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## Green function for linearized Navier-Stokes around boundary layers: away from critical layers

I’ve just submitted this new paper with E. Grenier (ENS de Lyon) on arxiv (scheduled to announce next Tuesday 1:00GMT), in which we construct the Green function for the classical Orr-Sommerfeld equations and derive sharp semigroup bounds for linearized Navier-Stokes equations around a boundary layer profile. This is part of the long program to understand the stability of classical Prandtl’s layers appearing in the inviscid limit of incompressible Navier-Stokes flows.

More precisely, let ${\nu}$ be the small viscosity and let ${L_\nu}$ be the linearized Navier-Stokes operator around a stationary boundary layer ${U = [U(y),0]}$ on the half-line ${y\ge 0}$, together with the classical no-slip boundary condition. Naturally, there are two cases: either ${U(y)}$ is spectrally stable or spectrally unstable to the corresponding Euler operator ${L_0}$. In this paper, we consider the unstable case, giving the existence of the maximal unstable eigenvalue ${\lambda_0}$, with ${\Re \lambda_0>0}$.

Our first main result in this paper is roughly as follows:

Theorem 1 There holds the sharp semigroup bound:

$\displaystyle \| e^{L_\nu t} \|_{\eta} \le C_\epsilon e^{( \Re \lambda_0 + \epsilon) t} , \qquad \forall~\epsilon>0$

uniformly in time ${t\ge 0}$ and uniformly in the inviscid limit ${\nu \rightarrow0}$. Here, ${\|\cdot \|_\eta}$ denotes some exponentially weighted ${L^\infty}$ norm.

Certainly, the key difficulty in such a theorem is to derive sharp bounds in term of time growth and uniform estimates in the inviscid limit. Standard energy estimates yield precisely a semigroup bound of order ${e^{\| \nabla U\|_{L^\infty} t}}$, which is far from being sharp.

In fact, such a sharp semigroup bound is a byproduct of our much delicate study on the Orr-Sommerfeld problem, the resolvent equations of the linearized Navier-Stokes. In order to accurately capture the behavior of (unbounded) vorticity on the boundary, we are obliged to derive pointwise estimates on the corresponding Green function. We follow the seminal approach of Zumbrun-Howard developed in their study of stability of viscous shocks in the system of conservation laws.

That is to say, our second main result of this paper is to provide uniform bounds on the Green function of the classical Orr-Sommerfeld problem and derive pointwise bounds on the corresponding Green function of Navier-Stokes.  This paper is the first in our program of deriving sharp semigroup bounds for Navier-Stokes around a boundary layer profile.