Global Small Solutions for the Navier–Stokes–Maxwell System

Abstract

We consider a full system of magnetohydrodynamic equations. The system does not enjoy any property of scaling invariance and, at least formally, has an energy estimate. Nevertheless, the existence of a global weak solution seems to remain an interesting open problem in both two and three space dimensions. In three dimensions, we show the existence of global small mild solutions. In two dimensions, we prove the same result in a space “close” to the energy space.