Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition

Lan Zeng,Yingying Li,Guoxi Ni

doi:10.3934/dcdsb.2019068

Abstract

In this paper, we consider the low Mach number limit of the full compressible MHD equations in a 3-D bounded domain with Dirichlet boundary condition for velocity field, Neumann boundary condition for temperature and perfectly conducting boundary condition for magnetic field. First, the uniform estimates in the Mach number for the strong solutions are obtained in a short time interval, provided that the initial density and temperature are close to the constant states. Then, we prove the solutions of the full compressible MHD equations converge to the isentropic incompressible MHD equations as the Mach number tends to zero.

Highlights

In this paper, we consider an initial boundary value problem for the following full compressible MHD equations in a bounded domain Ω ⊂ R3 with smooth boundary.ρt + div(ρu) = 0, (1)(ρu)t + div(ρu ⊗ u) − div(2μD(u) + ξdivuI) + ∇P + 4π × H = (2)
The main results of this paper is stated as follows, which shows the uniform estimates of strong solutions to (9)-(15), and the corresponding low Mach number limit
Applying Lemma 4.1 we can complete the proof of this lemma

Summary

Introduction

We consider an initial boundary value problem for the following full compressible MHD equations in a bounded domain Ω ⊂ R3 with smooth boundary. In [7], the authors have researched the low Mach number limit in the 2-D case, in which u satisfies the Dirichlet boundary condition. Motivated by the works mentioned above, in this paper, we consider the low Mach number limit of strong solution for the full compressible MHD system (1)(4) in a 3-D bounded domain with Dirichlet boundary condition for u, Neumann boundary condition for θ and perfectly conducting boundary condition for H. Because of the large parameter 1/ 2 in the momentum equation (2), this low Mach number limit process is singular, which makes it a challenging mathematical problem to obtain uniform estimates in Mach number. The main results of this paper is stated as follows, which shows the uniform estimates of strong solutions to (9)-(15), and the corresponding low Mach number limit. By using Lemma (4.1), we obtain the estimate of (24)

L2 and curlH

Estimates of

Findings

Boundary estimates of