Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1097497930, G7153082605
OpenAlex ID: W7222996719