Global existence and asymptotic behavior for the 3D generalized Hall-MHD system

In this paper we consider the global existence for the 3D generalized Hall-MHD equations with fractional dissipative terms (−Δ)αu and (−Δ)αb under small initial data in the setting of Sobolev norms with lower regularity. For the global existence we enlarge the range of dissipative exponents α=β from (1,76] to (1,32), which established in a recent work. In addition, the long time behavior and rates of decay for both the solutions and higher derivatives in different Sobolev spaces are obtained by using the Fourier splitting method, which extends the previous work by Chae and Schonbek.