Refined Liouville-Type Theorems for the Stationary Navier--Stokes Equations

TL;DR

This paper proves new Liouville-type theorems for stationary Navier--Stokes equations, showing solutions are trivial under certain refined decay and integrability conditions, advancing understanding of solution behavior in fluid dynamics.

Contribution

It introduces refined decay assumptions and integrability criteria that extend previous Liouville-type results for stationary Navier--Stokes solutions.

Findings

Solutions with bounded $L^p$ growth for $3/2 < p < 3$ are trivial.

Refined decay assumptions lead to new triviality results.

Analysis combines decay estimates, energy inequalities, and iteration schemes.

Abstract

We study smooth solutions to the three-dimensional stationary Navier--Stokes equations and establish new Liouville-type theorems under refined decay assumptions. Building on the work of Cho et al., we introduce a refinement to previously known integrability criteria and analyze the associated averaged quantities. Our main result shows that if the $L^p$ growth rate of a solution remains bounded for some $3/2 < p < 3$, then the solution must be trivial. The proof combines averaged decay estimates, energy inequalities, and an iteration scheme.