Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations

TL;DR

This paper establishes Liouville-type theorems for steady incompressible Euler and Navier-Stokes equations, showing under certain conditions that solutions must be trivial, thus contributing to the understanding of solution uniqueness and behavior at infinity.

Contribution

It introduces new Liouville-type results for steady Euler and Navier-Stokes equations with specific force and decay conditions, extending previous knowledge.

Findings

Solutions are trivial under single signedness force condition.

Decay conditions on velocity imply trivial solutions for Navier-Stokes.

Reproof of a known theorem using self-similar Euler equations.

Abstract

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $\Bbb R^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1 (\Bbb R^N)$ solution $(v,p)$ with $|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$, $q\in (\frac{3N}{N-1}, \infty)$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition $\int_{\Bbb R^3} |\Delta v|^{\frac65} dx<\infty$, which is stronger than the important D-condition, $\int_{\Bbb R^3} |\nabla v|^2 dx <\infty$, but both having the same scaling property, implies that $v=0$. In the appendix we reprove the Theorem 1.1(\cite{cha0}), using the self-similar Euler equations directly.