A local smoothness criterion for 3D generalized Hall-MHD equations

Abstract

This paper is devoted to studying the generalized incompressible Hall-MHD system in ${\mathbb{R}}^3$. Based on energy estimates and regularization of the initial data with the generalized heat semigroup, local existence for initial data $(u_0,b_0,v_0)$ in the critical spaces ${\dot{H}}^{\frac{5-4\alpha }{2}}({\mathbb{R}}^3)$ and global existence for the ${\dot{H}}^{\frac{5-4\alpha }{2}}$-norm of initial data $(u_0,b_0,v_0)$ sufficiently small are proved.