Separation for the stationary Prandtl equation

Dalibard, Anne-Laure; Masmoudi, Nader

doi:10.1007/s10240-019-00110-z

Article

Separation for the stationary Prandtl equation

Dalibard, Anne-Laure ; Masmoudi, Nader

Publications Mathématiques de l'IHÉS, Tome 130 (2019), pp. 187-297

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 $\partial_{y} u_{| y = 0} (x) \sim C \sqrt{x^{*} - x}$ 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.

MR

DOI : 10.1007/s10240-019-00110-z

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     author = {Dalibard, Anne-Laure and Masmoudi, Nader},
     title = {Separation for the stationary {Prandtl} equation},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {187--297},
     year = {2019},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {130},
     doi = {10.1007/s10240-019-00110-z},
     mrnumber = {4028516},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-019-00110-z/}
}

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Dalibard, Anne-Laure; Masmoudi, Nader. Separation for the stationary Prandtl equation. Publications Mathématiques de l'IHÉS, Tome 130 (2019), pp. 187-297. doi: 10.1007/s10240-019-00110-z

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