Separation for the stationary Prandtl equation

DOI10.1007/S10240-019-00110-ZzbMATH Open1427.35201arXiv1802.04039OpenAlexW2972263909WikidataQ127300636 ScholiaQ127300636MaRDI QIDQ2335505

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Publication date: 14 November 2019

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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 $x^{\ast }>0$ such that $p_y{u}_{y=0}(x)simCsqrt{x}^{\ast }-x$ as $xo{x}^{\ast }$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl equation in the domain $0<x<x^{\ast },y>0$. 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.

Full work available at URL: https://arxiv.org/abs/1802.04039

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