVert _{H^{2}}^{2} + \bigl\Vert \nabla \bigl( \hat{T}\sigma '\bigr) \bigr\Vert _{L^{2}}^{2} \leq \hat{C} \lVert{\vartheta } \rVert _{H^{1}}^{2} + C_{\Omega} \bigl( \lVert{\nabla \cdot v} \rVert _{H^{1}}^{2} + \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} + \lVert{\nabla b} \rVert _{L^{2}}^{2} \bigr). $$

(4.18)

Moreover, using (4.15), one can see easily that

$$\begin{aligned} & \lVert{\partial _{x_{3}}\nabla \cdot v} \rVert _{L^{2}}^{2} \\ & \quad \leq C_{\Omega} \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} + C_{\Omega} \Biggl(\sum _{i=1}^{2} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} + \lVert{\nabla b} \rVert _{L^{2}}^{2} + T_{0}^{2} \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \lVert{ \vartheta } \rVert _{H^{1}}^{2} \Biggr), \end{aligned}$$

and since σ (see (2.15)) has mean value equal to zero and $T_{0}$ is large enough, we get

$$ \bigl\lVert {\nabla \bigl(\bar{T}\sigma '\bigr)} \bigr\rVert _{L^{2}}^{2}\geq C_{ \Omega} \bigl(T_{0}^{2} \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} - \bigl\lVert {\sigma '} \bigr\rVert _{L^{2}}^{2} \bigr)\leq C'_{ \Omega }T_{0}^{2} \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}. $$

Taking into account these inequalities, it results that estimate (4.17) follows from (4.18), and this completes the proof of the lemma. □

Lemma 4.5

With the same assumptions as in Lemma 4.1, we have

$$\begin{aligned} \lVert{b} \rVert _{H^{2}}^{2}\leq C'_{5}\bigl( \lVert{ \nabla v} \rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2}\bigr). \end{aligned}$$

(4.19)

Proof

Since $\lVert{b} \rVert _{H^{2}}\leq C_{\Omega} \lVert{ \Delta b} \rVert _{L^{2}}$ for all $b\in H^{2}_{\natural}(\Omega )\cap H_{0}^{1}(\Omega )$, from equation (3.7) and boundary condition (2.14), we easily get (4.19). □

Lemma 4.6

Under the same assumptions as in Lemma 4.1, we have

$$\begin{aligned} & \sum_{i,j=1}^{2} \biggl({\frac{\mu}{R}} \lVert{ \nabla \partial _{x_{i}}\partial _{x_{j}} v} \rVert _{L^{2}}^{2} + { \frac{\mu +\lambda}{R}} \lVert{\partial _{x_{i}}\partial _{x_{j}} \nabla \cdot v} \rVert _{L^{2}}^{2} \biggr) + k\sum _{i,j=1}^{2} \bigl\Vert \bigl(\hat{T}\hat{\varrho}^{-1}\bigr)^{\frac{1}{2}}\partial _{x_{i}} \partial _{x_{j}}\sigma \bigr\Vert _{L^{2}}^{2} \\ & \quad \leq k\sum_{i,j=1}^{2} \bigl\Vert \bigl(\hat{T} \hat{\varrho}^{-1}\bigr)^{\frac{1}{2}} \partial _{x_{i}}\partial _{x_{j}} \sigma ' \bigr\Vert _{L^{2}}^{2}+ C'_{6}\bigl( \lVert{v} \rVert _{H^{2}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} \bigr)+ \tilde{C}_{5} \lVert{ \vartheta } \rVert _{H^{2}}^{2} \\ & \qquad + C'_{6} \bigl( \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2} \bigr). \end{aligned}$$

(4.20)

Proof

We apply the differential operator $\partial _{x_{i}}\partial _{x_{j}}$ $(i, j=1,2)$ to equations (3.6) and (3.8), and we multiply them by $R^{ -1}\partial _{x_{i}}\partial _{x_{j}}v$ and $\hat{T}\hat{\varrho}^{-1}\partial _{x_{i}}\partial _{x_{j}}\sigma$ respectively, then we integrate the resulting equations on Ω. By integrating them by parts, using the fact that $\partial _{x_{i}}\partial _{x_{j}}v$ satisfies (2.12), we obtain

$$\begin{aligned} &{\frac{\mu}{R}} \lVert{\nabla \partial _{x_{i}} \partial _{x_{j}} v} \rVert _{L^{2}}^{2} + { \frac{\mu + \lambda}{R}} \lVert{\partial _{x_{i}}\partial _{x_{j}} \nabla \cdot v} \rVert _{L^{2}}^{2} + {\frac{k}{2}} \biggl\Vert \biggl({ \frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\partial _{x_{i}} \partial _{x_{j}} \sigma \biggr\Vert _{L^{2}}^{2} \\ & \quad + {\frac{k}{2}} \biggl\Vert \biggl({\frac{\bar{T}}{\bar{\varrho}}} \biggr)^{\frac{1}{2}}(\partial _{x_{i}}\partial _{x_{j}} \sigma -\partial _{x_{i}} \partial _{x_{j}}\sigma ) \biggr\Vert _{L^{2}}^{2} = {\frac{k}{2}} \biggl\Vert \biggl({ \frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\partial _{x_{i}} \partial _{x_{j}} \sigma ' \biggr\Vert _{L^{2}}^{2} + \sum_{i=1}^{8}I_{i}, \end{aligned}$$

