Year: 2026
Type: preprint
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 $\ell\ge1$, we construct smooth initial data for which separation occurs at a finite location $x=x^\ast$, with \[ \partial_y u(x,0) =\mathbf C_\ell(x^\ast-x)^{\ell/2}\bigl(1+o(1)\bigr), \qquad x\uparrow x^\ast, \] where $\mathbf C_\ell>0$ 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 $\ell=1$, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: 流体方程式における不安定性の励起・抑制メカニズムの数学解析, Deepening the Mathematical Theory of Plasma Boundary Layers, Asymptotic analysis of spatio-temporal nonuniform dynamic in conservation laws of quasilinear hyperbolic-parabolic systems, Evolution equations describing singularities of non-equilibrium systems beyond the linear-response regime and development of nonlocal-nonlinear analysis, Mathematical Analysis of Multilayer Structures in Fluid Equations
OpenAlex ID: W7216195001