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