On The Two and Three Dimensional Ideal Magnetic B\'enard Problem - Local Existence and Blow-up Criterion

TL;DR

This paper investigates the ideal magnetic Bénard problem in 2D and 3D, establishing local existence of solutions and a criterion for singularity formation based on the vorticity and current's BMO-norm.

Contribution

It provides the first local existence results for strong solutions in both 2D and 3D and generalizes the Beale-Kato-Majda criterion to this magnetohydrodynamic setting.

Findings

Proves local-in-time existence and uniqueness of strong solutions in $H^s$.

Derives a necessary condition for singularity development involving the BMO-norm.

Extends classical hydrodynamics results to magnetic Bénard problems.

Abstract

In this paper, we consider the ideal magnetic B\'{e}nard problem in both two and three dimensions and prove local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}+1, n = 2,3$. In addition, a necessary condition is derived for singularity development with respect to the $BMO$-norm of the vorticity and electrical current, generalising the Beale-Kato-Majda condition for ideal hydrodynamics.