Separation for the stationary Prandtl equation

Dalibard, Anne-Laure; Masmoudi, Nader

doi:10.1007/s10240-019-00110-z

Geodesic

Parcourir par

Article

Separation for the stationary Prandtl equation

Dalibard, Anne-Laure ; Masmoudi, Nader

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

Voir la notice de l'article provenant de la source Numdam

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.

DOI : 10.1007/s10240-019-00110-z

@article{PMIHES_2019__130__187_0,
     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},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {130},
     year = {2019},
     doi = {10.1007/s10240-019-00110-z},
     mrnumber = {4028516},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-019-00110-z/}
}

TY  - JOUR
AU  - Dalibard, Anne-Laure
AU  - Masmoudi, Nader
TI  - Separation for the stationary Prandtl equation
JO  - Publications Mathématiques de l'IHÉS
PY  - 2019
SP  - 187
EP  - 297
VL  - 130
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10240-019-00110-z/
DO  - 10.1007/s10240-019-00110-z
LA  - en
ID  - PMIHES_2019__130__187_0
ER  -

%0 Journal Article
%A Dalibard, Anne-Laure
%A Masmoudi, Nader
%T Separation for the stationary Prandtl equation
%J Publications Mathématiques de l'IHÉS
%D 2019
%P 187-297
%V 130
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://geodesic.mathdoc.fr/articles/10.1007/s10240-019-00110-z/
%R 10.1007/s10240-019-00110-z
%G en
%F PMIHES_2019__130__187_0

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

Cité par Sources :