Local separation profiles for the stationary Prandtl system
Abstract
We construct solutions to the two-dimensional stationary Prandtl system under a constant adverse pressure gradient exhibiting a family of distinct separation laws. More precisely, for every integer ℓ≥1, we construct smooth initial data for which separation occurs at a finite location x=x∗, with ∂y u(x,0) =Cℓ(x∗−x)ℓ/2(1+o(1)), x↑ x∗, where Cℓ>0 depends on the initial data. We determine the local asymptotic structure of the solution near separation through matched inner and outer expansions based on linearization around the ground state in von Mises variables. This description addresses Open Problem~5 posed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We also prove that the square-root separation regime, corresponding to ℓ=1, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.
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BibTeX
@misc{duong2026local,
title = {{Local separation profiles for the stationary Prandtl system}},
author = {Ky Giao Duong and Tej-Eddine Ghoul and Sameer Iyer and Yasunori Maekawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01689},
eprint = {2610.01689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01689},
}
RIS
TY - UNPB
TI - Local separation profiles for the stationary Prandtl system
AU - Ky Giao Duong
AU - Tej-Eddine Ghoul
AU - Sameer Iyer
AU - Yasunori Maekawa
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01689
UR - https://arxiv.org/abs/2610.01689
AB - We construct solutions to the two-dimensional stationary Prandtl system under a constant adverse pressure gradient exhibiting a family of distinct separation laws. More precisely, for every integer ℓ≥1, we construct smooth initial data for which separation occurs at a finite location x=x^∗, with ∂_y u(x,0) =C_ℓ(x^∗−x)^ℓ/2(1+o(1)), x↑ x^∗, where C_ℓ>0 depends on the initial data. We determine the local asymptotic structure of the solution near separation through matched inner and outer expansions based on linearization around the ground state in von Mises variables. This description addresses Open Problem~5 posed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We also prove that the square-root separation regime, corresponding to ℓ=1, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.
ER -