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 ℝ3 [J. Differential Equations 261 (2016) 5541-5560]. We show that their logarithmic improvement of the classical L9/2 condition is part of a substantially broader weighted framework. More precisely, we prove that a solution u∈Ḣ1(ℝ3) is necessarily trivial whenever ∫ℝ3|u(x)|9/2 ω(|u(x)|) dx< +∞, for every positive, nondecreasing and bounded weight ω 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.
Related
- Liouville theorems for the stationary Navier-Stokes equations under partial component conditions
- Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
- Beyond the L9/5 Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem
- Liouville Theorems for the Stationary Fractional Hall-Magnetohydrodynamic Equations
- Liouville Theorems for the Stationary Fractional Magnetohydrodynamic Equations
BibTeX
@misc{fierro2026remarks,
title = {{Remarks on a Liouville-type theorem by Chae and Wolf for stationary Navier-Stokes equations}},
author = {Raúl Fierro and Gaston Vergara-Hermosilla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06013},
eprint = {2609.06013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Remarks on a Liouville-type theorem by Chae and Wolf for stationary Navier-Stokes equations
AU - Raúl Fierro
AU - Gaston Vergara-Hermosilla
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06013
UR - https://arxiv.org/abs/2609.06013
AB - In this note, we revisit a Liouville-type theorem of Chae and Wolf for stationary Navier-Stokes equations in 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∈Ḣ^1(ℝ^3) is necessarily trivial whenever ∫_ℝ^3|u(x)|^9/2 ω(|u(x)|) dx< +∞, for every positive, nondecreasing and bounded weight ω 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.
ER -