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arXiv · 1407.7831

Blow-up criterion for the $3$D non-resistive compressible Magnetohydrodynamic equations

Shuai Xi·Shengguo Zhu
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Abstract

In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $\rho$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the $L^\infty$ norms of $(H,\rho)$ control the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which means that if a solution of the compressible isentropic non-resistive MHD equations is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the $L^\infty$ norm of $H$ or $\rho$ as the critical time approaches.

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Shuai Xi, Shengguo Zhu. 2014-07-29. Blow-up criterion for the $3$D non-resistive compressible Magnetohydrodynamic equations. https://arxiv.org/abs/1407.7831

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