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

摘要

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 (−Δ)α and (−Δ)β when α≥54,β≥74 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 H1-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.