Liouville Theorems Above the Critical $9/2$ Threshold for Stationary Navier-Stokes Equations

TL;DR

This paper extends Liouville theorems for stationary Navier-Stokes equations by weakening the integrability conditions from the classical $L^{9/2}$ space to variable exponent spaces, emphasizing asymptotic behavior.

Contribution

It introduces a novel approach using variable exponent Lebesgue spaces to relax the classical integrability conditions for trivial solutions.

Findings

Trivial solutions follow under weakened integrability conditions $u otin L^{9/2}( ext{R}^3)$ but in a broader variable exponent space.

A localized Liouville theorem is established, requiring integrability only at infinity.

The approach relies on a new uniqueness result in Lebesgue spaces with variable exponents.

Abstract

We establish new Liouville-type theorems for the stationary Navier-Stokes equations in $\mathbb{R}^3$. A central open problem in this context is whether the classical $L^{9/2}(\mathbb{R}^3)$ condition of G.Galdi can be relaxed. In this note we show that this global integrability requirement can indeed be weakened. More precisely, we prove that triviality already follows under assumptions of the form $u \in L^{9/2 + \varepsilon(\cdot)}(\mathbb{R}^3)$, where $\varepsilon(\cdot)>0$. As a consequence, we obtain a localized Liouville theorem: it is sufficient to impose this integrability condition only at infinity, with no additional assumptions on the behavior of $u$ inside a compact set. This highlights that the mechanism enforcing triviality is purely asymptotic. Our approach relies on a general uniqueness result in the framework of Lebesgue spaces with variable exponents, which naturally captures the coexistence of different integrability regimes across the domain.