Notes on the Liouville type problem for the stationary Navier-Stokes equations in $\Bbb R^3$

TL;DR

This paper investigates conditions under which solutions to the stationary Navier-Stokes equations in three-dimensional space must be trivial, deriving an asymptotic formula involving the head pressure to establish new criteria.

Contribution

It introduces a novel asymptotic formula for the head pressure integral, offering new sufficient conditions for the triviality of solutions to the stationary Navier-Stokes equations.

Findings

Derived an asymptotic formula for the head pressure integral.

Established new sufficient conditions for solution triviality.

Provided insights into the behavior of stationary Navier-Stokes solutions.

Abstract

In this paper we study the Liouville type problem for the stationary Navier-Stokes equations in $\Bbb R^3$. We deduce an asymptotic formula for an integral involving the head pressure, $Q=\frac12 |v|^2 +p$, and its derivative over domains enclosed by level surfaces of $Q$. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations.