Quantitative Hydrodynamic Stability for Couette Flow on Unbounded Domains with Navier Boundary Conditions

TL;DR

This paper establishes a stability threshold for 2D Navier-Stokes equations near Couette flow on unbounded domains, demonstrating inviscid damping, enhanced dissipation, and Taylor dispersion with precise decay scales, extending nonlinear stability results.

Contribution

It introduces new nonlinear analysis at low and intermediate frequencies for unbounded domains, providing the first enhanced dissipation results for fully nonlinear shear flows in such settings.

Findings

Inviscid damping of velocity for initial perturbations.

Enhanced dissipation at high frequencies with decay time-scale O(ν^{-1/3}|k|^{-2/3}).

Taylor and low frequency dispersion with specific decay scales.

Abstract

We prove a stability threshold theorem for 2D Navier-Stokes on three unbounded domains: the whole plane $\mathbb{R} \times \mathbb{R}$, the half plane $\mathbb{R} \times [0,\infty)$ with Navier boundary conditions, and the infinite channel $\mathbb{R} \times [-1, 1]$ with Navier boundary conditions. Starting with the Couette shear flow, we consider initial perturbations $\omega_{in}$ which are of size $\nu^{1/2}(1+\ln(1/\nu)^{1/2})^{-1}$ in an anisotropic Sobolev space with an additional low frequency control condition for the planar cases. We then demonstrate that such perturbations exhibit inviscid damping of the velocity, as well as enhanced dissipation at $x$-frequencies $|k| \gg \nu$ with decay time-scale $O(\nu^{-1/3}|k|^{-2/3})$. On the plane and half-plane, we show Taylor dispersion for $x$-frequencies $|k| \ll \nu$ with decay time-scale $O(\nu |k|^{-2})$, while on the channel we show low frequency dispersion for $|k| \ll \nu$ with decay time-scale $O(\nu^{-1})$. Generalizing the work of arXiv:2311.00141 done on $\mathbb{T} \times [-1,1]$, the key contribution of this paper is to perform new nonlinear computations at low frequencies with wave number $|k| \lesssim \nu$ and at intermediate frequencies with wave number $\nu \lesssim |k| \leq 1$, and to provide the first enhanced dissipation result for a fully-nonlinear shear flow on an unbounded $x$-domain. Additionally, we demonstrate that the results of this paper apply equally to solutions of the perturbed $\beta$-plane equations from atmospheric dynamics.