Remarks on a Liouville-type theorem for Beltrami flows

TL;DR

This paper provides simple, elementary proofs showing that Beltrami flows with finite energy or certain decay conditions must be trivial (zero), extending Liouville-type theorems for these flows.

Contribution

It offers new, straightforward proofs for Liouville-type theorems for Beltrami flows under various energy and decay conditions.

Findings

Beltrami flows with finite energy are zero.

Flows with specific decay rates are zero.

Simplified proofs for classical Liouville theorems.

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 $v\in L^\infty _{loc} (\Bbb R^3) \cap L^q(\Bbb R^3)$ with $q\in [2, 3)$, or $|v(x)|=O(1/|x|^{1+\varepsilon})$ for some $\varepsilon >0$, we provide a different, simple proof that $v=0$.