Incompressible Limit of the Compressible Magnetohydrodynamic Equations with Vanishing Viscosity Coefficients

Abstract

This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients and general initial data in the whole space $\mathbb{R}^d$ ($d=2$ or 3). It is rigorously showed that, as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient simultaneously go to zero, the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the ideal incompressible magnetohydrodynamic equations as long as the latter exists.