Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium

Abstract

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation.

Submitted to Advances in Mathematics on December 28, 2018

References in corpus (2)

• An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system
• Global Existence for Two Dimensional Incompressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity

Cited by in corpus (1)

• Global regularity for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion