A blow-up criterion for strong solutions to three-dimensional compressible magnetohydrodynamic equations

TL;DR

This paper establishes a criterion for the global existence of strong solutions to 3D compressible magnetohydrodynamic equations based on the boundedness of the velocity gradient, using energy and elliptic estimates.

Contribution

It provides a new blow-up criterion for strong solutions with density-dependent viscosity in compressible MHD equations, extending previous results.

Findings

Global existence of strong solutions under bounded velocity gradient

Energy estimates are crucial for the analysis

Density away from vacuum ensures well-posedness

Abstract

We are concerned with an initial boundary value problem for the compressible magnetohydrodynamic equations with viscosity depending on the density. It is show that for the initial density away from vacuum, the strong solution to the problem exists globally if the gradient of velocity satisfies $\|\nabla\mathbf{u}\|_{L^{2}(0,T;L^\infty)}<\infty$. Our method relies upon the delicate energy estimates and elliptic estimates.