On steady solutions of the Hall-MHD system in Besov spaces

In this paper, we investigate the well-posedness and ill-posedness issues for the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in R³. We first show the existence and uniqueness of solutions provided with the forces in B3/p-3p,r(R³) for 1≤ p <3 and $r=1$. Moreover, this result can be extended to any 1≤ r≤ ∞ whenever $p=2,$ without any additional assumption on the physical parameters. On the other hand, we establish some ill-posedness results for Hall-MHD system by using the discontinuity of the solution mapping of the three-dimensional stationary Navier-Stokes equations in critical function spaces Ḃ3/p-1p,r(R³) (p≥ 3).