On the Liouville theorem for weak Beltrami flows

TL;DR

This paper investigates weak Beltrami flows in stationary Euler equations, proving regularity and Liouville properties under weaker decay conditions, and establishing new formulas for velocity components.

Contribution

It introduces weaker decay conditions for Liouville property and derives new mean value formulas for weak solutions of stationary Euler equations.

Findings

Weak decay conditions imply trivial solutions.

Established mean value formulas for velocity components.

Proved regularity of weak Beltrami flows.

Abstract

We study Beltrami flows in the setting of weak solution to the stationary Euler equations in $\Bbb R^3$. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if tangential part of the velocity has certain decay property at infinity, then the solution becomes trivial. This decay condition of of the velocity is weaker than the previously known sufficient conditions for the Liouville property of the Betrami flows. For the proof we establish a mean value formula and other various formula for the tangential and the normal components of the weak solutions to the stationary Euler equations.