@article{a39343a686e24dcc9aa63f0ae2cc7f03,

title = "On singularity formation for the two-dimensional unsteady Prandtl system around the axis",

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.",

keywords = "analyticity, blow-up, blowup rate, Prandtl{\textquoteright}s equations, self-similarity, singularity, stability",

author = "Charles Collot and Ghoul, {Tej Eddine} and Slim Ibrahim and Nader Masmoudi",

note = "Funding Information: The authors would like to thank the anonymous referees for their comments that helped improve the presentation and results of this paper. Part of this work was done while C. Collot, T.-E. Ghoul and N. Masmoudi were visiting IH{\'E}S and they thank that institution. S. Ibrahim is grateful to New York University in Abu Dhabi for hosting him. The authors thank V. T. Nguyen for helpful comments. C. Collot is supported by the ERC-2014-CoG 646650 SingWave, S. Ibrahim is partially supported by NSERC Discovery grant # 371637-2019 and N. Masmoudi is supported by NSF grant DMS-1716466, and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. Funding Information: Funding. C. Collot is supported by the ERC-2014-CoG 646650 SingWave, S. Ibrahim is partially supported by NSERC Discovery grant # 371637-2019 and N. Masmoudi is supported by NSF grant DMS-1716466, and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. Publisher Copyright: {\textcopyright} 2022 European Mathematical Society.",

year = "2022",

doi = "10.4171/JEMS/1240",

language = "English (US)",

volume = "24",

pages = "3703--3800",

journal = "Journal of the European Mathematical Society",

issn = "1435-9855",

publisher = "European Mathematical Society Publishing House",

number = "11",

}