Green function for linearized Navier-Stokes around a boundary shear layer profile for long wavelengths

TL;DR

This paper derives pointwise estimates for the Green function of the Orr-Sommerfeld equations in the context of long wavelength perturbations around a boundary shear layer profile, advancing understanding of stability in fluid flows.

Contribution

It provides new pointwise Green function estimates for long wavelength perturbations near the instability threshold in boundary shear flows.

Findings

Green function estimates for long wavelengths are established.

Results clarify stability boundaries for monotonic shear profiles.

Advances in understanding linearized Navier-Stokes stability analysis.

Abstract

This paper is the continuation of a program, initiated in Grenier-Nguyen [8,9], to derive pointwise estimates on the Green function of Orr Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely horizontal wavenumbers $\alpha$ of order $\nu^{1/4}$, which correspond to the lower boundary of the instability area for monotonic profiles.