Local separation profiles for the stationary Prandtl system

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 $\mathrm{\ell }\ge 1$, we construct smooth initial data for which separation occurs at a finite location $x=x^{\ast }$, with $${\mathrm{\partial }}_{y}u(x,0)={\mathbf{C}}_{\mathrm{\ell }}(x^{\ast }-x{)}^{\mathrm{\ell }\mathrm{/}2}(1+o(1)),x\uparrow x^{\ast },$$ where ${\mathbf{C}}_{\mathrm{\ell }}>0$ 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 $\mathrm{\ell }=1$, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.