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Documents  Gérard, Patrick | enregistrements trouvés : 9

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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations

The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.
The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform ...

35B40 ; 35B15 ; 35Q55 ; 37K15 ; 47B35

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Research talks;Partial Differential Equations

The one dimensional half wave equation is an interesting example of a nonlinear wave equation with vanishing dispersion, displaying arbitrarily small mass solitons. I will discuss how, in some resonant regime, the interaction of two such solitons leads to long time transition to high frequencies.
This talk is issued from a jointwork with Enno Lenzmann, Oana Pocovnicu and Pierre Raphael.

35Qxx

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Research talks;Partial Differential Equations

Inspired by a recent result of Dodson-Luhrmann-Mendelson, who proved almost sure scattering for the energy-critical wave equation with radial data in four dimensions, we establish the analogous result for the Schrödinger equation.
This is joint work with R. Killip and J. Murphy.

35Q55 ; 35L05 ; 35R60

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Research talks;Partial Differential Equations;Mathematical Physics

It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of an energy impulsion tensor in the equation for the background metric. This non trivial contribution is of due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric. It has been conjectured by Burnett that the only tensors which can be obtained this way are massless Vlasov, and it has been proved by Green and Wald that the limit tensor must be traceless and satisfy the dominant energy condition. The known exemples of this phenomena are constructed under symmetry reductions which involve two Killing fields and lead to an energy impulsion tensor which consists in at most two dust fields propagating in null directions. In this talk, I will explain our construction, under a symmetry reduction involving one Killing field, which leads to an energy impulsion tensor consisting in N dust fields propagating in arbitrary null directions. This is a joint work with Jonathan Luk (Stanford). It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of an energy impulsion tensor in the equation for the background metric. This non trivial contribution is of due to the nonlinearities in Einstein equations, which involve products of derivatives ...

35Q75 ; 53C80 ; 83C05

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Research talks;Partial Differential Equations;Mathematical Physics

The talk will discuss a recent result showing that certain type II blow up solutions constructed by Krieger-Schlag-Tataru are actually stable under small perturbations along a co-dimension one Lipschitz hypersurface in a suitable topology. This result is qualitatively optimal.
Joint work with Stefano Burzio (EPFL).

35L05 ; 35B40

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Research talks;Partial Differential Equations;Mathematical Physics

Choosing favourable gauges is a crucial step in the study of nonlinear geometric dispersive equations. A very successful tool, that has emerged originally in work of Tao on wave maps, is the use of caloric gauges, defined via the corresponding geometric heat flows. The aim of this talk is to describe two such flows and their associated gauges, namely the harmonic heat flow and the Yang-Mills heat flow.

70S15 ; 35Q53 ; 35Q55

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- 188 p.
ISBN 978-2-86883-363-1

Savoir actuels

Localisation : Ouvrage RdC (ALIN)

opérateur pseudo-différentiel # théorème de Nash-Moser # analyse microlocale # théorie de Littlewood-Paley # inégalité d'énergie pour les équations hyperboliques # théorème de fonction implicite # estimation d'énergie

35-02 ; 35Sxx ; 47G30 ; 47H15 ; 47N20 ; 58G15

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- 168 p.
ISBN 978-0-8218-3454-1

Graduate studies in mathematics , 0082

Localisation : Collection 1er étage

opérateur pseudodifférentiel # fonction implicite # théorème de Nash-Moser # analyse microlocale # théorie de Littlewood-Paley # estimation d'énergie

35-02 ; 35Sxx ; 47G30 ; 47N20

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- vi; 112 p.
ISBN 978-2-85629-854-1

Astérisque , 0389

Localisation : Périodique 1er étage

équation de Szegö cubique # système intégrable # équation de Schrödinger non linéaire # opérateur de Hankel # analyse spectrale

35B15 ; 47B35 ; 37K15

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