Liouville type theorem for the stationary equations of magneto-hydrodynamics

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 $BM{O}^{-1}({\mathbb{R}}^3)$ 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.

arXiv comment7 pages