Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
Abstract
We study (−1)-homogeneous solutions of the steady Navier-Stokes equations in ℝn∖{0} with (−3)-homogeneous, locally Lipschitz forces. Previous work [2] established the existence of (−1)-homogeneous solutions without any smallness assumption in dimensions 4≤ n≤16. In this paper, we remove the dimension restriction and obtain the existence of self-similar solutions in all higher dimensions. The new ingredient is a dimension-independent a priori estimate for the total head pressure on the sphere, obtained by adapting the source-flux method and the weighted identity of the pressure of Wang and Yang [28] to the homogeneous setting. This combines with the structural relation between the radial velocity and the total head pressure used in [2] to close the a priori estimates for the solutions in all dimensions.
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BibTeX
@misc{gui2026selfsimilar,
title = {{Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions}},
author = {Changfeng Gui and Hao Liu and Yun Wang and Chunjing Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09840},
eprint = {2610.09840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09840},
}
RIS
TY - UNPB
TI - Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
AU - Changfeng Gui
AU - Hao Liu
AU - Yun Wang
AU - Chunjing Xie
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09840
UR - https://arxiv.org/abs/2610.09840
AB - We study (−1)-homogeneous solutions of the steady Navier-Stokes equations in ℝ^n∖{0} with (−3)-homogeneous, locally Lipschitz forces. Previous work [2] established the existence of (−1)-homogeneous solutions without any smallness assumption in dimensions 4≤ n≤16. In this paper, we remove the dimension restriction and obtain the existence of self-similar solutions in all higher dimensions. The new ingredient is a dimension-independent a priori estimate for the total head pressure on the sphere, obtained by adapting the source-flux method and the weighted identity of the pressure of Wang and Yang [28] to the homogeneous setting. This combines with the structural relation between the radial velocity and the total head pressure used in [2] to close the a priori estimates for the solutions in all dimensions.
ER -