Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion

Abstract

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 $\alpha >\frac{1}{2}$. 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.

13 pages, this version improves the previous one

References in corpus (1)

• Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system

Cited by in corpus (5)

• On 3D Hall-MHD equations with fractional Laplacians: global well-posedness
• Approximating three-dimensional magnetohydrodynamics system forced by space-time white noise
• Derivation of the Hall-MHD equations from the Navier-Stokes-Maxwell equations
• On the regularity of magneto-vorticity field and the global existence for the Hall magnetohydrodynamic equations
• Global small solutions to the inviscid Hall-MHD system