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Separation for the stationary Prandtl equation

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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 \(x^{*}>0\) 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 \(\{0< x< x^{*},\ y> 0\}\). 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.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, LJLL, Sorbonne Université, Université Paris-Diderot SPC, CNRS, 75005, Paris, France

    Anne-Laure Dalibard

  2. NYUAD campus, Saadiyat Island Abu Dhabi, UAE

    Nader Masmoudi

  3. Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, United States

    Nader Masmoudi

Authors
  1. Anne-Laure Dalibard
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  2. Nader Masmoudi
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Dalibard, AL., Masmoudi, N. Separation for the stationary Prandtl equation. Publ.math.IHES 130, 187–297 (2019). https://doi.org/10.1007/s10240-019-00110-z

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  • DOI: https://doi.org/10.1007/s10240-019-00110-z

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