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

TL;DR

This paper establishes a threshold theorem for 2D Navier-Stokes equations with Navier boundary conditions, demonstrating nonlinear enhanced dissipation and inviscid damping for nearly-Couette flows with perturbations independent of viscosity.

Contribution

It introduces a new singular integral operator to quantitatively analyze inviscid damping within a nonlinear hypocoercivity framework for nearly-Couette shear flows.

Findings

Proves nonlinear enhanced dissipation for small perturbations.

Demonstrates inviscid damping with a new singular integral operator.

Establishes stability threshold for nearly-Couette flows with Navier boundary conditions.

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.