Liouville Type Theorems for the Planar Stationary MHD Equations with Growth at Infinity

Abstract

For the two dimensional steady MHD equations, we prove that Liouville type theorems hold if the velocity is growing fast at infinity. The main obstacle comes from the nonlinear terms, since the vorticity system of the MHD equations has no maximum principle unlike the Navier–Stokes equations. As a corollary, we obtain that all solutions of the 2D Navier–Stokes equations satisfying $\nabla u\in L^p({\mathbb {R}}^2)$ with $1<p<\infty$ are constants, which is sharp since there exist some non-trivial linear solutions like the Couette flow in the sense of $\nabla u\in L^\infty ({\mathbb {R}}^2)$.