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

## TL;DR

This paper analyzes the inviscid 2D Prandtl system, providing sharp criteria for singularity formation, especially boundary layer separation, and introduces new Lagrangian formulas and generic blow-up profiles.

## Contribution

It offers a precise characterization of singularity onset in the inviscid Prandtl system and introduces novel Lagrangian methods and self-similar profiles for understanding boundary layer separation.

## Key findings

- Singularities occur only at the boundary or zero vorticity set.
- A specific self-similar profile is generically observed in blow-up solutions.
- The results connect boundary layer separation with the Van-Dommelen and Shen singularity.

## Abstract

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 connexion 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.