Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability
Abstract
This paper studies time-dependent semi-similar Prandtl flows with outer flow U(x)=xm, where m≥1. 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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BibTeX
@misc{gao2026semisimilar,
title = {{Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability}},
author = {Chen Gao and Tao Tao and Zhouping Xin and Shing-Tung Yau and Chuankai Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25807},
eprint = {2610.25807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25807},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability
AU - Chen Gao
AU - Tao Tao
AU - Zhouping Xin
AU - Shing-Tung Yau
AU - Chuankai Zhao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25807
UR - https://arxiv.org/abs/2610.25807
AB - This paper studies time-dependent semi-similar Prandtl flows with outer flow U(x)=x^m, where m≥1. 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.
ER -