(4.21)

where

$$\begin{aligned} & I_{1}: = - \int _{\Omega}{\frac{\hat{T}}{\hat{\varrho}}}\bigl( \partial _{x_{i}}\partial _{x_{j}}\sigma - \partial _{x_{i}} \partial _{x_{j}} \sigma '\bigr)\partial _{x_{j}} \partial _{x_{i}}\nabla \cdot ( \hat{\varrho}v)\,dx, \\ & I_{2}: = - \int _{\Omega}\bigl[(\partial _{x_{i}}\hat{T}) \partial _{x_{j}}\sigma ' + (\partial _{x_{j}}\hat{T}) \partial _{x_{i}} \sigma ' + \sigma ' \partial _{x_{j}} \partial _{x_{i}}\hat{T}\bigr] \partial _{x_{j}} \partial _{x_{i}}\nabla \cdot v \,dx, \\ & I_{3} := - \int _{\Omega}\partial _{x_{i}} \bigl[\hat{T} \hat{\varrho}^{-1} \bigl(\partial _{x_{i}}\partial _{x_{j}}(v \cdot \nabla \hat{\varrho}) + (\partial _{x_{i}}\partial _{x_{j}} \hat{\varrho})\nabla \cdot v \bigr) \bigr]\partial _{x_{j}}\sigma ' \,dx, \\ & I_{4}: =- \int _{\Omega}\partial _{x_{i}} \bigl[\hat{T} \hat{\varrho}^{-1} \bigl( (\partial _{x_{i}}\hat{\varrho})\partial _{x_{j}} \nabla \cdot v + (\partial _{x_{j}}\hat{\varrho}) \partial _{x_{i}}\nabla \cdot v \bigr) \bigr]\partial _{x_{j}} \sigma ' \,dx, \\ & I_{5}: = -{\frac{g}{R}} \int _{\Omega}\bigl( \partial _{x_{i}}\sigma '\bigr)\partial _{x_{i}}^{2} \partial _{x_{j}}v_{3} \,dx, \\ & I_{6}: = \int _{\Omega} \bigl(-(\hat{B}\cdot \nabla ) \partial _{x_{j}} \partial _{x_{i}}v + (\hat{B}\cdot \nabla \partial _{x_{j}} \partial _{x_{i}}v) \bigr)\cdot \partial _{x_{i}}\partial _{x_{j}}b \,dx, \\ & I_{7}: = \int _{\Omega} \bigl[(\partial _{x_{j}} \partial _{x_{i}}\nabla \cdot v)\partial _{x_{i}}\partial _{x_{j}}( \hat{\varrho}\vartheta ) - R^{-1}\bigl(\partial _{x_{i}}^{2} \partial _{x_{j}}v\bigr) \cdot \partial _{x_{j}}F' \bigr]\,dx, \\ & I_{8}: = - \int _{\Omega }{\frac{\hat{T}}{\hat{\varrho}}}( \partial _{x_{i}} \partial _{x_{j}}\sigma ) \partial _{x_{i}}\partial _{x_{j}} \nabla \cdot (v\sigma )\,dx. \end{aligned}$$

By the same arguments as in Lemma 4.2, we can estimate the terms $I_{i}$ ($i=1,\ldots ,7$) so that

$$\begin{aligned} I_{1}+\cdots +I_{7}&\leq {\frac{k}{2}}\bigg\| \biggl({\frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}(\partial _{x_{i}} \partial _{x_{j}} \sigma -\partial _{x_{i}}\partial _{x_{j}}\sigma '\bigg\| _{L^{2}}^{2} + { \frac{\mu}{2R}} \lVert{\nabla \partial _{x_{i}} \partial _{x_{j}}v} \rVert _{L^{2}}^{2} \\ &\quad +{\frac{\mu +\lambda}{2R}} \lVert{\partial _{x_{i}} \partial _{x_{j}}\nabla \cdot v} \rVert _{L^{2}}^{2} + C_{ \Omega}\bigl( \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} + \lVert{v} \rVert _{H^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2}\bigr)\\ &\quad + \hat{C} \lVert{\vartheta } \rVert _{H^{2}}^{2}. \end{aligned}$$

We estimate the term $I_{8}$ in a similar way as in (4.12), to get

$$ I_{8}\leq C_{\Omega} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2}. $$

By adding the above estimates to (4.21), we obtain (4.20), and the proof of the lemma is completed. □

