Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

Brunovský, Pavol; Joly, Romain; Raugel, Geneviève

doi:10.1007/s10884-019-09813-7

Cited by 2 publications

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References 60 publications

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“…We emphasize that genericity of both hyperbolicity and the Morse-Smale property has been proved for scalar unidimensional semilinear equations, which respectively imply the local and global stability of the dynamics with respect to perturbations of the system. See [4,31,41,48,7]. These results should remain true for fully nonlinear equations.…”

Section: Proof Of Main Results 21 Backgroundmentioning

confidence: 93%

Sturm attractors for fully nonlinear parabolic equations

Lappicy¹

2021

Preprint

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We explicitly construct global attractors of fully nonlinear parabolic equations. The attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and forwards in time). In particular, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria, which is accompanied by a method that compute such conditions.

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Section: Proof Of Main Results 21 Backgroundmentioning

confidence: 93%

Sturm attractors for fully nonlinear parabolic equations

Lappicy¹

2021

Preprint

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“…Following a research program initiated with Jack Hale -the generalization to PDEs of known results for differential equations in finite dimension -she strove to describe qualitatively the dynamics of generic PDEs. With Pavol Brunovský and Romain Joly, she was the first to establish the genericity of the Morse-Smale property for a damped wave equation or for a PDE without gradient structure [35][36][37][38]. On the other hand, with Nicolas Burq and Wilhelm Schlag [39], she obtained a dichotomy between finite time blow-up and convergence to an equilibrium for a damped Klein-Gordon equation, thus improving on what was known by the classical methods of PDE analysis only.…”

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confidence: 99%

In Memoriam: Geneviève Raugel

2022

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Geneviève Raugel was born in 1951 in Strasbourg. She was the second child of Marguerite Huchler, a housewife, and Henri Michel Raugel, a school teacher. Strasbourg is the capital of Alsace and one of the major cities in Eastern France, at the meeting point of French and German cultural influences in the heart of Europe.Geneviève spent her childhood and adolescence in Strasbourg. She attended the Lycée International des Pontonniers and, after her baccalaureate, the preparatory class at the Lycée Kléber. In 1972, she left Strasbourg to enter the École Normale Supérieure of Fontenay-aux-Roses, near Paris. This distinguished graduate school, founded in 1880, was still reserved for women at the time Geneviève was admitted. Like her, many female mathematicians have studied in this school, which eventually became co-educational in 1981.In 1972, shortly after she began her studies in the Paris area, Geneviève met her future husband Gérard Laumon. They got married in 1980 and settled permanently in 1988 in Fontenay-aux-Roses, in a nice apartment in the city center located 500 m away from the former ÉNS.After her studies at the ÉNS, Geneviève joined the CNRS as an associate researcher and was assigned to the IRMAR laboratory at the University of Rennes 1. There, she began her research activity under the direction of Michel Crouzeix. Her PhD thesis, defended in 1978, is devoted to the numerical analysis of partial differential equations and is entitled Résolution numérique de problèmes elliptiques dans des domaines avec coins (Numerical resolution of elliptic problems in domains with corners). Still under the direction of M. Crouzeix, she defended a thèse d'État (equivalent to habilitation) on the numerical approximation of nonlinear problems in 1984, one year after being promoted to a permanent research position at the CNRS. Two years later, she was appointed to the Centre de Mathématique de l'École Polytechnique, in Palaiseau, and in 1989 she joined the department of mathematics at Orsay, where she worked as a research director until she passes away in 2019.

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Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

Cited by 2 publications

References 60 publications

Sturm attractors for fully nonlinear parabolic equations

Sturm attractors for fully nonlinear parabolic equations

In Memoriam: Geneviève Raugel

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