New Liouville type theorems for 3D steady incompressible MHD equations and Hall-MHD equations

Abstract: In this paper, we study Liouville type results for the three-dimensional stationary incompressible MHD equations and Hall-MHD equations. By a new iteration argument, we establish Liouville type theorems if the velocity and magnetic field satisfy certain growth conditions of Lebesgue norms on the annulus. In our iteration procedure, the $L^2$ norms of the gradients of the velocity and magnetic field and the $L^6$ norms of the velocity and magnetic field are iterated together. The conditions imposed on the magnetic field are weaker than the velocity field in certain sense. As a consequence, we show that the velocity and magnetic field are trivial provided that they belong to some Lebesgue spaces or satisfy some decay conditions at infinity. Our results extend and improve the recent works of Chae-Lee (2024 Nonlinearity 37 095006) and Cho-Neustupa-Yang (2024 Nonlinearity 37 035007).