Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

TL;DR

This paper develops a regularity theory for suitable weak solutions to the incompressible MHD equations, extending Navier-Stokes methods and showing that singular points are measure-zero, thus advancing understanding of solution smoothness.

Contribution

It introduces a novel regularity framework for MHD equations using scaling analysis and harmonic function properties, extending classical Navier-Stokes techniques.

Findings

Singular points have zero one-dimensional parabolic Hausdorff measure.

Regularity results extend to MHD equations with external forces.

Methodology employs iterative sequences and harmonic function monotonicity.

Abstract

This paper establishes a regularity theory for the magnetohydrodynamics (MHD) equations with external forces through scaling analysis. Inspired by the existing methodology, we utilize linearized approximations and the monotonicity property of harmonic functions to construct iterative sequences capturing scaling properties. This work successfully extends Navier-Stokes techniques to MHD coupling and demonstrates that the one dimensional parabolic Hausdorff measure of the possible singular points for the suitable weak solutions is zero.