On the wellposedness of the Navier-Stokes-Maxwell system

TL;DR

This paper investigates the mathematical well-posedness of the coupled Navier-Stokes-Maxwell system, establishing local existence for large data and global existence for small initial data, advancing understanding of magneto-hydrodynamic models.

Contribution

It proves local and global well-posedness results for the full Magneto-Hydro-Dynamic system with large initial data for local solutions and small data for global solutions.

Findings

Local existence of mild solutions for large data

Global solutions for small initial data

Solutions in scale-invariant function spaces

Abstract

We study the local and global wellposedness of a full system of Magneto-Hydro-Dynamic equations. The system is a coupling of the forced (Lorentz force) incompressible Navier-Stokes equations with the Maxwell equations through Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier-Stokes. These solutions are global if the initial data are small enough.