New Liouville type theorems for the stationary Navier-Stokes equations

TL;DR

This paper establishes new Liouville type theorems for stationary Navier-Stokes equations, including fractional cases, by deriving a formula linking the Dirichlet integral to solution behavior at the origin, leading to uniqueness results.

Contribution

It introduces a novel formula for the Dirichlet integral and extends Liouville theorems to fractional Navier-Stokes equations, including the case s=5/6.

Findings

New formula for Dirichlet integral of solutions

Liouville theorems for classical stationary Navier-Stokes

Extension to fractional Navier-Stokes equations for 1/2 ≤ s < 1

Abstract

We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$.