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arXiv · 2107.02348

Serrin-type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

Hui Chen·Chenyin Qian·Ting Zhang
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Abstract

In this paper, we consider the Cauchy problem to the 3D MHD equations. We show that the Serrin--type conditions imposed on one component of the velocity $u_{3}$ and one component of magnetic fields $b_{3}$ with $$ u_{3} \in L^{p_{0},1}(-1,0;L^{q_{0}}(B(2))),\ b_{3} \in L^{p_{1},1}(-1,0;L^{q_{1}}(B(2))), $$ $\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1$ and $3<q_{0},q_{1}<+\infty$ imply that the suitable weak solution is regular at $(0,0)$. The proof is based on the new local energy estimates introduced by Chae-Wolf (Arch. Ration. Mech. Anal. 2021) and Wang-Wu-Zhang (arXiv:2005.11906).

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Hui Chen, Chenyin Qian, Ting Zhang. 2021-07-06. Serrin-type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component. https://arxiv.org/abs/2107.02348

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