Singularities, separation, and generic self-similar behaviour for the inviscid unsteady Prandtl boundary layer

Abstract: The inviscid unsteady Prandtl system in two dimensions describes an incompressible non viscous fluid close to a boundary. First, we will prove that the boundary layer separates off the wall if and only if the solution becomes singular away from it. Second, we will present a method to find explicitly backward self-similar solutions forming finite time singularities. Finally, we will show that one of such self-similar solution is the attractor for singular solutions near blow-up time, in a generic sense (for a dense an open set). This explains the generic appearance of the so-called Van Dommelen and Shen singularity, and describes completely and rigorously the associated separating structure. The talk will combine ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation. This is joint work with T.-E. Ghoul and N. Masmoudi.

analysis of PDEs

Audience: researchers in the topic

Open PDE and analysis seminar and lectures