Separation for the stationary Prandtl equation
Résumé
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 $\{00\}$. 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.
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