Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 91, pp. 1-26.

Title: Local existence and blow-up criterion for the two and three dimensional 
       ideal magnetic Benard problem
       
Authors: Utpal Manna (Indian Inst. of Sci. Educ. and Research, Kerala, India)
         Akash Ashirbad Panda (Indian Inst. of Sci. Educ. and Research, Kerala, India)

Abstract:
 In this article, we consider the ideal magnetic Benard problem in both two and
 three dimensions and prove  the existence and uniqueness of strong local-in-time
 solutions, in H<sup>s</sup> for s &gt; (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.
 
Submitted June 3, 2018. Published September 07, 2020.
Math Subject Classifications: 76D03, 35B44, 35A01.
Key Words: Magnetic Benard problem; commutator estimates;
           blow-up criterion; logarithmic Sobolev inequality.