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

TL;DR

This paper establishes Liouville type theorems for 2D stationary incompressible MHD equations, showing that under certain magnetic field conditions, only trivial solutions exist, extending classical results from Navier-Stokes to MHD.

Contribution

It proves new Liouville theorems for 2D stationary MHD equations under magnetic field smallness conditions, generalizing known results from Navier-Stokes equations.

Findings

No non-trivial solutions under small magnetic field conditions

Finite Dirichlet integral or $L^p$ norm implies triviality

Results extend classical Navier-Stokes Liouville theorems to MHD

Abstract

This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger \cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak \cite{KNSS}, to the MHD setting.