Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions

Jacob Bedrossian; Siming He; Sameer Iyer; Fei Wang

Paper

Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions

Abstract

We consider the 2D, incompressible Navier-Stokes equations near the Couette flow,

, set on the channel

, supplemented with Navier boundary conditions on the perturbation,

. We are simultaneously interested in two asymptotic regimes that are classical in hydrodynamic stability: the long time,

, stability of background shear flows, and the inviscid limit,

in the presence of boundaries. Given small (

, but independent of

) Gevrey 2- datum,

, that is supported away from the boundaries

, we prove the following results: \begin{align*} & \|ω^{(ν)}(t) - \frac{1}{2π}\int ω^{(ν)}(t) dx \|_{L^2} \lesssim εe^{-δν^{1/3} t}, & \text{(Enhanced Dissipation)} \\ & \langle t \rangle \|u_1^{(ν)}(t) - \frac{1}{2π} \int u_1^{(ν)}(t) dx\|_{L^2} + \langle t \rangle^2 \|u_2^{(ν)}(t)\|_{L^2} \lesssim εe^{-δν^{1/3} t}, & \text{(Inviscid Damping)} \\ &\| ω^{(ν)} - ω^{(0)} \|_{L^\infty} \lesssim ενt^{3+η}, \quad\quad t \lesssim ν^{-1/(3+η)} & \text{(Long-time Inviscid Limit)} \end{align*} This is the first nonlinear asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries. The techniques we develop represent a major departure from prior works on nonlinear inviscid damping as physical space techniques necessarily play a central role. In this paper, we focus on the primary nonlinear result, while tools for handling the linearized parabolic and elliptic equations are developed in our separate, companion work.