Lemma 4.7

Under the assumptions of Lemma 4.1, we have

$$\begin{aligned} (\bar{\kappa}+1)C_{0}T_{0}^{2} \sum_{i=1}^{2} \lVert{\partial _{x_{3}}\partial _{x_{i}}\sigma } \rVert _{L^{2}}^{2} &\leq (\bar{\kappa}- 1)C_{0}T_{0}^{2}\sum _{i=1}^{2} \bigl\lVert {\partial _{x_{3}}\partial _{x_{i}}\sigma '} \bigr\rVert _{L^{2}}^{2} \\ & \quad + C'_{7}\sum_{j=1}^{2} \lVert{\nabla \partial _{x_{i}} \partial _{x_{j}} v} \rVert _{L^{2}}^{2} \\ &\quad + C'_{7} \bigl(T_{0}^{-2} \lVert{v} \rVert _{H^{2}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}+ \lVert{b} \rVert _{H^{2}}^{2}\bigr) \\ &\quad + \tilde{C}_{6} \bigl( \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \lVert{\vartheta } \rVert _{H^{2}}^{2} \bigr) + C'_{7} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2}, \end{aligned}$$

(4.22)

where κ̄ and $C_{0}$ are given by (4.2) and (4.14) respectively.

Proof

We differentiate equation (3.8) by applying the operator $\partial _{x_{3}}\partial _{x_{i}}$, $(i=1,2)$ on both sides of the equation and multiply the resulting equation by $\partial _{x_{3}}\partial _{x_{i}}\sigma$. We then obtain (4.22) by using the same arguments of Lemma 4.3. □

Lemma 4.8

Under the assumptions of Lemma 4.1, one has

$$\begin{array}{rll}\sum _{i=1}^2({\parallel {\partial }_{x_i}v\parallel }_{H^2}^2+T_0^2{\parallel \mathrm{\nabla }{\partial }_{x_i}{\sigma }^{\prime }\parallel }_{L^2}^2)& \le C_8^{\prime }\sum _{i,j=1}^2{\parallel \mathrm{\nabla }{\partial }_{x_i}{\partial }_{x_j}v\parallel }_{L^2}^2+{\tilde{C}}_7{\parallel \vartheta \parallel }_{H^2}^2& \\ & +C_8^{\prime }\sum _{i=1}^2{T}_0^2{\parallel {\partial }_{x_3}{\partial }_{x_i}{\sigma }^{\prime }\parallel }_{L^2}^2\\ & +C_8^{\prime }({\parallel v\parallel }_{H^2}^2+{\parallel b\parallel }_{H^2}^2+{\parallel F^{\prime }\parallel }_{H^1}^2+{\parallel \mathrm{\nabla }{\sigma }^{\prime }\parallel }_{L^2}^2).& \end{array}$$

(4.23)

Proof

We apply to equation (3.6) the differential operator $\partial _{x_{i}}$ $(i = 1,2)$, and we rewrite the obtained equation as the following Stokes problem:

$$\begin{array}{rl}& -\mu \mathrm{\Delta }({\partial }_{x_i}v)+R\mathrm{\nabla }\left({\partial }_{x_i}\left(\hat{T}{\sigma }^{\prime }\right)\right)\\ & =-(\mu +\lambda ){\partial }_{x_i}\mathrm{\nabla }\mathrm{\nabla }\cdot v-R\mathrm{\nabla }{\partial }_{x_i}(\hat{\varrho }\vartheta )-g{\partial }_{x_i}{\sigma }^{\prime }{e}_3+{\partial }_{x_i}((\hat{B}\cdot \mathrm{\nabla })b-\mathrm{\nabla }(\hat{B}\cdot b))+{\partial }_{x_i}{F}^{\prime },& \\ & \mathrm{\nabla }\cdot {\partial }_{x_i}v=\mathrm{\nabla }\cdot {\partial }_{x_i}v,\end{array}$$

with the same boundary conditions on $\partial _{x_{i}}v$ as (2.12). We obtain by the same arguments of the proof of Lemma 4.4 that

$$\begin{aligned} & \lVert{\partial _{x_{i}}v} \rVert _{H^{2}}^{2} + \bigl\Vert \nabla \partial _{x_{i}}\bigl(\hat{T}\sigma '\bigr) \bigr\Vert _{L^{2}}^{2} \\ &\quad \leq C_{\Omega} \bigl[ \lVert{\nabla \cdot \partial _{x_{i}} v} \rVert _{H^{1}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} \bigr] + \hat{C} \lVert{ \vartheta } \rVert _{H^{2}}^{2}. \end{aligned}$$

(4.24)

In addition, let us notice that

$$ \lVert{\partial _{x_{i}}\nabla \cdot v} \rVert _{H^{1}}^{2} \leq C_{\Omega} \Biggl[\sum _{j=1}^{2} \lVert{\partial _{x_{i}} \partial _{x_{j}}\nabla \cdot v} \rVert _{L^{2}}^{2} + \lVert{\partial _{x_{i}}\partial _{x_{3}}\nabla \cdot v} \rVert _{L^{2}}^{2} + \lVert{v} \rVert _{H^{2}}^{2} \Biggr]$$

