On classical solutions of the compressible magnetohydrodynamic equations with vacuum

TL;DR

This paper proves the local existence of unique classical solutions to 3-D compressible MHD equations with vacuum and large initial data, and establishes a blow-up criterion based on the deformation tensor of velocity gradients.

Contribution

It introduces the first local existence result for classical solutions with vacuum and large data, and links blow-up to the deformation tensor norm.

Findings

Existence of unique local classical solutions with vacuum.

Blow-up occurs if the deformation tensor norm becomes unbounded.

The blow-up criterion aligns with known criteria for related fluid equations.

Abstract

In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is firstly established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we prove that the $L^\infty$ norm of the deformation tensor of velocity gradients controls the possible blow-up (see \cite{olga}\cite{zx}) for classical (or strong) solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by the losing the bound of the deformtion tensor as the critical time approches. Our result (see (1.12) is the same as Ponce's criterion for $3$-D incompressible Euler equations \cite{pc} and Huang-Li-Xin's blow-up criterion for the $3$-D compressible Navier-stokes equations \cite{hup}.