Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain

In this paper we establish the uniform estimates of strong solutions with respect to the Mach number and the dielectric constant to the full compressible Navier-Stokes-Maxwell system in a bounded domain. Based on these uniform estimates, we obtain the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared data.

In [8,9], Kawashima and Shizuta established the global existence of smooth solutions for small data [11] and studied its zero dielectric constant limit ǫ 2 → 0 in the whole space R 2 . Recently, Jiang and Li [6] studied the zero dielectric constant limit ǫ 2 → 0 to the system (1.1)-(1.5) and obtained the convergence of the system (1.1)- (1.5) to the full compressible magnetohydrodynamic equations in T 3 , see also [7] on the similar results to the invisid case of (1.1)- (1.5). In [10], Li and Mu study the low Mach number limit ǫ 1 → 0 to the system (1.1)-(1.5) and obtained the convergence of the system (1.1)-(1.5) to the incompressible Navier-Stokes-Maxwell system in the torus T 3 .
It should be pointed out that no boundary effect is considered in the references mentioned above. The purpose of this paper is to invistigate the singular limit ǫ 1 , ǫ 2 → 0 to the system (1.5)-(1.5) in a bounded domain. For simplicity, we shall take ǫ 1 = ǫ 2 = ǫ and consider the case that the fluid is a polytropic ideal gas, that is e := C V T , p := RρT (1.6) with C V > 0 and R being the specific heat at constant volume and the generic gas constant, respectively.
To state the main result of this paper, we denote the density and temperature variations by σ ǫ and θ ǫ : Then we can rewrite the system (1.1)-(1.5) as follows: Here we have added the superscript ǫ on the unknowns (σ, u, θ, E, b) to emphasise the dependence of ǫ. The system (1.8)-(1.12) are supplemented with the following initial and boundary conditions: where n is the unit outer normal vector to the smooth boundary ∂Ω.

the density equation and the other quantities are defined by an analogous way.
Based on the uniform estimates of the solutions, we can prove the following convergence result by applying the Arzelá-Ascolis theorem in a standard way.
) with the following initial and boundary conditions: The remainder of this paper is devoted to the proof of Theorem 1.1 which will be given in next section.

Proof of Theorem 1.1
In this section we shall prove Theorem 1.1 by combining the ideas developed in [1,3,4,11]. First, by taking the similar arguments to that [1,11], we know that in order to prove (1.19), it suffices to show the following inequality for ∀t ∈ [0,T ] and some given positive nondecreasing continuous functions C 0 (·) and C(·).
Below we shall omit the spatial domain Ω in the integrals and drop the superscript "ǫ" of ρ ǫ , σ ǫ , u ǫ , θ ǫ , etc. for the sake of simplicity; moreover, we write M ǫ (t) and M ǫ (0) as M and M 0 , respectively. Since the physical constants κ, C V , and R do not bring any essential difficulties in our arguments, we shall take κ = C V = R = 1.
We will also use the following two inequalities: for any u ∈ H s (Ω) with s ≥ 1, which were obtained in [2] and [12] respectively.
Because the local existence for the problem (1.8)-(1.14) with fixed ǫ > 0 is essential similar to that in [13], we only need to prove (2.1). We will use the methods developed in [3,4].
First, by the same calculations as that in [3], we get Now we use the same method as that in [4] to prove some a priori estimates on (E, b).
Testing (1.11) and (1.12) by E and b, respectively, and summing up the results, we see that Integrating the above inequality over (0, t), we find that Using (1.14) and the formula we infer that rot b × n = 0 on ∂Ω. (2.8) Taking rot to (1.11) and (1.12), testing the results by rot E and rot b, respectively, summing up the results, and using (2.8) and integration by parts, we have Integrating the above inequality over (0, t), we have Taking div to (1.11) and testing the result by div E, we infer that Integrating the above inequality over (0,t), we deduce that Taking ∂ t to (1.11) and (1.12), testing the results by E t and b t , respectively, summing up the results, we get Integrating the above inequality over (0, t), we get Integrating the above inequality over (0, t), we have Taking ∇div to (1.11), testing the result by ∇div E, we have Integrating the above inequality over (0, t), we obtain (2.14) Taking ∂ t rot to (1.11) and (1.12), testing the results by ∂ t rot E and ∂ t rot b, respectively, summing up the results, and using (2.8), we obtain Integrating the above inequality over (0, t), we obtain Applying ∂ t div to (1.11), testing the result by div E t , we have Integrating the above inequality over (0, t), we have Now we use the method in [3] to prove some a priori estimates on (σ, u, θ).