TY - JOUR

T1 - Separation for the stationary Prandtl equation

AU - Dalibard, Anne Laure

AU - Masmoudi, Nader

N1 - Publisher Copyright:
© 2019, IHES and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - 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 ∂yu|y=0(x)∼Cx∗−x as x→ x∗ for some positive constant C, where u is the solution of the stationary Prandtl equation in the domain {00}. 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.

AB - 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 ∂yu|y=0(x)∼Cx∗−x as x→ x∗ for some positive constant C, where u is the solution of the stationary Prandtl equation in the domain {00}. 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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U2 - 10.1007/s10240-019-00110-z

DO - 10.1007/s10240-019-00110-z

M3 - Article

AN - SCOPUS:85073798128

SN - 0073-8301

VL - 130

SP - 187

EP - 297

JO - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

JF - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

IS - 1

ER -