Liouville theorem of D-solutions to the stationary magnetohydrodynamics system in a slab

Abstract

In this paper, we study Liouville theorems of D-solutions to the stationary magnetohydrodynamic system in a slab. We will prove trivialness of the velocity and the magnetic field with various boundary conditions. In some boundary conditions, the additional assumption that the horizontal angular component(s) of the velocity or (and) the magnetic field is (are) axially symmetric is needed. More precisely, five types of boundary conditions will be considered: the vertical periodic boundary condition for the velocity and the magnetic field, the Navier-slip boundary condition for the velocity, the perfectly conducting or insulating boundary condition for the magnetic field, the non-slip boundary condition for the velocity, and the perfectly conducting or insulating boundary condition for the magnetic field. One of our innovations is that we do not impose finite Dirichlet integral assumption on the magnetic field compared with previous works.