Global well-posedness for the incompressible Hall-magnetohydrodynamic system in critical Fourier–Besov spaces

We investigate the initial value problem for the incompressible magnetohydrodynamic system with the Hall-effect in the whole space R³. In this paper, we focus on a solution as a perturbation from a constant equilibrium state (0 , B¯) , where 0 ∈ R³ is the zero velocity and B¯ ∈ R³ is a constant magnetic field. Our goal is to establish the existence of a global-in-time solution in L^p-type critical Fourier–Besov spaces. In order to prove our results, we establish various type product estimates in space-time mixed spaces and smoothing estimates for the solution of the linear equation, which has a non-symmetric diffusion derived from the Hall-term. Our results hold in the case of small initial data. However, the result can cover initial velocity fields whose high frequency part is highly oscillating.