Global Strong and Weak Solutions to the Initial-boundary-value Problem of 2D Compressible MHD System with Large Initial Data and Vacuum

TL;DR

This paper proves the global existence of strong and weak solutions for the 2D compressible MHD system with large initial data and vacuum, under specific viscosity conditions, in a general bounded domain.

Contribution

It is the first to establish global strong solutions for the 2D compressible MHD system with large initial data and vacuum in general bounded domains.

Findings

Global strong and weak solutions exist without initial data size restrictions.

Solutions are valid even with vacuum regions in the initial density.

The results apply to general 2D bounded simply connected domains.

Abstract

In this paper, we study the barotropic compressible magnetohydrodynamic equations with the shear viscosity being a positive constant and the bulk one being proportional to a power of the density in a general two-dimensional bounded simply connected domain. For initial density allowed to vanish, we prove that the initial-boundary-value problem of 2D compressible MHD system admits the global strong and weak solutions without any restrictions on the size of initial data provided the shear viscosity is a positive constant and bulk one is $\lambda=\rho^\beta$ with $\beta>4/3$. As we known, this is the first result concerning the global existence of strong solutions to the compressible MHD system in general two-dimensional bounded domains with large initial data and vacuum.