Almost sure well-posedness for Hall MHD

TL;DR

This paper proves that the Hall MHD system is well-posed almost surely for supercritical initial data by employing a randomization technique, establishing local and global results in critical Sobolev spaces.

Contribution

It introduces a novel probabilistic approach to establish well-posedness for the Hall MHD system with supercritical initial data.

Findings

Almost sure local well-posedness in critical Sobolev space

Almost sure global well-posedness for small data

Use of randomization to handle supercritical initial conditions

Abstract

We consider the magnetohydrodynamics system with Hall effect accompanied with initial data in supercritical Sobolev space. Via an appropriate randomization of the supercritical initial data, both local and small data global well-posedness for the system are obtained almost surely in critical Sobolev space.