Remarks on a Liouville-type theorem by Chae and Wolf for stationary Navier-Stokes equations

Abstract

In this note, we revisit a Liouville-type theorem of Chae and Wolf for stationary Navier-Stokes equations in ${\mathbb{R}}^3$ [J. Differential Equations 261 (2016) 5541-5560]. We show that their logarithmic improvement of the classical $L^{9/2}$ condition is part of a substantially broader weighted framework. More precisely, we prove that a solution $u\in {\dot{H}}^1({\mathbb{R}}^3)$ is necessarily trivial whenever $${\int }_{{\mathbb{R}}^3}|u(x){|}^{9/2}\omega (|u(x)|)dx<+\infty ,$$ for every positive, nondecreasing and bounded weight $\omega$ satisfying a mild growth condition near the origin. This structural condition encompasses the logarithmic weight due to Chae and Wolf, as well as a hierarchy of iterated-logarithmic weights and (genuinely) non-logarithmic examples, including a dyadic weight. Our result identifies a broader class of weighted integrability conditions under which the triviality of stationary Navier-Stokes solutions follows, and shows that the mechanism underlying the Chae-Wolf improvement is not intrinsically tied to a specific logarithmic weight or to a single logarithmic scale.