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

Emmanuel Grenier and I have just submitted this 84-page! long paper, also posted on arxiv (arXiv:1705.05323). This work is a continuation and completion of the program (initiated in Grenier-Toan1 and Grenier-Toan2) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a stationary boundary layer profile.

In this program, we study the linear problem

$\displaystyle (\partial_t - L)\omega =0, \qquad L\omega : = \sqrt\nu \Delta \omega - U\partial_x \omega - v_2 U''$

in which ${\omega = \Delta \phi}$ denotes the fluid vorticity and ${v = \nabla^\perp \phi}$ the fluid velocity. We solve the problem with no-slip boundary conditions ${v=0}$ on ${z= 0}$. We are aimed to derive uniform estimates in the inviscid limit ${\nu \rightarrow 0}$. Observe that the fluid vorticity is unbounded, but localized, near the boundary, and therefore pointwise bounds on the Green function are needed to study the precise convolution with the boundary layer behavior.

As ${L}$ is a compact perturbation of the Laplacian ${\sqrt \nu \Delta}$ (say in the usual ${L^2}$ space), ${L}$ is sectorial, has discrete unstable spectrum, and the corresponding semigroup can be described by the Dunford’s integral:

$\displaystyle e^{L t} \omega = {1 \over 2 i \pi} \int_\Gamma e^{\lambda t} (\lambda - L)^{-1} \omega \, d \lambda$

where ${\Gamma}$ is a contour on the right of the spectrum of ${L}$.

In estimating the semigroup, we can move the contour across the discrete spectrum by adding corresponding projections on the eigenfunction. However, we cannot move the contour of integration ${\Gamma}$ across the Euler continuous spectrum (or equivalently, the phase velocity ${c}$ is near the range of ${U}$ and hence critical layers appear). In addition, there are unstable eigenvalues that exist near the critical layers and that vanish in the inviscid limit (Grenier-Guo-Toan, justifying the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among others).

One of the contributions of this paper is to carefully study the contour integral near the critical layers and thus to provide sharp bounds on the Navier-Stokes semigroup.

The first step is to study the resolvent solutions ${(\lambda-L)^{-1}}$, or equivalently, solutions to the classical Orr-Sommerfeld equations (a ${4^{th}}$ ODEs), corresponding to each wavenumber ${\alpha\in \mathbb{R}}$ and each phase velocity ${c\in \mathbb{C}}$:

$\displaystyle -\epsilon \Delta_\alpha^2 \phi + (U-c) \Delta_\alpha \phi - U'' \phi =0, \qquad \epsilon = \sqrt \nu / i\alpha,$

with zero boundary conditions on ${\phi}$ and ${\phi'}$. Here, ${\Delta_\alpha = \partial_z^2 - \alpha^2}$. We need to study the Green function of the Orr-Sommerfeld problem.

When ${c}$ is away from the range of ${U}$ (or equivalently, ${\lambda}$ is away from the continuous spectrum of Euler), solutions of Orr-Sommerfeld are regular and consist of two slow modes linked to the Rayleigh equations ${(U-c) \Delta_\alpha \phi - U'' \phi =0}$ and two fast modes linked to the Airy equations ${ -\epsilon \Delta_\alpha^2 \phi + (U-c) \Delta_\alpha \phi =0}$. This is studied carefully in Grenier-Toan1.

When ${c}$ is near the range of ${U}$, we have to deal with critical layers, points ${z=z_c}$ at which ${c = U(z_c)}$. The presence of critical layers greatly complicates the analysis of constructing Orr-Sommerfeld solutions and deriving uniform bounds for the corresponding Green function. Roughly speaking, there are two independent solutions to the Orr-Sommerfeld equations that are approximated by the Rayleigh solutions, whose solutions experience a singularity of the form ${(z-z_c)\log (z-z_c)}$. We thus need to analyze the smoothing effect of the Airy operator, and design precise function spaces to capture the ${z\log z}$ singularity near the critical layers.

Finally, to capture the unbounded vorticity near the boundary, we study the semigroup in the boundary layer norms that were developed recently in Grenier-Toan2.