Liouville type theorem for the stationary equations of magneto-hydrodynamics

TL;DR

This paper proves that smooth solutions to the stationary magneto-hydrodynamics equations that are in both L^6 and BMO^{-1} spaces must be trivial, extending previous results by relaxing integrability conditions.

Contribution

It establishes a Liouville type theorem for stationary MHD equations under weaker integrability assumptions than prior work.

Findings

Any smooth solution in L^6 and BMO^{-1} is zero.

Extends previous results requiring finite Dirichlet integral.

Provides new conditions for triviality of solutions.

Abstract

We show that any smooth solution $(\mathbf{u},\mathbf{H})$ to the stationary equations of magneto-hydrodynamics (MHD) belonging to both spaces $L^6 (\mathbb{R}^3)$ and $BMO^{-1}(\mathbb{R}^3)$ must be identically zero. This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption $\nabla \mathbf{u}, \nabla \mathbf{H} \in L^2 (\mathbb{R}^3) $, i.e., finite Dirichlet integral.