Incompressible limit of the compressible magnetohydrodynamic equations with ill-prepared data in a perfectly conducting container

Abstract

We study the low Mach number limit of the compressible magnetohydrodynamic equations in a bounded domain $\Omega \subset R^3$ with ill-prepared initial data. The velocity field satisfies the Navier-slip boundary conditions and the magnetic field satisfies the perfectly conducting boundary conditions. By performing energy estimate in the conormal Sobolev space and proving the maximum principle to the equations satisfied by $(\nabla \times v^{ϵ},\nabla \times B^{ϵ})$, we overcome the difficulties caused by the simultaneous occurrence of fast oscillation and boundary layer. As a consequence, the uniform existence and the convergence of solutions are obtained.