Three dimensional stationary solutions of the Electron MHD equations

TL;DR

This paper constructs non-trivial steady-state weak solutions for 3D Electron MHD equations using convex integration, revealing solutions not possible in classical Navier-Stokes equations.

Contribution

It introduces a novel convex integration approach to find stationary weak solutions of 3D Electron MHD equations in low regularity spaces.

Findings

Existence of non-trivial stationary solutions in $H^s$ for small $s$

Solutions are not realizable in classical 3D Navier-Stokes equations

Method extends convex integration techniques to EMHD context

Abstract

The goal of this paper is to construct non-trivial steady-state weak solutions of the three dimensional Electron Magnetohydrodynamics equations in the class of $H^s(\mathbb T^3)$ for some small $s > 0$. By exploiting the formulation of the stationary EMHD equations one can treat them as generalized Navier-Stokes equations with half Laplacian. Therefore with convex integration scheme we obtained such stationary weak solutions, which is not yet realizable in the case of classical 3D Navier-Stokes equations.