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Enhanced dissipation and stability threshold for the Navier--Stokes equations near Poiseuille flow

  • Tao Liang
  • Cuili Zhai
  • Xiaoping Zhai
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Abstract

We study the nonlinear stability of the quadratic plane Poiseuille flow U(y)=(y2,0) for the two-dimensional incompressible Navier--Stokes equations on 𝕋×ℝ. We prove that for initial perturbations of size O(ν2/3) in a ν-dependent weighted Sobolev space, the flow is globally nonlinearly stable. In particular, the nonzero streamwise modes experience enhanced dissipation and decay exponentially on the characteristic time scale O(ν−1/2), while the zero mode remains uniformly controlled. Thus, in this weighted Sobolev topology, O(ν2/3) is a sufficient stability scale without logarithmic loss.

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Cite

BibTeXRIS
BibTeX
@misc{liang2026enhanced,
  title = {{Enhanced dissipation and stability threshold for the Navier--Stokes equations near Poiseuille flow}},
  author = {Tao Liang and Cuili Zhai and Xiaoping Zhai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35185},
  eprint = {2609.35185},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35185},
}
RIS
TY  - UNPB
TI  - Enhanced dissipation and stability threshold for the Navier--Stokes equations near Poiseuille flow
AU  - Tao Liang
AU  - Cuili Zhai
AU  - Xiaoping Zhai
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35185
UR  - https://arxiv.org/abs/2609.35185
AB  - We study the nonlinear stability of the quadratic plane Poiseuille flow U(y)=(y^2,0) for the two-dimensional incompressible Navier--Stokes equations on 𝕋×ℝ. We prove that for initial perturbations of size O(ν^2/3) in a ν-dependent weighted Sobolev space, the flow is globally nonlinearly stable. In particular, the nonzero streamwise modes experience enhanced dissipation and decay exponentially on the characteristic time scale O(ν^−1/2), while the zero mode remains uniformly controlled. Thus, in this weighted Sobolev topology, O(ν^2/3) is a sufficient stability scale without logarithmic loss.
ER  -