(4.25)

and, in applying $\partial _{x_{i}}$ $(i=1,2)$ to (4.15), it follows that

$$\begin{aligned} \lVert{\partial _{x_{i}}\partial _{x_{3}}\nabla \cdot v} \rVert _{L^{2}}^{2} &\leq \hat{C} \lVert{\vartheta } \rVert _{H^{2}}^{2} + C_{\Omega} \Biggl[ \lVert{b} \rVert _{H^{2}}^{2} \\ &\quad + \sum_{j=1}^{2} \lVert{\nabla \partial _{x_{i}}\partial _{x_{j}}v} \rVert _{L^{2}}^{2} + T_{0}^{2} \bigl\lVert {\partial _{x_{3}} \partial _{x_{i}}\sigma '} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} \Biggr]. \end{aligned}$$

(4.26)

By substituting (4.25) and (4.26) into (4.24) and taking into account

$$\begin{aligned} \bigl\lVert {\nabla \partial _{x_{i}}\bigl(\hat{T}\sigma '\bigr)} \bigr\rVert _{L^{2}}^{2}\geq C_{\Omega} \bigl[T_{0}^{2} \bigl\lVert {\nabla \partial _{x_{i}}\sigma '} \bigr\rVert _{L^{2}}^{2} - \bigl\lVert { \sigma '} \bigr\rVert _{H^{1}}^{2} \bigr]\geq C_{\Omega} \bigl[T_{0}^{2} \bigl\lVert {\nabla \partial _{x_{i}}\sigma '} \bigr\rVert _{L^{2}}^{2} - \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} \bigr], \end{aligned}$$

we obtain (4.23), which completes the proof of the lemma. □

Lemma 4.9

Under the assumptions of Lemma 4.1, one has

$$\begin{aligned} (\bar{\kappa}+ 1)C_{0}T_{0}^{2} \lVert{\Delta \sigma } \rVert _{L^{2}}^{2} &\leq (\bar{\kappa}- 1)C_{0}T_{0}^{2} \bigl\lVert { \Delta \sigma '} \bigr\rVert _{L^{2}}^{2} + C'_{9}T_{0}^{-2} \lVert{v} \rVert _{H^{2}}^{2} \\ & \quad + \tilde{C}_{8} \bigl( \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \lVert{\vartheta } \rVert _{H^{2}}^{2} \bigr) \\ &\quad + C'_{9} \lVert{v} \rVert _{H^{3}} \lVert{\sigma } \rVert _{H^{2}}^{2} + C'_{9}\bigl( \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}+ \lVert{b} \rVert _{H^{2}}^{2}\bigr), \end{aligned}$$

(4.27)

where κ̄ and $C_{0}$ are given by (4.2) and (4.14) respectively.

Proof

We apply the Laplacian operator to equation (3.8) and multiply the resulting equation by Δσ. By using the same arguments of Lemmas 4.2 and 4.3, we obtain (4.27). □

Lemma 4.10

Under the same assumptions of Lemma 4.1, one has

$$\begin{aligned} \lVert{v} \rVert _{H^{3}}^{2} + T^{2}_{0} \bigl\lVert { \nabla \sigma '} \bigr\rVert _{H^{1}}^{2}& \leq C'_{10} \Biggl(\sum_{i=1}^{2} \lVert{\partial _{x_{i}}v} \rVert _{H^{2}}^{2} + \lVert{v} \rVert _{H^{1}}^{2} \Biggr) \\ & \leq C'_{10}T^{2}_{0} \Biggl( \bigl\lVert {\Delta \sigma '} \bigr\rVert _{L^{2}}^{2} + \sum_{i=1}^{2} \bigl\lVert {\nabla \partial _{x_{i}} \sigma '} \bigr\rVert _{L^{2}}^{2} \Biggr) \\ &\quad + C'_{10} \bigl( \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}+ \lVert{b} \rVert _{H^{2}}^{2} \bigr) + \tilde{C}_{9} \lVert{\vartheta } \rVert _{H^{2}}^{2}. \end{aligned}$$

(4.28)

Proof

As in Lemma 4.4, according the well-known theory on the estimates of the Stokes problem, one can deduce from (4.18) with boundary conditions (2.12) that

$$\begin{aligned} \lVert{v} \rVert _{H^{3}}^{2} + \bigl\Vert \nabla \bigl(\hat{T} \sigma '\bigr) \bigr\Vert _{H^{1}}^{2} \leq \hat{C} \lVert{\vartheta } \rVert _{H^{2}}^{2} + C_{\Omega} \bigl( \lVert{\nabla \cdot v} \rVert _{H^{2}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2}+ \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} \bigr). \end{aligned}$$

(4.29)

Let us first notice that

$$ \lVert{\nabla \cdot v} \rVert _{H^{2}}^{2}\leq \lVert{ \partial _{x_{3}}\nabla \cdot v} \rVert _{H^{1}}^{2} + \sum_{i=1}^{2} \lVert{\partial _{x_{i}}v} \rVert _{H^{2}}^{2} + \lVert{v} \rVert _{H^{1}}^{2}. $$

