The inhomogeneous incompressible Hall-MHD system with only bounded density

This paper is dedicated to the global-in-time existence and uniqueness issues of solutions for the inhomogeneous incompressible Hall-magnetohydrodynamics (MHD) system with merely bounded density. In a three-dimensional case, assuming that the initial density is a small perturbation of a positive constant in the L infinity norm, we prove global well-posedness for small initial velocity and magnetic fields in critical Besov spaces. Next, we consider the so-called 212D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2{1\over{2}}\rm{D}$$\end{document} flows for the inhomogeneous Hall-MHD system (that is 3D flows independent of the vertical variable) and establish the global existence of strong solutions by only assuming that the initial magnetic field is small in critical spaces and the initial density is bounded and bounded away from zero. In particular, those solutions allow piecewise constant density with jumps so that a mixture of fluids can be considered. Compared with inhomogeneous incompressible Navier-Stokes equations, the new difficulties of proving these results come from the additional so-called Hall term, which endows the magnetic equation with a quasi-linear character. In order to overcome them, we reformulate the system by taking advantage of the curl form of the magnetic equation and develop some new maximal regularity estimates for the Stokes system with just bounded coefficients.