On Liouville type theorems for three-dimensional stationary MHD and Hall-MHD equations

Abstract

In this paper, we study the Liouville type theorems of the MHD (magnetohydrodynamics) equation and Hall-MHD equation in three-dimensional steady states. In the framework of local Morrey space, we overcome the estimations of the pressure term with an infinite energy solution and obtain some sufficient conditions to guarantee the existence of a unique trivial solution for these two kinds of systems. Then, by using the embedding relations between the Lorentz spaces and local Morrey spaces, we give the Liouville type theorems for the two kinds of systems in the Lorentz spaces. The obtained results do not need the condition of finite Dirichlet integral, and hence generalize and improve some existing results about the two kinds of systems in the framework of Lebesgue space.