Singularities of Rayleigh equation

The Rayleigh equation, which is the linearized Euler equations near a shear flow in vorticity formulation, is a key ingredient in the study of the long time behavior of solutions of linearized Euler equations, in the study of the linear stability of shear flows for Navier–Stokes equations and in particular in the construction of the so called Tollmien–Schlichting waves. It is also a key ingredient in the study of vorticity depletion. In this article we locally describe the solutions of Rayleigh equation near critical points of any order of degeneracy, link their values on the boundary with their behaviors at infinity and describe the Green function of Rayleigh equation. By combining these various results, we can get an accurate description of the solution of the Rayleigh equation with an arbitrary given forcing term.