[Paper Review] Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers

Emmanuel Grenier, Toan T. Nguyen|arXiv (Cornell University)|May 15, 2017

Navier-Stokes equation solutions10 references3 citations

TL;DR

This paper establishes sharp pointwise bounds on the Green function and semigroup estimates for the linearized Navier-Stokes equations near monotonic, spectrally stable boundary layer profiles, focusing on the critical layer regime where the phase speed matches the base flow velocity. Using spectral analysis and detailed study of the Orr-Sommerfeld and Rayleigh equations, it derives exponential growth bounds in time with rate $\nu^{1/4}$, resolving the instability mechanism in the inviscid limit.

ABSTRACT

This is a continuation and completion of the program (initiated in \cite{GrN1,GrN2}) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a generic stationary boundary layer profile. This is done via a spectral analysis approach and a careful study of the Orr-Sommerfeld equations, or equivalently the Navier-Stokes resolvent operator $(λ- L)^{-1}$. The earlier work (\cite{GrN1,GrN2}) treats the Orr-Sommerfeld equations away from critical layers: this is the case when the phase velocity is away from the range of the background profile or when $λ$ is away from the Euler continuous spectrum. In this paper, we study the critical case: the Orr-Sommerfeld equations near critical layers, providing pointwise estimates on the Green function as well as carefully studying the Dunford's contour integral near the critical layers. As an application, we obtain pointwise estimates on the Green function and sharp bounds on the semigroup of the linearized Navier-Stokes problem near monotonic boundary layers that are spectrally stable to the Euler equations, complementing \cite{GrN1,GrN2} where unstable profiles are considered.

Motivation & Objective

• To complete the spectral analysis program initiated in [9, 10] by addressing the critical layer case in linearized Navier-Stokes around boundary layer profiles.
• To derive pointwise estimates on the Green function and sharp semigroup bounds for the linearized Navier-Stokes operator near monotonic, spectrally stable boundary layers.
• To analyze the behavior of the Orr-Sommerfeld and Rayleigh equations in the vicinity of critical layers where the phase speed matches the base flow velocity.
• To establish uniform-in-$\nu$ bounds on the semigroup $e^{Lt}$ with exponential growth rate $\nu^{1/4}$, capturing the destabilizing effect of viscosity in the inviscid limit.
• To extend previous results on unstable profiles to the stable case, providing a complete picture of linearized Navier-Stokes dynamics near critical layers.

Proposed method

• Employ a spectral approach via the resolvent operator $(\lambda - L)^{-1}$, analyzing the Orr-Sommerfeld equation for the linearized Navier-Stokes problem.
• Study the Rayleigh equation and its solutions near critical layers using asymptotic expansions and approximate Green kernels for $\alpha \ll 1$, $\alpha \approx 1$, and $\alpha \gg 1$.
• Introduce a modified Airy operator to model the fourth-order differential structure near critical layers and derive convolution estimates and smoothing effects.
• Use Dunford's contour integral representation to express the semigroup $e^{Lt}$ and carefully deform the contour to avoid unstable eigenvalues, gaining exponential decay factors.
• Decompose the solution into slow and fast modes based on the behavior of the Green function, with distinct pointwise bounds for each.
• Apply weighted $L^1$ and Gevrey-type norms to control the vorticity and derive uniform estimates in the small viscosity limit $\nu \ll 1$.

Experimental results

Research questions

• RQ1 How do the Green function and semigroup behave for linearized Navier-Stokes near critical layers in spectrally stable boundary layers?
• RQ2 What are the precise pointwise estimates on the Green function when the phase speed $c$ equals the base flow velocity $U(z)$ at some $z$?
• RQ3 How does the addition of a small positive real part $\gamma_1\nu^{1/4}$ to the contour of integration affect the stability and decay of the semigroup?
• RQ4 What is the role of the Rayleigh equation and its solutions in capturing the critical layer behavior for small $\alpha$?
• RQ5 Can sharp $L^1$-type semigroup bounds be derived that reflect the $\nu^{1/4}$ growth rate observed in the unstable case, even for stable profiles?

Key findings

• The paper establishes pointwise bounds on the Green function for the Orr-Sommerfeld equation near critical layers, with separate estimates for slow and fast modes.
• For small $\alpha$, the Green function satisfies $|G_{\alpha,c,s}(x,y)| \leq C_0(1 + \alpha|U(x) - c|^{-1})e^{-\theta_0\alpha|x-y|}$ and $|G_{\alpha,c,f}(x,y)| \leq C_0\delta_{cr}e^{-\frac{3}{4}\nu^{-1/4}\sqrt{\gamma}|x-y|}e^{-\theta_0\sqrt{\max\{|X|,|Y|\}}|X-Y|}$.
• The semigroup $e^{Lt}$ satisfies the sharp bound $\|e^{Lt}v_0\|_X \leq C_0\nu^{-1/4}e^{\theta_0\nu^{1/4}t}\|v_0\|_X$ uniformly in $\nu \ll 1$ and for all $t > 0$, with $\theta_0 > 0$.
• The $\nu^{-1/4}$ loss in the bound arises from the singularity of $(U - c)^{-1}$ in the slow mode, which is controlled via the $\nu^{1/8}$ localization gain from the contour deformation.
• The authors derive a gain of $e^{-\theta_0|y-z|/\nu^{1/8}}$ in the integral estimates due to the addition of $\gamma_1\nu^{1/4}$ to the contour, improving decay and controlling growth.
• The semigroup decomposition $e^{L_{\alpha}t} = \mathcal{S}_{\alpha} + \mathcal{R}_{\alpha}$ allows uniform control of the slow and remainder components, with $\|\mathcal{S}_{\alpha}\| \lesssim \nu^{-1/4}e^{\gamma_1\nu^{1/4}t}$ and $\|\mathcal{R}_{\alpha}\|$ bounded in $L^1$-type norms.