Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows
Abstract
We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier--Stokes equations. More precisely, every smooth solution on ℝ5∖{0} satisfying supR>0R−2∫BR|u|3 dx<∞ is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.
Related
- The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition
- Stationary varifolds with singularities II
- Rigidity of stationary black holes with small ADM angular momentum
- On rigidity of stationary black holes under a nontrapping assumption
- Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher
BibTeX
@misc{chen2026critical,
title = {{Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows}},
author = {Yubo Chen and Wendong Wang and Xiao Wang and Guoxu Yang and Jianbo Yu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29664},
eprint = {2608.29664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29664},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows
AU - Yubo Chen
AU - Wendong Wang
AU - Xiao Wang
AU - Guoxu Yang
AU - Jianbo Yu
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29664
UR - https://arxiv.org/abs/2608.29664
AB - We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier--Stokes equations. More precisely, every smooth solution on ℝ^5∖{0} satisfying sup_R>0R^−2∫_B_R|u|^3 dx<∞ is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.
ER -