  We are concerned with the 3D incompressible Hall-magnetohydro-dynamic system
(Hall-MHD). Our first aim is to provide the reader with an elementary proof of
a global well-posedness result for small data with critical Sobolev regularity,
in the spirit of Fujita-Kato's theorem [10] for the Navier-Stokes equations.
Next, we investigate the long-time asymptotics of global solutions of the
Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong
uniqueness statement is also proven. Finally, we consider the so-called 2 1/2 D
flows for the Hall-MHD system, and prove the global existence of strong
solutions, assuming only that the initial magnetic field is small.
