Separation for the stationary 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 $\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}$ as $x\to x^{*}$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl equation in the domain $\{0 0\}$. 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 and comparison principle techniques to handle some of the nonlinear terms.