Separation for the stationary Prandtl equation

2018/02/12 by Dalibard, Anne-Laure, Masmoudi, Nader · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.04039

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)∼ C √(x^* -x) as x→ 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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