On Liouville-type theorems for the 2D stationary MHD equations

TL;DR

This paper proves new Liouville-type theorems for 2D stationary incompressible MHD equations, showing that under certain boundedness conditions, solutions must be trivial, using maximum principles for associated drift-diffusion equations.

Contribution

It introduces novel Liouville-type theorems for 2D stationary MHD equations based on bounded Dirichlet integrals and maximum principles for the stream function.

Findings

Solutions are trivial under bounded Dirichlet integral conditions.

The stream function satisfies a drift-diffusion equation with a maximum principle.

The results extend Liouville theorems to the MHD context.

Abstract

We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift-diffusion equation for which a maximum principle is available.