On partial regularity for the steady Hall magnetohydrodynamics system

TL;DR

This paper investigates the partial regularity of solutions to the steady Hall magnetohydrodynamics equations, establishing bounds on the size and structure of potential singularities in three-dimensional space.

Contribution

It proves that the singular set of suitable weak solutions has Hausdorff dimension at most one and is compact in the whole space case, advancing understanding of solution regularity.

Findings

Singular set has Hausdorff dimension at most one.

In , singularities form a compact set.

Provides new bounds on the structure of singularities.

Abstract

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $\Omega \subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $\Omega=\Bbb R^3$, we show that the set of possible singularities is compact.