On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

TL;DR

This paper studies the mathematical well-posedness of the 3D Hall-magnetohydrodynamics system in critical function spaces, establishing global existence, uniqueness, and regularity results under small initial data conditions.

Contribution

It provides new global well-posedness results for the Hall-MHD system in critical Besov spaces, including cases with equal viscosity and resistivity, and extends local existence criteria for large data.

Findings

Global existence and uniqueness for small initial data in critical Besov spaces.

Global well-posedness when viscosity equals resistivity with small data in larger Besov spaces.

Local existence and continuation criteria for large data when r=1 in Besov spaces.

Abstract

We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity $u_0$ and magnetic field $B_0$ in critical regularity spaces.In the case where $u_0,$ $B_0$ and the current $J_0:=\nabla\times B_0$ belong to the homogeneous Besov space $\dot B^{\frac 3p-1}_{p,1},$ $\:1\leq p<\infty,$ and are small enough, we establish a global result and the conservation of higher regularity.If the viscosity is equal to the magnetic resistivity, then we obtain the global well-posedness provided $u_0,$ $B_0$ and $J_0$ are small enough in the \emph{larger} Besov space $\dot B^{\frac12}_{2,r},$ $r\geq1.$If $r=1,$ then we also establish the local existence for large data, and exhibit continuation criteria for solutions with critical regularity. Our results rely on an extended formulation of the Hall-MHD system, that has some similarities with the incompressibleNavier-Stokes equations.