On singularity formation for the two dimensional unsteady Prandtl's system

Charles Collot, Tej-Eddine Ghoul, Slim Ibrahim, Nader Masmoudi

Published 2018-08-17Version 1

We consider the two dimensional unsteady Prandtl's system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative along the transversal axis solves a closed one dimensional equation. We give a precise description of singular solutions for this reduced problem. A stable blow-up pattern and a countable family of other unstable solutions are found. The blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. The proof uses modulation techniques and different energy estimates in the various zones of interest.