Some Liouville-Type Results for the 3D Incompressible MHD Equations

In this paper, we investigate some Liouville-type properties for the stationary incompressible MHD equations in three dimensions. Liouville-type properties are concerned with the uniqueness of trivial solutions to the stationary MHD equations. We show that the velocity field u and the magnetic field b are vanishing provided that u ∈ BMO - 1 ( R 3 ) and b ∈ L p , ∞ ( R 3 ) with 2 < p ≤ 6 . Moreover, we also prove that if then u and b are vanishing for the stationary 3D MHD equations with partial viscosity and partial diffusivity.