Existence and uniqueness in critical spaces for the magnetohydrodynamical system in $\mathbb{R}^n$

TL;DR

This paper studies the magnetohydrodynamical system in n-dimensional space, proving global existence for small initial data and local existence for large data in critical spaces, along with solution uniqueness.

Contribution

It introduces a novel exterior derivative formulation and establishes existence and uniqueness results in critical function spaces for the MHD system.

Findings

Global solutions exist for small initial data

Local solutions exist for large initial data

Uniqueness of solutions in certain critical spaces

Abstract

We give a description of a magnetohydrodynamical system in $n$ dimension using the exterior derivative. We then prove existence of global solutions for small initial data and local existence for arbitrary large data in two classes of critical spaces -- $L^q_tL^p_x$ and $\mathcal{C}_tL^p_x$, as well as uniqueness for solutions in $\mathcal{C}_tL^p_x$.