Almost sure existence of global weak solutions for supercritical electron MHD

TL;DR

This paper proves the almost sure existence of global weak solutions for the supercritical electron MHD equations with rough initial data by employing a randomization technique.

Contribution

It introduces a method to establish global weak solutions for supercritical electron MHD with rough initial data using probabilistic approaches.

Findings

Global weak solutions exist almost surely for supercritical electron MHD.

The method handles rough initial data in negative Sobolev spaces.

Randomization of initial data is key to the existence proof.

Abstract

We consider the Cauchy problem for the electron magnetohydrodynamics model in the supercritical regime. For rough initial data in $\mathcal H^{-s}(\mathbb T^n)$ with $s>0$, we obtain global in time weak solutions almost surely via an appropriate randomization of the initial data.