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

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 Bedrossian et al. (Stability threshold of nearly-couette shear flows with Navier boundary conditions in 2d, 2311.00141, 2023) 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 \(\nu \lesssim |k| \le 1\), 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.