Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

TL;DR

This paper proves the global regularity of solutions for 2D incompressible MHD equations with mixed partial dissipation and magnetic diffusion, addressing the development of singularities and establishing conditions for weak solutions.

Contribution

It introduces new results on global regularity and uniqueness for 2D MHD equations with partial dissipation, expanding understanding of solution behavior under less restrictive conditions.

Findings

Global regularity for equations with mixed partial dissipation

Existence and uniqueness of weak solutions with magnetic diffusion

Conditional regularity results for solutions without full dissipation

Abstract

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.