  The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from
kinetic models, are useful in describing many physical phenomena in geophysics
and astrophysics. This paper studies the local well-posedness of classical
solutions to the Hall-MHD equations with the magnetic diffusion given by a
fractional Laplacian operator, $(-\Delta)^\alpha$. Due to the presence of the
Hall term in the Hall-MHD equations, standard energy estimates appear to
indicate that we need $\alpha\ge 1$ in order to obtain the local
well-posedness. This paper breaks the barrier and shows that the fractional
Hall-MHD equations are locally well-posed for any $\alpha>\frac12$. The
approach here fully exploits the smoothing effects of the dissipation and
establishes the local bounds for the Sobolev norms through the Besov space
techniques. The method presented here may be applicable to similar situations
involving other partial differential equations.
