In this talk, I will describe some recent results about the separation phenomenon within the stationary Prandtl equation.
The Prandtl equation describes the behavior of a fluid with small viscosity close to the boundary. In the presence of an adverse pressure gradient, it has been observed experimentally that the boundary layer may detach itself from the boundary. This separation has been described in a formal way by Goldstein in 1958. 
With Nader Masmoudi, we gave a rigorous proof of the validity of the Goldstein singularity, and a precise, quantitative description of the behaviour of the solution in the vicinity of the separation point.
I will also tackle some issues related to the well-posedness of the Prandtl system in the presence of a recirculation bubble, i.e. downstream of the separation point. This topic is a work in progress with Frédéric Marbach and Jean Rax.

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will focus on semilinear cases (and therefore we will not address the Prandtl equation in these lectures). We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid… We will also tackle some semilinear problems, adding for instance a convection term to the previous examples.