Title: The magnetohydrodynamic equations with partial or fractional dissipation

Abstract:
The magnetohydrodynamic (MHD) equations govern the motion of electrically conducting fluids such as plasmas, liquid metals, and electrolytes. They consist of a coupled system of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. Besides their wide physical applicability, the MHD equations are also of great interest in mathematics. In the last few years there have been substantial developments on the global regularity problem concerning the MHD equations, especially when there is only partial or fractional dissipation. The talk presents recent results on the global well-posedness and stability problem for the MHD equations with various partial or fractional dissipation.