Long-wave instabilities

This paper aims to give a detailed presentation of long-wave instabilities of shear layers for NavierStokes equations,and in particular to give a simple and easy-to-read presentation of the study of the OrrSommerfeld 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 fuids,slightly compressible fuids,or fuids with Navier boundary conditions,under smallness conditions.