Logarithmic improvement of a Liouville-type theorem for the stationary Navier--Stokes equations

TL;DR

The paper proves a Liouville-type theorem for smooth solutions to the stationary Navier--Stokes equations in $\mathbb{R}^3$ under a refined $L^p$-growth condition with a logarithmic factor. It develops a robust energy-method framework, defining an energy functional $E(\rho)$ with a radial cutoff and deriving precise differential identities, then obtains local energy inequalities and gradient bounds via a sequence of technical lemmas. By connecting $L^p$-growth on annuli $S(\rho)$ to energy decay and using an iterative argument, the authors show that either the solution must vanish or violate the derived energy growth bounds, ultimately concluding $u\equiv 0$. This refines previous results by incorporating a logarithmic correction, extending Liouville-type rigidity results for the stationary Navier--Stokes system. The approach hinges on fine energy estimates, a divergence-correcting operator, and a careful handling of the $L^p$-growth regime with $\tfrac{3}{2}<p<3$. The findings contribute to the understanding of decay and rigidity of stationary flows at infinity and sharpen the known thresholds for zero solutions.

Abstract

We establish a new Liouville-type theorem for the stationary Navier--Stokes equations in $\mathbb{R}^3$. The main result is an improvement of the previous result with a logarithmic factor based on an understanding of $L^p$ growth of the velocity field near infinity.

Theorems & Definitions (11)

• Theorem 1
• Remark 1
• Definition 1
• Lemma 1: Lemma 1 of MR4354995
• Lemma 2: Lemma 8 of MR3720841
• Definition 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6