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

This paper is the continuation of a program, initiated in Grenier and Nguyen [SIAM J. Math. Anal. 51 (2019); J. Differential Equations 269 (2020)], to derive pointwise estimates on the Green function of Orr–Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely horizontal wave numbers \alpha of order \nu^{1/4} , which correspond to the lower boundary of the instability area for monotonic profiles.