Regularity of the $3D$ stationary Hall magnetohydrodynamic equations on the plane

TL;DR

This paper proves that all weak solutions to the 3D stationary Hall MHD equations on the plane are smooth and establishes a Liouville theorem, advancing understanding of solution regularity in magnetohydrodynamics.

Contribution

It demonstrates the smoothness of weak solutions and introduces a Liouville type theorem for the stationary Hall MHD equations on the plane.

Findings

Weak solutions are smooth.

Liouville type theorem established.

Enhanced understanding of solution behavior in Hall MHD.

Abstract

We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $ \Bbb R^2$. We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations.