Separation for the stationary Prandtl equation

TL;DR

This paper proves that the stationary Prandtl equation exhibits flow separation under adverse pressure gradients, demonstrating the Goldstein singularity and providing detailed analysis of boundary layer behavior.

Contribution

It establishes the occurrence of separation for the stationary Prandtl equation with adverse pressure gradient and boundary data, confirming the Goldstein singularity with rigorous energy and maximum principle methods.

Findings

Flow separation occurs at a finite point x*

The boundary shear rate behaves like a square root near separation

The proof introduces a new formulation and energy estimates for the Prandtl equation

Abstract

In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at $x=0$.We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at $x=0$, there exists $x^*>0$ such that $\p\_y u\_{y=0}(x)\sim C \sqrt{x^* -x}$ as $x\to x^*$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl equation in the domain $\{0<x<x^*,\ y>0\}$. Our proof relies on three main ingredients: the computation of a "stable" approximate solution, using modulation theory arguments, a new formulation of the Prandtl equation, for which we derive energy estimates, relying heavily on the structure of the equation, and maximum principle techniques to handle nonlinear terms.