### 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 ∂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 {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.

Original language | English (US) |
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Pages (from-to) | 187-297 |

Number of pages | 111 |

Journal | Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques |

Volume | 130 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2019 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques*,

*130*(1), 187-297. https://doi.org/10.1007/s10240-019-00110-z