In addition, taking into account (4.15), we have

$$\begin{aligned} & \lVert{\partial _{x_{3}}\nabla \cdot v} \rVert _{H^{1}}^{2} \\ &\quad \leq C_{\Omega} \lVert{b} \rVert _{H^{2}}^{2}+ C_{\Omega} \Biggl(\sum_{i=1}^{2} \lVert{\partial _{x_{i}}v} \rVert _{H^{2}}^{2} + T_{0}^{2} \bigl\lVert {\nabla \partial _{x_{3}} \sigma '} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {F'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} \Biggr) + \hat{C} \lVert{\vartheta } \rVert _{H^{2}}^{2} . \end{aligned}$$

Since

$$\begin{aligned} & \bigl\lVert {\nabla \bigl(\hat{T}\sigma '\bigr)} \bigr\rVert _{H^{1}}^{2} \geq C_{\Omega} \bigl(T_{0}^{2} \bigl\lVert {\nabla \sigma '} \bigr\rVert _{H^{1}}^{2} - \bigl\lVert {\sigma '} \bigr\rVert _{H^{1}}^{2} \bigr)\geq C_{\Omega} \bigl(T_{0}^{2} \bigl\lVert {\nabla \sigma '} \bigr\rVert _{H^{1}}^{2} - \bigl\lVert {\nabla \sigma '} \bigr\rVert _{L^{2}}^{2} \bigr) \\ & \bigl\lVert {\nabla \partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} \leq \bigl\lVert {\Delta \sigma '} \bigr\rVert _{L^{2}}^{2} + \sum _{i=1}^{2} \bigl\lVert {\nabla \partial _{x_{i}}\sigma '} \bigr\rVert _{L^{2}}^{2}, \end{aligned}$$

by adding these inequalities to (4.29), we obtain (4.28). □

5 Fixed point of the operator S

Having proved Lemmas 4.1–4.10, we are now in a position to establish that the operator S has a fixed point. We recall that following Remark 3.3 and Lemma 3.4, it only remains to show that the crucial point (3.35) holds, which is the goal in the following lemma.

We recall that the nonlinear operator $S : V\to V$, $u' = (\vartheta ',v',b',\sigma ')$, $S(u') = u$, where $u= (\vartheta ,v,b,\sigma )\in V$ is the unique solution of system (3.5)–(3.8).

Lemma 5.1

There is a norm $|\cdot |_{V}$ equivalent (see (3.3)) to $\|\cdot \|_{V}$ such that

$$ S\bigl(B_{V}^{a}\bigr)\subseteq B_{V}^{a},\quad B_{V}^{a}= \bigl\{ u=(\vartheta ,v,b, \sigma )\in V: \vert u \vert _{V}\leq a \bigr\} .$$

(5.1)

Proof

Let $\lambda _{1}, \lambda _{2}, \ldots , \lambda _{9}$ be positive numbers that will be suitably chosen thereafter. We set

$$\begin{aligned} & \nu _{1} := C_{0}(\bar{\kappa}+1)\lambda _{9}T_{0}^{2},\qquad \nu _{2} := C_{0}(\bar{\kappa}+1)\lambda _{7}T_{0}^{2}, \qquad \nu _{3}: = k \lambda _{6}, \\ &\nu _{4} := C_{0}(\bar{\kappa}+1)\lambda _{3}T_{0}^{2},\qquad \nu _{5} := k\lambda _{2},\qquad \nu _{6}: = k\lambda _{1}, \end{aligned}$$

where $C_{0}$ is given in (4.14) and κ̄ in (4.2).

For any $\varphi \in H^{2}$, we set

$$\begin{aligned} \vert \varphi \vert _{2,\Omega}^{2} &= \nu _{1} \lVert{\Delta \varphi } \rVert _{L^{2}}^{2} + \nu _{2}\sum_{i=1}^{2} \lVert{\partial _{x_{i}}\partial _{x_{3}}\varphi } \rVert _{L^{2}}^{2} + \nu _{3}\sum _{i,j=1}^{2} \biggl\Vert { \biggl({ \frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\partial _{x_{i}} \partial _{x_{j}} \varphi} \biggr\Vert _{L^{2}}^{2} \\ & \quad + \nu _{4} \lVert{\partial _{x_{3}}\varphi } \rVert _{L^{2}}^{2} + \nu _{5}\sum _{i=1}^{2} \biggl\Vert { \biggl({ \frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\partial _{x_{i}} \varphi} \biggr\Vert _{L^{2}}^{2} + \nu _{6} \biggl\Vert { \biggl({\frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\varphi} \biggr\Vert _{L^{2}}^{2}. \end{aligned}$$

(5.2)

It is clear that $|\cdot |_{2,\Omega}$ is equivalent to the $H^{2}$-norm. Moreover, if we set

$$ \vert u \vert _{V}^{2}: = \lVert{v} \rVert _{H^{3}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} + \lVert{\vartheta } \rVert _{H^{2}}^{2} + \vert \sigma \vert _{2,\Omega}^{2}, \quad u=( \vartheta ,v,b,\sigma )\in V,$$

(5.3)

we obtain (see (3.3)) a norm equivalent to $\|\cdot \|_{V}$.

