PreprintCC BY 4.0arXiv
Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
Abstract
We study Liouville-type results for the stationary Navier--Stokes equations in ℝ3. We prove that any Ḣ1(ℝ3) solution is trivial under an integrability condition imposed only on the radial component of the velocity, namely uρ(x) ∈ Lp(ℝ3) with 3/2 < p ≤ 3. We also establish a uniqueness result in a variable-exponent setting, where an L6-type condition is required only on a bounded region, while the exponent approaches the critical value $3$ at infinity. Our analysis reveals that the rigidity of the stationary Navier--Stokes system can be driven by localized and radial integrability properties, rather than uniform global conditions.
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BibTeX
@misc{vergarahermosilla2026liouville,
title = {{Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component}},
author = {Gaston Vergara-Hermosilla},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.05647},
eprint = {2605.05647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05647},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
AU - Gaston Vergara-Hermosilla
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2605.05647
UR - https://arxiv.org/abs/2605.05647
AB - We study Liouville-type results for the stationary Navier--Stokes equations in ℝ^3. We prove that any Ḣ^1(ℝ^3) solution is trivial under an integrability condition imposed only on the radial component of the velocity, namely u_ρ(x) ∈ L^p(ℝ^3) with 3/2 < p ≤ 3. We also establish a uniqueness result in a variable-exponent setting, where an L^6-type condition is required only on a bounded region, while the exponent approaches the critical value $3$ at infinity. Our analysis reveals that the rigidity of the stationary Navier--Stokes system can be driven by localized and radial integrability properties, rather than uniform global conditions.
ER -