Global well-posedness for the two dimensional compressible MHD equations with large data

TL;DR

This paper proves the existence and uniqueness of global classical solutions for the two-dimensional compressible MHD equations with large initial data, under specific viscosity conditions, extending previous Navier-Stokes results.

Contribution

It extends the global well-posedness results from compressible Navier-Stokes equations to compressible MHD equations with large data and specific viscosity dependencies.

Findings

Unique global classical solutions exist under given conditions.

The results apply to periodic boundary conditions on the torus.

New techniques overcome coupling challenges between velocity and magnetic field.

Abstract

In this paper we are concerned with the global well-posedness for the compressible MHD equations with large data. We show that if the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda$ is the power function of the density, that is, $\lambda(\rho)=\rho^{\beta}$ with $\beta>3$, then the two dimensional compressible MHD system with the periodic boundary conditions on the torus $\mathbb{T}^2$ have a unique global classical solution $(\rho, u,H)$. In this work we extended the results about compressible Navier-Stokes equations in \cite{Karzhikhov} to compressible MHD equations by applying several new techniques to overcome the coupling between velocity and magnetic field.