Liouville-type theorem for steady helically symmetric MHD system in ${\mathbb{R}}^3$

Abstract

We show that any bounded smooth helically symmetric solution $(\Bu, \Bh)$ in ${\mathbb{R}}^3$ must be constant vectors. This is an extension of previous result \cite[Theorem 1.1]{HWXAHE} from Navier-Stokes system to MHD system. The proof relies on establishing a Saint-Venant type estimate to characterize the growth of Dirichlet integral of nontrivial solutions.