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 u0 and magnetic field B0 in critical regularity spaces. In the case where u0, B0 and the current belong to the homogeneous Besov space 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 u0, B0 and J0 are small enough in the larger Besov space If 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 incompressible Navier–Stokes equations.