On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction

TL;DR

This paper investigates conditions under which solutions to the 3D full compressible magnetohydrodynamic equations with zero heat conduction remain regular or develop singularities, highlighting criteria independent of magnetic field influence.

Contribution

It establishes a new global existence criterion for strong solutions with vacuum initial data, based on deformation tensor and pressure bounds, independent of magnetic field effects.

Findings

Global existence criterion involving deformation tensor and pressure.

The criterion is independent of magnetic field influence.

Key use of logarithm-type estimates and energy methods.

Abstract

We are concerned with the formation of singularity and breakdown of strong solutions to the Cauchy problem of the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction. It is proved that for the initial density allowing vacuum, the strong solution exists globally if the deformation tensor $\mathfrak{D}(\mathbf{u})$ and the pressure $P$ satisfy $\|\mathfrak{D}(\mathbf{u})\|_{L^{1}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In particular, the criterion is independent of the magnetic field. The logarithm-type estimate for the Lam{\'e} system and some delicate energy estimates play a crucial role in the proof.