Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients

TL;DR

This paper rigorously analyzes the limit where compressible magnetohydrodynamic equations become incompressible as viscosity and magnetic diffusion vanish, showing convergence to ideal incompressible MHD solutions.

Contribution

It establishes the simultaneous vanishing viscosity and magnetic diffusion limit for compressible MHD equations in the whole space.

Findings

Weak solutions converge to strong incompressible solutions

Results hold for general initial data in =2 or 3

Convergence occurs as Mach number, viscosity, and magnetic diffusion go to zero

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 solution of the compressible magnetohydrodynamic equations converges to the strong solution of the ideal incompressible magnetohydrodynamic equations as long as the latter exists.