Local existence and blow-up criterion for the two and three dimensional ideal magnetic Benard problem

Abstract

In this article, we consider the ideal magnetic Bénard problem in both two and three dimensions and prove the existence and uniqueness of strong local-in-time solutions, in Hs for s > 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, generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.