Let now

$$ S\bigl(u'\bigr) = u = (\vartheta ,v,b,\sigma ), \quad u'\in B_{V}^{a}.$$

(5.4)

If we multiply the estimates on the solution u established in Lemmas 4.1–4.10 by

$$ \lambda _{1}, \lambda _{2}, \ldots , \lambda _{8}, \lambda _{9} \quad \text{and} \quad \lambda _{10} = 1 $$

respectively, then adding them and taking into account (3.10), we obtain

$$\begin{aligned} & \lVert{v} \rVert _{H^{3}}^{2} + \lVert{ \vartheta } \rVert _{H^{2}}^{2} + \Lambda _{4} \lVert{b} \rVert _{H^{2}}^{2} + \vert \sigma \vert _{2,\Omega}^{2} + \Lambda _{1} \Biggl[ \sum _{i=1}^{2} \lVert{\partial _{x_{i}}v} \rVert _{H^{2}}^{2} + T_{0}^{2}\sum_{i=1}^{2} \bigl\Vert \nabla \partial _{x_{i}}\sigma ' \bigr\Vert _{L^{2}}^{2} \Biggr] \\ &\quad + \Lambda _{2}\sum_{i,j=1}^{2} \lVert{\nabla \partial _{x_{i}} \partial _{x_{j}}v} \rVert _{L^{2}}^{2} + \Lambda _{3} \lVert{v} \rVert _{H^{2}}^{2} + \Lambda _{5}\sum _{i=1}^{2} \lVert{\nabla \partial _{x_{i}}v} \rVert _{L^{2}}^{2} \\ &\quad + \Lambda _{6}\bigl( \lVert{\nabla v} \rVert _{L^{2}}^{2} + \lVert{\nabla b} \rVert _{L^{2}}^{2}\bigr) + \Lambda _{7} \bigl\Vert \nabla \sigma ' \bigr\Vert _{L^{2}}^{2} + \Lambda _{8} \bigl\lVert { \sigma '} \bigr\rVert _{L^{2}}^{2}\leq N\bigl(\sigma '\bigr) \\ &\quad + \tilde{C} \bigl( \bigl\lVert {G'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {H'} \bigr\rVert _{H^{1}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{H^{2}}^{2} \bigr) \end{aligned}$$

(5.5)

with some positive constant C̃ while $N(\sigma ')$ is given by

$$\begin{aligned} N\bigl(\sigma '\bigr) &:= \bigl(C'_{10} + (\bar{\kappa}- 1) C_{0} \lambda _{9}\bigr)T_{0}^{2} \bigl\lVert { \Delta \sigma '} \bigr\rVert _{L^{2}}^{2} \\ &\quad + \bigl(C'_{8}\lambda _{8} + C_{0}(\bar{\kappa}- 1)\lambda _{7} \bigr)T_{0}^{2} \sum_{i=1}^{2} \bigl\lVert {\partial _{x_{i}} \partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + k\lambda _{6} \sum _{i,j=1}^{2} \biggl\Vert \biggl({ \frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\partial _{x_{i}}\partial _{x_{j}} \sigma ' \biggr\Vert _{L^{2}}^{2} \\ &\quad + \bigl(C'_{4}\lambda _{4} + C_{0}(\bar{\kappa}-1)\lambda _{3} \bigr)T_{0}^{2} \bigl\lVert {\partial _{x_{3}}\sigma '} \bigr\rVert _{L^{2}}^{2} + k\lambda _{2}\sum _{i=1}^{2} \biggl\Vert \biggl({ \frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}}\partial _{x_{i}}\sigma ' \biggr\Vert _{L^{2}}\\ &\quad + k \lambda _{1} \biggl\Vert \biggl({\frac{\hat{T}}{\hat{\varrho}}} \biggr)^{\frac{1}{2}} \sigma ' \biggr\Vert _{L^{2}}^{2} , \end{aligned}$$

