Global regularity for 3D generalized Hall magneto-hydrodynamics equations

For the 3D incompressible Hall magneto-hydrodynamics equations, global regularity of the weak solutions is not established so far. The major difficulty is that the dissipation given by the Laplacian operator is insufficient to control the nonlinearities. Wan obtained the global regularities of the 3D generalized Hall-MHD equations with critical and subcritical hyperdissipation in (Global regularity for generalized Hall-magnetohydrodynamics systems, Electron. J. Differential Equations, 2015, 2015(179), 1-18). We improve this slightly by making logarithmic reductions in the dissipation and still obtain the global regularity.