On global strong solutions to the 3D MHD ﬂows with density-temperature-dependent viscosities

In this paper, we establish the global existence of strong solutions for the 3D viscous, compressible, and heat conducting magnetohydrodynamic (MHD) ﬂows with density-temperature-dependent viscosities in a bounded domain. We essentially show that for the initial boundary value problem with initial density allowed to vanish, the strong solution exists globally under some suitable small conditions. As a byproduct, we obtain the nonlinear exponential stability of the solution.

When the initial density can contain vacuum, Feireisl [6] investigated the full compressible Navier-Stokes equations with temperature-dependent heat conductivity coefficient and obtained the existence of "variational" weak solutions for large initial data with vacuum. Bresch and Desjardins [1] studied the Cauchy problem of system (1.5) with densitydependent viscosities and obtained the global stability of weak solutions. Recently, Yu and Zhang [25] considered the three-dimensional full compressible Navier-Stokes equations with density-temperature-dependent viscosities and proved the existence of global strong solutions in a bounded domain in R 3 .
For the compressible MHD system (1.1), Chen and Wang in [2] investigated the nonlinear MHD equations with general initial data and obtained the global existence and uniqueness of solutions with large initial data. Hu and Wang [10] investigated the compactness of weak solution of 3D full compressible MHD equations with density-dependent-heat conductivity and the magnetic coefficient with vacuum. Later, Huang and Li [13] studied the mechanism of blowup and structure of possible singularities of strong solutions to system (1.1) and obtained a blowup criterion, which is analogous to the well-known Serrin blowup criterion for the Cauchy problem and the initial boundary value one of system (1.1). Due to the physical importance, complexity, and mathematical challenges, our main aim in this paper is to investigate the global existence of strong solutions to the 3D MHD flows with density-temperature-dependent viscosities in a bounded domain.
Before stating our main result, we define q by for some 0 < β ≤ 1, and thus q ∈ (4,6]. For simplicity, we denote Now we are in a position to formulate our main results.
for some positive constantρ and that the following compatibility conditions hold: for some g 1 , g 2 ∈ L 2 . Then there exists a positive constant ε, depending only on , g 1 , g 2 , κ, ν, R, c v , and ρ 0 , such that if Moreover, for any t ≥ 0, we have that with positive constant C depending only on , ρ 0 , κ, ν, R, c v , g 1 , and g 2 .
Now we make some comments on the analysis of this paper. Note that for the Cauchy problem with constant viscosities satisfying (1.7)-(1.8), the local existence of strong solutions to the compressible MHD equations (1.1) with large initial data has been recently established [5]. Thus, to extend the strong solutions globally in time, we need global a priori estimates on smooth solutions for (ρ, u, θ , B). Some of the main new difficulties are due to the appearance of the density-temperature-dependent viscosities and the bounded domain. It turns out that the key issue of this paper is to derive the time-uniform upper bounds for the gradient of the density to bound ∇ 2 u L q and ∇ 2 B L q . We start with the a priori hypothesis on ∇ρ L q and initial layer analysis and succeed in deriving an estimate of ∇u t L 1 (0,T;L 2 ) and time-weighted estimates on the gradient of u t , θ t , and B t . Another difficulty caused by the bounded domain can be overcome by the energy method.
The rest of the paper is organized as follows. In Sect. 2, we establish estimates of the global strong solutions, which are independent of time t, to the initial boundary value problem (1.1)-(1.4). With the help of global (uniform) estimates at hand, in Sect. 3, we prove Theorem 1.1. In Sect. 4, we give some declarations of this paper.

Preliminaries
In this section, we recall some known facts and elementary inequalities. Before stating the results, we denote We first begin with the following local existence result of the initial-boundary value problem (1.1)-(1.4), which is obtained on a small time interval in [5]. Next, we give the Korn inequality, which can be found in [14].
Then there exists a constant ε > 0 such that Proof The proof of Proposition 3.1 will be done by a series of lemmas below.
We start with the following uniform estimates for (ρ, u, θ , B) under conditions (3.1).
Proof Equation (1.1) 3 , together with (3.1), (2.2), and (2.4), yields that and thus On the other hand, from the Sobolev and Poincaré inequalities it follows that which, together with (3.3), leads to (3.2). The proof of the lemma is therefore completed.
where q is defined as in (1.6).
Proof Thanks to the bounded domain , we obtain On the other hand, it follows from Lemma 2.2, (2.2), (2.4), and (3.1) that and thus Similarly, and thus which gives (3.4). The proof of the lemma is therefore completed.

Lemma 3.3
There exist a constant ε 1 > 0 such that

2), and (3.4) we obtain
The proof of the lemma is therefore completed.

Lemma 3.5 Let
Proof We multiply (1.1) 2 by u t and integrate the result over : The right-hand side terms of (3.15) can be estimated as follows. By (3.4) and the fact that and thus

Lemma 3.8 Let
Proof Thus we immediately obtain (3.35). The proof of the lemma is therefore completed.