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

Xin Zhong

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arXiv · 1705.06606

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

Xin Zhong

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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{é} system and some delicate energy estimates play a crucial role in the proof.

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Xin Zhong. 2017-09-14. On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction. https://arxiv.org/abs/1705.06606

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