## Abstract

We consider the two-dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a closed one-dimensional equation. First, we give a precise description of singular solutions for this reduced problem. A stable blow-up pattern is found, in which the blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. Second, in the case where, for a general analytic solution, this trace of the derivative on the axis follows the stable blow-up pattern, we show persistence of analyticity around the axis up to the blow-up time, and establish a universal lower bound of .T - t /^{7=4} for its radius of analyticity.

Original language | English (US) |
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Pages (from-to) | 3703-3800 |

Number of pages | 98 |

Journal | Journal of the European Mathematical Society |

Volume | 24 |

Issue number | 11 |

DOIs | |

State | Published - 2022 |

## Keywords

- Prandtl’s equations
- analyticity
- blow-up
- blowup rate
- self-similarity
- singularity
- stability

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics