On partial regularity for the $3D$ non-stationary Hall magnetohydrodynamics equations on the plane

TL;DR

This paper investigates the partial regularity of weak solutions to the 3D non-stationary Hall magnetohydrodynamics equations on the plane, establishing bounds on the size of singularity sets.

Contribution

It proves the existence of weak solutions with singularity sets of Hausdorff dimension at most two, advancing understanding of solution regularity in Hall MHD equations.

Findings

Singularity set has Hausdorff dimension at most two.

Existence of weak solutions with controlled singularities.

Partial regularity results for 3D Hall MHD equations.

Abstract

We study partial regularity of weak solutions of the 3D valued non-stationary Hall magnetohydrodynamics equations on $ \Bbb R^2$. In particular we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two.