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

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.