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PreprintCC BY 4.0arXiv

Nonlinear stability of 3D Couette flow on 𝕋×ℝ2 under critical transverse low-frequency control

  • Yubo Chen
  • Guoxu Yang
  • Jianbo Yu
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Abstract

We establish nonlinear stability of three-dimensional Couette flow for the incompressible Navier--Stokes equations on 𝕋×ℝ2. Divergence-free perturbations satisfying |uin|H2+|u⊥0,in|Ḃ−22,∞≤ c*ν generate unique global H2 solutions that remain close to Couette flow, where u⊥0,in denotes the streamwise average of the transverse velocity. This result retains the O(ν) stability threshold despite the absence of a spanwise spectral gap. The nonzero streamwise modes exhibit enhanced dissipation, their wall-normal component satisfies a viscosity-uniform time-integrated inviscid-damping estimate, and the transverse zero mode decays in L2 at rate (1+νt)−1. The endpoint space Ḃ−22,∞ controls the transverse low frequencies, accommodates divergence-form nonlinear forcing, and is critical for linear lift-up from the wall-normal zero mode. The proof combines propagation of this endpoint norm, Fourier splitting, and a two-level roll-to-streak representation with an adaptation of the nonzero-mode framework of Wei--Zhang (CPAM, 2021).

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Cite

BibTeXRIS
BibTeX
@misc{chen2026nonlinear,
  title = {{Nonlinear stability of 3D Couette flow on $\mathbb T\times\mathbb R^2$ under critical transverse low-frequency control}},
  author = {Yubo Chen and Guoxu Yang and Jianbo Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08288},
  eprint = {2610.08288},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08288},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nonlinear stability of 3D Couette flow on 𝕋×ℝ^2 under critical transverse low-frequency control
AU  - Yubo Chen
AU  - Guoxu Yang
AU  - Jianbo Yu
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08288
UR  - https://arxiv.org/abs/2610.08288
AB  - We establish nonlinear stability of three-dimensional Couette flow for the incompressible Navier--Stokes equations on 𝕋×ℝ^2. Divergence-free perturbations satisfying |u_in|_H^2+|u^⊥_0,in|_Ḃ^−2_2,∞≤ c_*ν generate unique global H^2 solutions that remain close to Couette flow, where u^⊥_0,in denotes the streamwise average of the transverse velocity. This result retains the O(ν) stability threshold despite the absence of a spanwise spectral gap. The nonzero streamwise modes exhibit enhanced dissipation, their wall-normal component satisfies a viscosity-uniform time-integrated inviscid-damping estimate, and the transverse zero mode decays in L^2 at rate (1+νt)^−1. The endpoint space Ḃ^−2_2,∞ controls the transverse low frequencies, accommodates divergence-form nonlinear forcing, and is critical for linear lift-up from the wall-normal zero mode. The proof combines propagation of this endpoint norm, Fourier splitting, and a two-level roll-to-streak representation with an adaptation of the nonzero-mode framework of Wei--Zhang (CPAM, 2021).
ER  -