A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity

We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain.


Introduction
In this paper, we consider the 3D full compressible magnetohydrodynamic equations in a bounded domain Ω ⊂ R 3 :

2)
C V [∂ t (ρθ) + div (ρuθ)] − κ∆θ + pdiv u = µ 2 |∇u + ∇u t | 2 + λ(div u) 2 + ν|rot b| 2 , Here the unknowns ρ, u, p, θ, and b stand for the density, velocity, pressure, temperature, and magnetic field, respectively. The physical constants µ and λ are the shear viscosity and bulk viscosity of the fluid and satisfy µ > 0 and λ + 2 3 µ ≥ 0. C V > 0 is the specific heat at constant volume and κ > 0 is the heat conductivity. ν > 0 is the magnetic diffusivity. ∇u t denotes the transpose of the matrix ∇u. We assume that Ω is a bounded and simply connected domain in R 3 with smooth boundary ∂Ω. We use n to denote the outward unit normal vector to ∂Ω.
The full compressible magnetohydrodynamic equations (1.1)-(1.4) can be rigorous derivation from the compressible Navier-Stokes-Maxwell system [14]. Due to the physical importance of the magnetohydrodynamics, there are a lot of literature on the system (1.1)-(1.4), among others, we mention [8] on the local strong solutions, [4,9,10] on the global weak solutions, [15,16] on low Mach number limit, and [19] on the time decay of smooth small solutions.
Assume that the pressure take the form p = Rρθ with R being the generic gas constant.
We mention that when taking b = 0 in the system (1.1)-(1.4), it is reduced to the full compressible Navier-Stokes system and a lot of regularity criteria can be found in [5,20,23] and the references cited therein.
The remainder of this paper is devoted to the proof of Theorem 1.1. We give some preliminaries in section 2 and present the proof of Theorem 1.1 in section 3. Below we shall use the letter C to denote the positive constant which may change from line to line.

Preliminaries
First, we consider the boundary value problem for the Lamé operator L . It is well known that the system (2.1) is a strongly elliptic system, thus there exists a unique weak solution U ∈ H 1 0 (Ω) for F ∈ W −1,2 (Ω).
Lemma 2.1. Let q ∈ (1, ∞) and U be a solution of (2.1). There exists a constant C depending only on λ, µ, q and Ω such that the following estimates hold: Proof. The estimates (2.2) and (2.3) are classical for strongly elliptic systems, see for example [2]. The estimate (2.4) can be proved by a duality argument with the help of (2.2).
We need an endpoint estimate for L in the case q = ∞. Let BM O(Ω) stand for the John-Nirenberg space of bounded mean oscillation whose norm is defined by Here Ω r (x) := B r (x)∩Ω, B r (x) is a ball with center x and radius r, d is the diameter of Ω and |Ω r (x)| denotes the Lebesque measure of Ω r (x).
Let us conclude this section by recalling a variant of the Brezis-Waigner inequality [3]. Then there holds In the following proofs, we will use the Poincaré inequality [17]: for any b ∈ H 1 (Ω) with b · n = 0 or b × n = 0 on ∂Ω. We will also use the inequality [22]: for any b ∈ H 1 (Ω) with b · n = 0 or b × n = 0 on ∂Ω.

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1, we only need to show a priori estimates. For simplicity, we will take ν = C V = R = 1.
Testing (1.2) by u, (1.4) by b, summing up the results and using (1.1) and (1.10), we see that 1 2 which gives (ρ|u| 2 + |b| 2 )dx + T 0 (|∇u| 2 + |rot b| 2 )dxdt ≤ C. By the same calculations as that in [12], we get and w := u − v. Thanks to Lemma 2.1, for any 1 < r < ∞, there hold It is easy to see that w satisfies Then it follows from Lemma 2.1 that Let E be the specific energy defined by Then ∂ t ρE + |b| 2 2 + div (ρEu + pu + |b| 2 u) Testing (1.2) by u t and using (1.1) and denotingḟ := f t + u · ∇f , we deduce that 1 2 We remark that And according to (3.8) and (1.1), for any small 0 < δ 1 , δ 2 and δ 3 . Here we have used the Gagliardo-Nirenberg inequality Observing that the last term of (3.9) can be bounded as On the other hand, testing (1.4) by b t − ∆b, we get Here we have used the inequality 3.14) and the Gagliardo-Nirenberg inequality Inserting (3.10), (3.11) and (3.12) into (3.9) and combining (3.13) and choosing δ 1 , δ 2 and δ 3 suitably small and using the Gronwall inequality, we have Now we are in a position to give a high order regularity estimates of the solutions. The calculations were motivated by [20]. First of all, we rewrite the equation (1.2) as Testing the above equation byu and using (1.1), we have = (p t divu + (u · ∇)u · ∇p)dx + (g t + div (g ⊗ u))udx. (3.17) As in [20], one can estimate the second and third terms in above equation as follows.
Direct calculations show that d dt ∇ρ L q ≤ C(1 + ∇u L ∞ ) ∇ρ L q + C ∇ 2 u L q , (3.24) and d dt We bound the last term of (3.25) as follows.