(5.6)

and the numbers $\Lambda _{i}$ ($i=1,2, \ldots ,8$) are given by

$$\begin{aligned} &\Lambda _{1}: = \lambda _{8} - C'_{10}, \qquad \Lambda _{2} := {\frac{\mu}{R}}\lambda _{6} - C'_{7}\lambda _{7} - C'_{8} \lambda _{8}, \\ & \Lambda _{3} := \lambda _{4} - C'_{6} \lambda _{6} - C'_{7}T_{0}^{-2} \lambda _{7} - C'_{8}\lambda _{8} - C'_{9}T_{0}^{-2} \lambda _{9} - C'_{10}, \\ & \Lambda _{4} := \lambda _{5} - C'_{6} \lambda _{6} - C'_{7}\lambda _{7} - C'_{8}\lambda _{8} - C'_{9}\lambda _{9} - C'_{10}, \\ & \Lambda _{5} := {\frac{\mu}{R}}\lambda _{2} - C'_{3} \lambda _{3} - C'_{4}\lambda _{4}, \qquad \Lambda _{6}: = {\frac{\mu}{R}}\lambda _{1} - C'_{3}\lambda _{3} - C'_{5} \lambda _{5}, \\ & \Lambda _{7}:=C'_{4}\lambda _{4}T_{0}^{2} - C'_{3} \lambda _{3} - C'_{6} \lambda _{6} - C'_{7}\lambda _{7} - C'_{8}\lambda _{8} - C'_{9} \lambda _{9} - C'_{10}, \\ & \Lambda _{8} := C'_{1}T_{0} \lambda _{1} - C'_{2}\lambda _{2}. \end{aligned}$$

If $T_{0}$ is large enough, we can easily see that it is possible to choose (see (1.10))

$$ \lambda _{i} = \lambda _{i}(\bar{T}_{0}), \quad i=9,8,\ldots ,1, $$

so that

$$ \Lambda _{1}, \Lambda _{2},\Lambda _{3},\Lambda _{5}\geq 0,\quad \Lambda _{4} \geq 1, \qquad N\bigl(\sigma '\bigr)\leq (1-\bar{t}) \bigl\vert \sigma ' \bigr\vert _{2}^{2} \quad \text{for some }\bar{t}\in\, ]0,1[ .$$

(5.7)

Indeed, if $\lambda _{j}$ for $j = i+1,\ldots ,9$ and $i = 8,\ldots ,1$ are given, as can be seen easily, considering constraints (5.7), we can choose $\lambda _{i}$ large enough so that the inequalities containing only $\lambda _{i}$ and $\lambda _{j}$ are satisfied (and so we can proceed to the choices of $\lambda _{i}$, starting from $\lambda _{9}$ and then choosing successively $\lambda _{i}$ for $i = 8,6,\ldots ,1$). The constants $\lambda _{i}= \lambda _{i}(\bar{T}_{0})$ ($i=9, \ldots , 1$) so determined imply in particular that $\Lambda _{7}> 0$ and $\Lambda _{8}> 0$ if $T_{0}$ is large enough, and we can choose $\lambda _{5}$ such that $\Lambda _{4} \geq 1$.

Recalling now inequality (5.5) and using (5.7), we obtain

$$\begin{aligned} \lVert{v} \rVert _{H^{3}}^{2} + \lVert{ \vartheta } \rVert _{H^{2}}^{2} + \lVert{b} \rVert _{H^{2}}^{2} + \vert \sigma \vert _{2,\Omega}^{2} &\leq (1-\bar{t}) \bigl\vert \sigma ' \bigr\vert _{2,\Omega}^{2} \\ &\quad + \tilde{C} \bigl( \bigl\lVert {F'} \bigr\rVert _{H^{1}}^{2} + \bigl\lVert {G'} \bigr\rVert _{L^{2}}^{2} + \bigl\lVert {H'} \bigr\rVert _{L^{2}}^{2} + \lVert{v} \rVert _{H^{3}} \lVert{ \sigma } \rVert _{H^{2}}^{2} \bigr), \end{aligned}$$

(5.8)

and given (3.36) (see also (5.3) and (5.4)), it follows from (5.8) that

$$ \bigl\vert S\bigl(u'\bigr) \bigr\vert _{V}^{2} \leq (1-\bar{t}) \bigl\vert u' \bigr\vert _{V}^{2} + c_{\Omega }a^{3} \leq (1- \bar{t})a^{2} + c_{\Omega }a^{3}\quad \forall u'\in B_{V}^{a}.$$

(5.9)

If a is small enough so that $a\leq {\frac{\bar{t}}{2c_{\Omega}}}$, it follows from (5.9) that

$$ \bigl\vert S\bigl(u'\bigr) \bigr\vert _{V}^{2} \leq \biggl(1-{\frac{\bar{t}}{2}} \biggr)a^{2} \quad \forall u'\in B_{V}^{a}, $$

which means that $S(u')\in B_{V}^{a}$ for every $u'\in B_{V}^{a}$, and hence we obtain (5.1) and equivalently (3.35), thus the lemma is proven. □

Remark 5.2

As we have already seen in Remark 1.1, if the perturbation $\varepsilon (x')$ is identically zero or is a nonzero constant function, then $(0,B_{rs},T_{rs},\varrho _{rs})$ given by (1.11) is the unique stationary solution to the system of equations (1.1)–(1.4) with boundary conditions (1.6)–(1.8). However, it is important to establish whether this system can still have a stationary solution $(0,B_{st},T_{st},\varrho _{ts})$ when $\varepsilon (x')$ is not identically zero and is not identically a nonzero constant function.

