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

Abstract

In this paper, we investigate the well-posedness and ill-posedness issues for the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in ${\mathbb{R}}^3.$ We first show the existence and uniqueness of solutions provided with the forces in ${\dot{B}}_{p,r}^{3/p-3}({\mathbb{R}}^3)$ for $1\le p<3$ and $r=1$. Moreover, this result can be extended to any $1\le r\le \infty$ 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 \emph{critical} function spaces ${\dot{B}}_{p,r}^{3/p-1}({\mathbb{R}}^3)$ ($p\ge 3$).

22 pages