Global existence of 2D electron MHD near a steady state

TL;DR

This paper proves the global existence and convergence of small smooth solutions in 2D electron MHD near shear steady states, highlighting the stability and long-term behavior of these solutions.

Contribution

It establishes the global existence and convergence rates for small smooth solutions in 2D electron MHD near shear steady states, a novel stability result.

Findings

Global in time existence of solutions

Convergence rate to steady state

Stability near shear steady states

Abstract

We study the electron magnetohydrodynamics (MHD) in two dimensional geometry, which has a rich family of steady states. In an anisotropic resistivity context, we show global in time existence of small smooth solution near a shear type steady state. Convergence rate of the solution to the steady state is also obtained.