Indeed, suppose that $\varepsilon (x')\not\equiv 0$ and is not identically a nonzero constant function, and suppose that there exists a stationary solution $(0,B_{st},T_{st},\varrho _{st})$ to equations (1.1)–(1.4) with boundary conditions (1.6)–(1.8). Hence, $(B_{st},T_{st},\varrho _{st})$ solves the system of equations

$$\begin{aligned} & \nu \nabla \nabla \cdot B_{st} - \nu \Delta B_{st}=0, \quad \nabla \cdot B_{st} = 0, \end{aligned}$$

(5.10)

$$\begin{aligned} & {-}\kappa \Delta T_{st} = \nu \vert \nabla \times B_{st} \vert ^{2}, \end{aligned}$$

(5.11)

$$\begin{aligned} & R\nabla (T_{st}\varrho _{st}) + g\varrho e_{3} = - (\nabla \times B_{st}) \times B_{st}, \end{aligned}$$

(5.12)

in $\Omega =\mathbb{T}^{2}\times ]0,h[$ with the boundary condition

$$\begin{aligned} & B_{st}\bigl(x',0\bigr) = B_{st}\bigl(x',h\bigr) = B_{0} e_{3}, \end{aligned}$$

(5.13)

$$\begin{aligned} & T_{st}\bigl(x',0\bigr) = T_{0} + \varepsilon \bigl(x'\bigr),\qquad T_{st} \bigl(x',h\bigr) = T_{0} + {\frac{\gamma -1}{\gamma R}}gh. \end{aligned}$$

(5.14)

Notice first that $B_{st}=B_{0}e_{3}$ is the unique solution of (5.10) with (5.13) and, since $\nabla \times B_{st}=0$, the boundary value problem (5.11) and (5.14) becomes

$$ \Delta T_{st} = 0,\qquad T_{st} \bigl(x',0\bigr) = T_{0} + \varepsilon \bigl(x'\bigr), \qquad T_{st}\bigl(x',h \bigr) = T_{0} - {\frac{\gamma -1}{\gamma R}}gh.$$

(5.15)

It is obvious that the unique solution of (5.15) is given by

$$ T_{st}\bigl(x',x_{3} \bigr) = T_{rs}(x_{3}) + \delta \bigl(x',x_{3} \bigr),\qquad T_{rs}(x_{3})= T_{0} - { \frac{\gamma -1}{\gamma R}}gx_{3},$$

(5.16)

where δ is the unique solution of the boundary value problem

$$ \Delta \delta = 0 \quad \text{in } \Omega = \mathbb{T}^{2}\times ]0,h[, \qquad \delta \bigl(x',0 \bigr) = \varepsilon \bigl(x'\bigr), \quad \delta \bigl(x',h\bigr) = 0.$$

(5.17)

Given (5.12) and $B_{st}=B_{0}e_{3}$, by setting $P_{st} = RT_{st}\varrho _{st}$, we have

$$ \partial _{x_{i}}P_{st} = 0 \quad (i=1,2),\qquad \partial _{x_{3}} \log P_{st} = -{ \frac{g}{RT_{st}}}.$$

(5.18)

Hence, $P_{st}(x',x_{3})= p_{st}(x_{3})$, and this contradicts (5.18)₂ unless (see (5.16)) $T_{st}(x',x_{3})$ does not depend on $x'=(x_{1},x_{2})$. Thus, we get the following two cases for the only solution $\delta (x',x_{3})$ of the boundary value problem (5.17): Either

(i) $\delta (x',x_{3})$ is constant and therefore, given (5.17)₃, we get $\delta (x',x_{3})\equiv 0$ for which $\varepsilon (x') = 0$ or

(ii) $\delta (x',x_{3})$ is independent of $x'=(x_{1},x_{2})$, and this contradicts (5.17)₂ unless $\varepsilon (x') = \varepsilon _{0}$.

Hence, in both cases (i) and (ii), we get $\varepsilon (x')$ is zero or constant, which is a contradiction. Therefore, the system of equations (1.1)–(1.4) does not have a stationary solution $(0,B_{st},T_{st},\varrho _{st})$ when $\varepsilon (x')$ is not identically zero or a nonzero constant function.

6 Conclusion

To our knowledge, there are not results in the literature on the study of stationary full magnetohydrodynamic flows for viscous, compressible, and heat-conducting fluids. Our work aims to contribute to the mathematical modeling of magnetoconvection by analyzing a system of partial differential equations describing the motion of full magnetohydrodynamic equations. As a first contribution to this study, we considered a simple model of three-dimensional plane-parallel atmosphere made up of ideal gas in the presence of a magnetic field. More precisely, we consider a plane-parallel polytropic atmosphere between $x_{3} = 0$ (bottom of the layer) and $x_{3} = h$ (top of the layer) with a large temperature gradient across the layer, and in the presence of a vertical magnetic field. We then focus on the existence of a magnetoconvective steady flow close to the equilibrium state. As an ongoing research, we are currently studying the stability of such a stationary solution, which is inspired from the works [28, 36, 46].