On singularity formation for the two-dimensional unsteady Prandtl system around the axis
type: Item
instance of: scholarly article
title: On singularity formation for the two-dimensional unsteady Prandtl system around the axis
zbMATH Open document ID: 1501.35312
DOI: 10.4171/JEMS/1240
author: Tej-Eddine Ghoul
author: Slim Ibrahim
author: Charles Collot
author: Nader Masmoudi
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1808.05967
review text: Summary: 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.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D10
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35B36
Mathematics Subject Classification ID: 35C06
zbMATH DE Number: 7608241
zbMATH Keywords: Prandtl's equations
zbMATH Keywords: blow-up
zbMATH Keywords: singularity
zbMATH Keywords: self-similarity
zbMATH Keywords: stability
zbMATH Keywords: analyticity
zbMATH Keywords: blow-up rate
MaRDI profile type: Publication
arXiv ID: 1808.05967
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description: scientific article