Blow-up criterion for the $3$D non-resistive compressible Magnetohydrodynamic equations

TL;DR

This paper establishes a criterion for the blow-up of strong solutions to 3D non-resistive compressible MHD equations, showing that unbounded magnetic field or density norms lead to singularity formation.

Contribution

It proves a new blow-up criterion based on the $L^ abla$ norms of magnetic field and density for the 3D compressible isentropic MHD equations without magnetic diffusion.

Findings

Blow-up occurs if the $L^ abla$ norms of $H$ or $ ho$ become unbounded.

The criterion links loss of regularity to magnetic field and density norms.

Provides conditions under which solutions remain regular or develop singularities.

Abstract

In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $\rho$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the $L^\infty$ norms of $(H,\rho)$ control the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which means that if a solution of the compressible isentropic non-resistive MHD equations is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the $L^\infty$ norm of $H$ or $\rho$ as the critical time approaches.