Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher
Abstract
For n≥5, let (u,p) be a smooth solution of the stationary incompressible Navier-Stokes equations in Rn∖{x'=0}, where {x'=0} is the xn-axis. We prove that the scale-invariant transverse bound |u(x)|≤ C |x'|−1 forces u≡0. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of u and ∇ u in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-Růžička weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case |u(x)|≤ C |x|−1 in [1], yields the same expansion in exterior domains under the condition |u(x)|≤ C |x'|−1.
Related
- Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions
- Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows
- Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations
- Homogeneous solutions of the stationary Navier-Stokes equations in arbitrary dimension
- Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
BibTeX
@misc{gui2026rigidity,
title = {{Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher}},
author = {Changfeng Gui and Hao Liu and Yun Wang and Chunjing Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26323},
eprint = {2609.26323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26323},
}
RIS
TY - UNPB
TI - Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher
AU - Changfeng Gui
AU - Hao Liu
AU - Yun Wang
AU - Chunjing Xie
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26323
UR - https://arxiv.org/abs/2609.26323
AB - For n≥5, let (u,p) be a smooth solution of the stationary incompressible Navier-Stokes equations in R^n∖{x'=0}, where {x'=0} is the x_n-axis. We prove that the scale-invariant transverse bound |u(x)|≤ C |x'|^−1 forces u≡0. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of u and ∇ u in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-Růžička weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case |u(x)|≤ C |x|^−1 in [1], yields the same expansion in exterior domains under the condition |u(x)|≤ C |x'|^−1.
ER -