On the existence and smoothness problem of the magnetohydrodynamics system

Abstract

Fluid mechanics plays a pivotal role in engineering application to daily lives. The prominently famous fluid dynamics partial differential equations (PDE) due to its remarkable utility is the Navier-Stokes equations of which its mathematical and physical significance is so highly regarded that it has become one of the seven Millennium Prize problems declared by the Clay Research Institute. We study closely related systems of partial differential equations with focus on the magnetohydrodynamics system, of which its special case is the Navier-Stokes equations. Other systems of PDEs of our concern include the surface quasi-geostrophic equations, incompressible porous media equation governed by Darcy's law, Boussinesq system, Leray, Lans-alpha models, micropolar and magneto-micropolar fluid models. We discuss the properties of solutions to these systems such as the global regularity issue with fractional Laplacians, logarithmic supercriticality, component reduction results of regularity criteria.