Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces

TL;DR

This paper establishes local well-posedness for the compressible viscous magnetohydrodynamic equations with large initial data and analyzes the low Mach number limit, proving convergence to incompressible equations with rates.

Contribution

It proves local well-posedness for large data and rigorously justifies the low Mach number limit in critical spaces for the first time.

Findings

Solutions exist locally for large initial data.

Solutions of compressible equations converge to incompressible ones as Mach number approaches zero.

Convergence rates are explicitly derived.

Abstract

The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained.