Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions
Abstract
Recently, the existence of regular solutions (u,p) to the stationary incompressible Navier--Stokes equations on ℝn with bounded compactly supported forces f for 5≤ n≤15 was established by Li--Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for dimensions n>15. Here we answer this question and obtain the existence of regular solutions to the stationary Navier--Stokes equations in arbitrarily high dimensions. The key innovation is to combine an averaged signed pressure-potential identity with L2-BMO estimates for the Bernoulli source flux, yielding a Bernoulli supremum bound independent of the size of the skew coefficient representing the drift. Together with the energy estimate for the comparison solution, this yields a sublinear bound for the Newtonian potential of the positive Bernoulli function, removing the upper dimension restriction in the earlier existence argument. We also construct periodic solutions and whole-space solutions for forces in L∞∩ L2n/(n+2). For compactly supported forces, we record two far-field profiles, their velocity, gradient and pressure remainders, and the corresponding cancellation criteria.
Related
- Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
- Sharp sign criterion for global existence and blow-up of strong solutions with arbitrarily large initial data in the 3D spherically symmetric compressible Navier-Stokes equations
- Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher
- Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations
- Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations
BibTeX
@misc{wang2026regular,
title = {{Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions}},
author = {Wendong Wang and Guoxu Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24206},
eprint = {2609.24206},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24206},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions
AU - Wendong Wang
AU - Guoxu Yang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24206
UR - https://arxiv.org/abs/2609.24206
AB - Recently, the existence of regular solutions (u,p) to the stationary incompressible Navier--Stokes equations on ℝ^n with bounded compactly supported forces f for 5≤ n≤15 was established by Li--Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for dimensions n>15. Here we answer this question and obtain the existence of regular solutions to the stationary Navier--Stokes equations in arbitrarily high dimensions. The key innovation is to combine an averaged signed pressure-potential identity with L^2-BMO estimates for the Bernoulli source flux, yielding a Bernoulli supremum bound independent of the size of the skew coefficient representing the drift. Together with the energy estimate for the comparison solution, this yields a sublinear bound for the Newtonian potential of the positive Bernoulli function, removing the upper dimension restriction in the earlier existence argument. We also construct periodic solutions and whole-space solutions for forces in L^∞∩ L^2n/(n+2). For compactly supported forces, we record two far-field profiles, their velocity, gradient and pressure remainders, and the corresponding cancellation criteria.
ER -