Singularity Formation of the Non-barotropic Compressible Magnetohydrodynamic Equations Without Heat Conductivity

Abstract

We study the singularity formation of strong solutions to the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction in a bounded domain. We show that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$, the magnetic field $\mathbf{b}$, and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^{\infty})} + \|\mathbf{b}\|_{L^{\infty}(0,T;L^6)} + \|P\|_{L^{\infty}(0,T;L^{\infty})} \lt \infty$ and the coefficients of viscosity verity $3\mu \gt \lambda$. This extends the corresponding results in Duan (2017), Fan et al. (2018) [1,2] where a blow-up criterion in terms of the upper bounds of the density, the magnetic field and the temperature was obtained under the condition $2\mu \gt \lambda$. Our proof relies on some delicate energy estimates.