Boundary-layer separation beyond the square-root law

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 $\ell \ge 1$, 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 proposed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We prove the nonlinear stability of the square-root separation law ($\ell =1$) under small admissible perturbations of the initial data in a suitable topology.