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 $\Vert \mathfrak{D}(\mathbf{u}){\Vert }_{L^1(0,T;L^{\infty })}+\Vert P{\Vert }_{L^{\infty }(0,T;L^{\infty })}<\infty$. In particular, the criterion is independent of the magnetic field. The logarithm-type estimate for the Lam{é} system and some delicate energy estimates play a crucial role in the proof.

to appear in Indiana University Mathematics Journal. arXiv admin note: text overlap with arXiv:1705.05161

References in corpus (4)

• Global solutions to the three-dimensional full compressible magnetohydrodynamic flows
• On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum
• A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system
• Singularity formation to the 2D Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction

Cited by in corpus (1)

• Singularity formation to the Cauchy problem of the two-dimensional non-baratropic magnetohydrodynamic equations without heat conductivity