Separation for the stationary Prandtl equation

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 ∂yu|y=0(x)∼Cx∗−x as x→ 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 and comparison principle techniques to handle some of the nonlinear terms.