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arXiv · 2610.25807

Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability

Chen Gao·Tao Tao·Zhouping Xin·Shing-Tung Yau·Chuankai Zhao
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Abstract

This paper studies time-dependent semi-similar Prandtl flows with outer flow $U(x)=x^m$, where $m\geq1$. The normal coordinate is scaled by the local boundary-layer thickness, and time by the associated diffusion scale. Every time-monotone solution in the Oleinik class extends globally in time. The Falkner--Skan boundary layer globally attracts every such unsteady semi-similar profile. The convergence rate is exponential for $m=1$ and algebraic for $m>1$. We also prove nonlinear stability of each admissible time-dependent semi-similar trajectory under small compatible perturbations that break the semi-similar ansatz. Together, these results provide a dynamical justification of the time-dependent semi-similar class within the full Prandtl theory. For semi-similar flows, the Crocco transformation reduces the problem to a one-dimensional degenerate parabolic equation. Combined with weighted barriers, comparison principles propagate time monotonicity and lead to global continuation. Zero-order convergence and a one-sided gradient estimate then produce uniform strict concavity after a transient period; this concavity supplies the damping needed for derivative decay. The same damping mechanism extends to the full ultraparabolic perturbation equation, allowing us to close the nonlinear stability estimates.

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Chen Gao, Tao Tao, Zhouping Xin, Shing-Tung Yau, Chuankai Zhao. 2026-10-19. Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability. https://arxiv.org/abs/2610.25807

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