The 3D incompressible Hall magneto-hydrodynamics equations with partial hyperdissipation

Similarly as the 3D Navier-Stokes equation, the regularity and uniqueness of weak solutions for the 3D standard Hall-MHD equations remain completely open. Wan obtained the global smooth solutions for the incompressible Hall-MHD equations with hyperdissipation (-Delta)(alpha) and (-Delta)(beta) when alpha >= 5/4, beta >= 7/4 in (Global regularity for generalized Hall-magnetohydrodynamics systems, Wan (2015) [ 21]). We obtained the global regularity for the Hall-MHD equations with a logarithmic reduction in the dissipation in (Global regularity for a class of generalized Hall-magnetohydrodynamics equations, Yuan and Li (2018)). In this paper, we prove the global existence and uniqueness in the H-1-functional setting by energy method for the three-dimensional incompressible Hall-MHD equations with fractional partial dissipation, which improve Wan's result by making a different type of reduction in the dissipation. (C) 2019 Elsevier Inc. All rights reserved.