Skip to content
DraconiticResearch
Search Sources API
190,869 studies31 sourcesUpdated 9 min ago
PreprintarXiv licencearXiv

A Hartman-Grobman Theorem for the Navier-Stokes Equation on 𝕋3

  • Rongchang Liu
  • Kening Lu
  • Lin Shi
Open at sourcePDFDOICite

Abstract

We prove a local Hartman-Grobman theorem at zero for the unforced Navier-Stokes equation on the three-dimensional torus in the strong topology of Hs for every s>7/2. More precisely, we construct a homeomorphism between open positively invariant neighborhoods of zero that conjugates the Navier-Stokes semiflow to the heat semigroup. We develop an infinite-dimensional boundary-time scheme built on a nested hierarchy of exact asymptotic dynamical factors. Their fiber geometry yields Sobolev block coordinates with autonomous finite-dimensional factor systems, on which boundary-time topological conjugacies can be constructed at every finite level and then assembled on a common neighborhood into the desired Hs conjugacy. The infinite-dimensional assembly relies on new spectral estimates for the divergence-free transport structure, which control derivative loss and the blockwise tails of the finite-level conjugacies and their inverses without an additional spectral-gap assumption.

Related

  1. A Hartman-Grobman theorem on Banach bundles
    Benjamin Rogoll, Stefan Siegmund
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsDynamical systemsarXiv
  2. Moments propagation in the Vlasov equation and application to the Navier-Stokes-Vlasov equations
    Yanqing Wang, Wei Wei, Yulin Ye +1
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. Polynomial mixing for infinite-energy solutions of the white-forced Navier--Stokes equation in an infinite pipe
    Meng Zhao
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. 3D Navier-Stokes Equation with Multivalued Friction
    Simon ZONGO, Bila Adolphe Kyelem
    Annals of Mathematics and Computer SciencePeer-reviewedCC BY-NC-NDFull text5 citedGeneral mathematicsOpenAlex
  5. Remarks on a Liouville-type theorem by Chae and Wolf for stationary Navier-Stokes equations
    RaΓΊl Fierro, Gaston Vergara-Hermosilla
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{liu2026hartmangrobman,
  title = {{A Hartman-Grobman Theorem for the Navier-Stokes Equation on $\mathbb T^3$}},
  author = {Rongchang Liu and Kening Lu and Lin Shi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08058},
  eprint = {2610.08058},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08058},
}
RIS
TY  - UNPB
TI  - A Hartman-Grobman Theorem for the Navier-Stokes Equation on 𝕋^3
AU  - Rongchang Liu
AU  - Kening Lu
AU  - Lin Shi
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08058
UR  - https://arxiv.org/abs/2610.08058
AB  - We prove a local Hartman-Grobman theorem at zero for the unforced Navier-Stokes equation on the three-dimensional torus in the strong topology of H^s for every s>7/2. More precisely, we construct a homeomorphism between open positively invariant neighborhoods of zero that conjugates the Navier-Stokes semiflow to the heat semigroup. We develop an infinite-dimensional boundary-time scheme built on a nested hierarchy of exact asymptotic dynamical factors. Their fiber geometry yields Sobolev block coordinates with autonomous finite-dimensional factor systems, on which boundary-time topological conjugacies can be constructed at every finite level and then assembled on a common neighborhood into the desired H^s conjugacy. The infinite-dimensional assembly relies on new spectral estimates for the divergence-free transport structure, which control derivative loss and the blockwise tails of the finite-level conjugacies and their inverses without an additional spectral-gap assumption.
ER  -