Zero-Mach Limit of the Viscous Compressible Magnetohydrodynamic Flows Inside a Perfectly Conducting Wall for All Time

Abstract

We consider the two-dimensional viscous compressible magnetohydrodynamic flows inside a perfectly conducting wall with a nonslip boundary condition. By virtue of the exponential decay of strong solutions to the incompressible system, we establish all-time existence of strong solutions to the initial-boundary value problem for slightly compressible magnetohydrodynamic flows without smallness restrictions on the initial velocity and magnetic field. Furthermore, we prove that solutions of the compressible system uniformly converge to that of the incompressible system for all time as the Mach number approaches zero.