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

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 $J_0:=\mathrm{\nabla }\times B_0$ belong to the homogeneous Besov space ${\dot{B}}_{p,1}^{\frac{3}{p}-1},$ $1\le p<\mathrm{\infty },$ 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 ${\dot{B}}_{2,r}^{\frac{1}{2}},$ $r\ge 1.$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.