Homogeneous solutions of the stationary Navier-Stokes equations in arbitrary dimension
Abstract
For every integer n≥4, we prove the existence of a (−1)-homogeneous solution of the stationary incompressible Navier-Stokes equations on ℝn∖{0} with an arbitrary locally Lipschitz (−3)-homogeneous force. The pressure is (−2)-homogeneous, and the solution is regular away from the origin. This extends the homogeneous existence theorem of Bang, Gui, Liu, Wang, and Xie beyond dimension sixteen. The principal estimate combines a subcritical localized pressure identity with an elementary dyadic estimate for an auxiliary tangential vector field in a suitable finite Lebesgue space. Testing the Bernoulli equation at the homogeneous cancellation exponent then controls its positive part in L((n−2)(n−1))/(2(n−3))(Sn−1). The resulting estimate closes with the energy inequality in every fixed dimension. We also establish uniqueness throughout the classical homogeneous class for sufficiently small force. Using the same ideas, we also give a simplified proof of the known existence theorem for bounded compactly supported forces on ℝn, n≥5. A general finite-exponent estimate for a scalar equation with divergence-free drift, together with a Euclidean version of the dyadic kernel estimate, bounds the positive Bernoulli function without BMO estimates or an iteration to obtain its supremum norm.
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BibTeX
@misc{ma2026homogeneous,
title = {{Homogeneous solutions of the stationary Navier-Stokes equations in arbitrary dimension}},
author = {Qi Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11301},
eprint = {2610.11301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11301},
}
RIS
TY - UNPB
TI - Homogeneous solutions of the stationary Navier-Stokes equations in arbitrary dimension
AU - Qi Ma
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11301
UR - https://arxiv.org/abs/2610.11301
AB - For every integer n≥4, we prove the existence of a (−1)-homogeneous solution of the stationary incompressible Navier-Stokes equations on ℝ^n∖{0} with an arbitrary locally Lipschitz (−3)-homogeneous force. The pressure is (−2)-homogeneous, and the solution is regular away from the origin. This extends the homogeneous existence theorem of Bang, Gui, Liu, Wang, and Xie beyond dimension sixteen. The principal estimate combines a subcritical localized pressure identity with an elementary dyadic estimate for an auxiliary tangential vector field in a suitable finite Lebesgue space. Testing the Bernoulli equation at the homogeneous cancellation exponent then controls its positive part in L^((n−2)(n−1))/(2(n−3))(S^n−1). The resulting estimate closes with the energy inequality in every fixed dimension. We also establish uniqueness throughout the classical homogeneous class for sufficiently small force. Using the same ideas, we also give a simplified proof of the known existence theorem for bounded compactly supported forces on ℝ^n, n≥5. A general finite-exponent estimate for a scalar equation with divergence-free drift, together with a Euclidean version of the dyadic kernel estimate, bounds the positive Bernoulli function without BMO estimates or an iteration to obtain its supremum norm.
ER -