Publications mathématiques de l'IHÉS

, Volume 130, Issue 1, pp 187–297 | Cite as

Separation for the stationary Prandtl equation

  • Anne-Laure Dalibard
  • Nader Masmoudi


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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Copyright information

© IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Anne-Laure Dalibard
    • 1
  • Nader Masmoudi
    • 2
    • 3
  1. 1.Laboratoire Jacques-Louis Lions, LJLLSorbonne Université, Université Paris-Diderot SPC, CNRSParisFrance
  2. 2.NYUAD campusSaadiyat Island Abu DhabiUAE
  3. 3.Courant InstituteNew York UniversityNew YorkUnited States

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