New energy functionals for the incompressible Hall-MHD system

We show the existence and uniqueness of solutions to the threedimensional incompressible Hall-magnetohydrodynamic (Hall-MHD) system with initial data in critical Sobolev spaces. Our result works for general physical parameters, thus improves the partial result obtained very recently by Danchin and the author in [13] and fully answers a problem proposed by Chae and Lee in the Remark 2 of [6]. Considering the so-called 2.5D flows for the Hall-MHD system (that is 3D flows independent of the vertical variable), we show that under the sole assumption that the initial magnetic field is small in the critical Sobolev space leads to a global well-posedness statement