Global Wellposedness and Large Time Behavior of Solutions to the Hall-Magnetohydrodynamics Equations

Abstract

We prove the global solutions to the incompressible Hall-magnetohydrodynamics (Hall-MHD) equations with small or some large initial data in ${\mathbb{R}}^n(n\ge 3)$. Especially, Fujita–Kato type initial data as the incompressible Navier–Stokes equations are allowed. We also study the large time behavior of the solutions and obtain an optimal decay rate in the general Besov spaces. Different from all the previous wellposedness results in [6, 7, 21], the initial energy we considered here may not be finite.