Regularity criteria for suitable weak solutions to the four dimensional incompressible magneto-hydrodynamic equations near boundary

TL;DR

This paper establishes new regularity criteria for suitable weak solutions to four-dimensional incompressible magneto-hydrodynamic equations, demonstrating conditions under which solutions are regular near boundaries and analyzing the measure of singular points.

Contribution

It introduces two novel $ ext{ε}$-regularity criteria for these equations, one based on the smallness of the $L^{p,q}$ norm of velocity and another on the $L^2$ norm of its gradient, with boundary regularity implications.

Findings

Two $ ext{ε}$-regularity criteria are established.

Solutions are shown to be regular near boundaries under certain smallness conditions.

The set of singular points has zero two-dimensional Hausdorff measure near the boundary.

Abstract

In this paper, we consider suitable weak solutions of the four dimensional incompressible magneto-hydrodynamic equations. We give two different kind $\varepsilon$-regularity criteria. One only requires the smallness of scaling $L^{p,q}$ norm of $u$, another requires the smallness of scaling space time $L^2$ norm of $\nabla u$ and boundedness of scaling norm of $H$ or $\nabla H$. And as an application of the second kind criteria, we also prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero.