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. Using the energy method and an iteration argument, we establish Liouville type theorems if Lebesgue norms of the velocity and magnetic field on the annulus satisfy certain growth conditions. Furthermore, by establishing new energy estimates and developing some novel differential inequality techniques, we relax the growth conditions by logarithmic factors and obtain logarithmic improvement version of Liouville type theorems. For the MHD equations, the assumptions imposed on the magnetic field are weaker and wider than that of the velocity field in certain sense. Our results extend and improve several recent works.