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 for its radius of analyticity.
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Charles Collot, Tej-Eddine Ghoul, Slim Ibrahim, Nader Masmoudi, On singularity formation for the two-dimensional unsteady Prandtl system around the axis. J. Eur. Math. Soc. 24 (2022), no. 11, pp. 3703–3800DOI 10.4171/JEMS/1240