Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations
Abstract
We construct nonzero compactly supported stationary distributional solutions u∈ L2(ℝ3;ℝ3) of the unforced incompressible Navier--Stokes equations, together with compactly supported pressures in L1, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized h-principle: nonzero stationary solutions are strongly dense in the localized Lp classes for every 1≤ p<2 and in H−1, and weakly dense at p=2, while strong L2 density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive L2 norms, periodize the construction to 𝕋3, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.
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BibTeX
@misc{zhao2026compactly,
title = {{Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations}},
author = {Xuanxuan Zhao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.17383},
eprint = {2608.17383},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17383},
}
RIS
TY - UNPB
TI - Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations
AU - Xuanxuan Zhao
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.17383
UR - https://arxiv.org/abs/2608.17383
AB - We construct nonzero compactly supported stationary distributional solutions u∈ L^2(ℝ^3;ℝ^3) of the unforced incompressible Navier--Stokes equations, together with compactly supported pressures in L^1, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized h-principle: nonzero stationary solutions are strongly dense in the localized L^p classes for every 1≤ p<2 and in H^−1, and weakly dense at p=2, while strong L^2 density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive L^2 norms, periodize the construction to 𝕋^3, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.
ER -