Singularities and unsteady separation for the inviscid two-dimensional Prandtl system

We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott-Smith-Cowley and Cassel-Smith-Walker. One particular profile is at the heart of the so-called Van-Dommelen and Shen singularity, and we prove its generic appearance (that is, for an open and dense set of blow-up solutions) for any prescribed Eulerian outer flow. We comment on the connection between these results and the full viscous Prandtl system. This paper combines ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation, in a novel manner, and designs a new way to study singularities for quasilinear transport equations.

Automated summary

This paper studies the inviscid unsteady Prandtl system in two dimensions and provides a sharp expression for the maximal time of existence of regular solutions. It shows that singularities occur only at the boundary or on the set of zero vorticity, corresponding to boundary layer separation. The paper also presents new Lagrangian formulae for backward self-similar profiles and explores a different approach to studying singularities for quasilinear transport equations.