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

TL;DR

This paper establishes regularity criteria for 3D MHD equations based on conditions on a single velocity component and a magnetic component, extending understanding of solution regularity with minimal component-wise assumptions.

Contribution

It introduces new Serrin-type regularity criteria for the 3D MHD equations using only one velocity and one magnetic component, based on recent local energy estimates.

Findings

Regularity at a point under component-wise conditions

Conditions involve integrability in Lorentz spaces

Extends previous criteria to minimal component assumptions

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).