[Paper Review] Inviscid damping and enhanced dissipation of the boundary layer for 2D Navier-Stokes linearized around Couette flow in a channel

Jacob Bedrossian, Siming He|arXiv (Cornell University)|Sep 16, 2019

Fluid Dynamics and Turbulent Flows64 references27 citations

TL;DR

This paper establishes inviscid damping and enhanced dissipation for the boundary layer in 2D Navier-Stokes linearized around Couette flow in a channel with no-slip walls, showing that vorticity in the boundary layer decays uniformly in viscosity ν when initial data is compactly supported away from the boundary. The key result is a ν-independent L∞_t L1_y estimate for the boundary layer vorticity, demonstrating that the boundary layer vorticity decays at the same rate as the free (unbounded) flow, a phenomenon not previously isolated in the literature.

ABSTRACT

We study the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)^t$ in the periodic channel $\mathbb T imes [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $ u o 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O( u^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimate of the boundary layer vorticity is independent of $ u$, provided the initial data is $H^1$. For $L^2$ data, the loss is only logarithmic in $ u$. Note both such estimates are false for the vorticity in the interior. To the authors' knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected $O(\exp(-\delta u^{1/3}\alpha^{2/3}t))$ enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for $H^1$ data that is against the boundary initially. For $L^2$ data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation.

Motivation & Objective

• To analyze the long-time behavior of the 2D Navier-Stokes equations linearized around Couette flow in a periodic channel with no-slip boundary conditions in the vanishing viscosity limit.
• To understand the dynamics of the boundary layer that forms due to the mismatch between viscous and inviscid boundary conditions.
• To establish whether the boundary layer vorticity exhibits inviscid damping and enhanced dissipation, similar to the free (unbounded) flow.
• To quantify the decay rates of velocity and vorticity in the boundary layer, especially in the L2 and L∞ settings.
• To resolve the open problem of fine dynamics for long-wave instabilities in bounded shear flows.

Proposed method

• Decompose the vorticity into a free evolution (on R) and a boundary corrector localized within an O(ν^{1/3}) layer near the walls.
• Use resolvent estimates and Green's function analysis via Airy function asymptotics to control the boundary corrector.
• Apply a novel variant of the method from [10] that leverages the free flow's known inviscid damping and enhanced dissipation to estimate the boundary layer.
• Introduce a weight function ηϵ,p that captures the spatial localization and decay of the boundary layer, with ϵ = α^{-1/3}ν^{1/3}.
• Establish uniform-in-ν estimates by proving the Evans function does not vanish in the relevant region, ensuring the invertibility of the resolvent.
• Use integration by parts and asymptotic analysis of special functions (e.g., A0(C)) to derive sharp bounds on the resolvent and Green's function.

Experimental results

Research questions

• RQ1 Does the boundary layer vorticity in 2D Navier-Stokes linearized around Couette flow exhibit inviscid damping in the vanishing viscosity limit?
• RQ2 Can enhanced dissipation and inviscid damping be quantified uniformly in ν for the boundary layer, even though the interior vorticity does not satisfy such estimates?
• RQ3 Is the boundary layer vorticity localized within an O(ν^{1/3}) layer, and does it decay at a rate independent of ν?
• RQ4 What is the precise decay rate of the velocity and vorticity in the boundary layer, and how does it compare to the free flow?
• RQ5 Can the boundary corrector be estimated using the free flow's dynamics, even in the presence of boundaries?

Key findings

• For initial vorticity compactly supported away from the boundary, the boundary layer vorticity satisfies ||e^{λt} ηϵ,p ωb(t, α, ·)||_{L^∞_t L^1_y} ≲ e^{-δ'α} ||ωin||_{L^2_y} with λ = (1−κ)α^2ν + δα^{2/3}ν^{1/3}, independent of ν.
• The boundary layer vorticity exhibits inviscid damping: ||e^{λt} ηϵ,p ωb||_{L^2_t L^p_y} ≲ e^{-δ'α} ||∂y ωin||_{L^2_y} for p ∈ [1,2], with ν-independent constants.
• For L2 data, the loss is only logarithmic: ||e^{λt} ηϵ,p ωb||_{L^2_t L^p_y} ≲ ⟨ln ν⟩ e^{-δ'α} ||ωin||_{L^2_y} for p ∈ [1,2].
• The velocity in the boundary layer satisfies ||e^{λt} u_b||_{L^2_t L^2_y} ≲ e^{-δ'α} ||ωin||_{L^2_y}, showing enhanced dissipation and inviscid damping uniformly in ν.
• The boundary corrector is exponentially localized within an O(ν^{1/3}) layer, and the Evans function is non-vanishing in the relevant region, ensuring the validity of the resolvent estimates.
• This work establishes, for the first time, that the boundary layer vorticity decays uniformly in ν, a phenomenon not observed in the interior and not previously isolated in the literature.