Long-wave instabilities

This paper aims to give a detailed presentation of long-wave instabilities of shear layers for Navier-Stokes equations, and in particular to give a simple and easy-to-read presentation of the study of the Orr-Sommerfeld equation and to detail the analysis of its adjoint. Using these analyses, we prove the existence of long-wave instabilities in the cases of slowly rotating fluids, slightly compressible fluids or Navier boundary conditions, under smallness conditions.