Remarks on a Liouville-type theorem for Beltrami flows

Abstract

We present a simple, short and elementary proof that if $v$ is a Beltrami flow with a finite energy in ${\mathbb{R}}^3$ then $v=0$. In the case of the Beltrami flows satisfying $v\in L_{loc}^{\infty }({\mathbb{R}}^3)\cap L^q({\mathbb{R}}^3)$ with $q\in [2,3)$, or $|v(x)|=O(1/|x{|}^{1+\epsilon })$ for some $\epsilon >0$, we provide a different, simple proof that $v=0$.

6 pages