On singularity formation for the two dimensional unsteady Prandtl system around the axis

TL;DR

This paper analyzes singularity formation in the 2D unsteady Prandtl system, identifying stable blow-up patterns and providing bounds on analyticity radius near blow-up.

Contribution

It introduces a detailed description of singular solutions and a universal lower bound for analyticity radius in the Prandtl system.

Findings

Stable blow-up pattern with blow-up point ejected to infinity

Persistence of analyticity up to blow-up time

Universal lower bound of (T-t)^{7/4} for analyticity radius

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.