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Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D

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arxiv 2311.00141 v1 pith:RJT2ZU4R submitted 2023-10-31 math.AP

Jacob Bedrossian , Siming He , Sameer Iyer , Fei Wang This is my paper · ORCID
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abstract

In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $\mathbb{T} \times [-1,1]$, supplemented with Navier boundary conditions $\omega|_{y = \pm 1} = 0$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $\nu$. On the other hand, the nonzero modes are assumed size $O(\nu^{\frac12})$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field

    math.AP 2025-05 conditional novelty 7.0 of 10

    For 3D MHD with a rationally aligned background magnetic field, the sharp stability threshold around Couette flow is gamma=1, with inviscid damping and a nu^{-1/3} magnetic amplification.

  2. Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane

    math.AP 2025-01 conditional novelty 7.0 of 10

    A proof that Couette flow in the stably stratified Boussinesq system on R^2 is asymptotically stable for Richardson number R>1/4, with explicit enhanced dissipation, Taylor dispersion, and inviscid damping rates.

  3. Stability of the Couette flow for 3D Navier-Stokes equations with rotation

    math.AP 2024-12 conditional novelty 6.0 of 10

    For the 3D Navier-Stokes-Coriolis equations at rotation strength matching the Couette shear rate, perturbations of size δ Re^{-2} in H^σ with σ>9/2 remain bounded, yielding a γ=2 stability threshold.

  4. Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$

    math.AP 2024-11 reject novelty 6.0 of 10

    Near Poiseuille flow in the plane, the paper claims enhanced dissipation for high x-frequencies and a nonlinear stability threshold of order ν^{7/3}, but a stream-function estimate in the proof is not justified.

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