A blow-up criterion of strong solutions to the 2D compressible magnetohydrodynamic equations

TL;DR

This paper establishes a criterion for the blow-up of strong solutions to 2D compressible MHD equations, showing that the solution's density becoming unbounded causes blow-up, regardless of velocity or magnetic field behavior.

Contribution

It introduces a density-dependent blow-up criterion for strong solutions to 2D compressible MHD equations, allowing vacuum states and independent of velocity and magnetic field.

Findings

Blow-up occurs when the density's $L^{\infty}$-norm tends to infinity.

The criterion is independent of velocity and magnetic field.

Vacuum states are permitted in the solutions.

Abstract

This paper establishes a blow-up criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More precisely, once the strong solutions blow up, the $L^{\infty}$-norm for the density tends to infinity. In particular, the vacuum in the solutions is allowed.