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Fourier Analysis For/And Partial Differential Equations

Description du projet

Analyse harmonique à l’interface de l’analyse de Fourier et des équations aux dérivées partielles

Les harmoniques sont des oscillations d’un signal dans le temps avec une fréquence qui est un multiple entier positif de la fréquence dite fondamentale. Parmi les exemples de mouvement harmonique simple, on peut citer un pendule oscillant et une corde de guitare ou un tympan vibrant. L’analyse de ce phénomène décompose le mouvement harmonique, en déterminant les harmoniques individuelles qui, ensemble, créent la forme d’onde finale. L’analyse de Fourier, l’expression d’une onde complexe comme une somme de sinus et de cosinus, et les équations aux dérivées partielles (EDP) constituent des outils essentiels dans l’étude des harmoniques. Le projet FAnFArE, financé par le Conseil européen de la recherche, étudiera les problèmes à l’interface de l’analyse de Fourier et des EDP, en faisant progresser de manière systématique les théories relatives aux fréquences, aux oscillations et aux résonances spatio-temporelles.

Objectif

"This project aims to develop the field of Harmonic Analysis, and more precisely to study problems at the interface between Fourier Analysis and PDEs (and also some Geometry).
We are interested in two aspects of the Fourier Analysis :

(1) The Euclidean Fourier Analysis, where a deep analysis can be performed using specificities as the notion of ``frequencies'' (involving the Fourier transform) or the geometry of the Euclidean balls. By taking advantage of them, this proposal aims to pursue the study and bring novelties in three fashionable topics : the study of bilinear/multilinear Fourier multipliers, the development of the ``space-time resonances'' method in a systematic way and for some specific PDEs, and the study of nonlinear transport equations in BMO-type spaces (as Euler and Navier-Stokes equations).

(2) A Functional Fourier Analysis, which can be performed in a more general situation using the notion of ``oscillation'' adapted to a heat semigroup (or semigroup of operators). This second Challenge is (at the same time) independent of the first one and also very close. It is very close, due to the same point of view of Fourier Analysis involving a space decomposition and simultaneously some frequency decomposition. However they are quite independent because the main goal is to extend/develop an analysis in the more general framework given by a semigroup of operators (so without using the previous Euclidean specificities). By this way, we aim to transfer some results known in the Euclidean situation to some Riemannian manifolds, Fractals sets, bounded open set setting, ... Still having in mind some applications to the study of PDEs, such questions make also a connexion with the geometry of the ambient spaces (by its Riesz transform, Poincaré inequality, ...). I propose here to attack different problems as dispersive estimates, ""L^p""-version of De Giorgi inequalities and the study of paraproducts, all of them with a heat semigroup point of view."

Régime de financement

ERC-STG - Starting Grant

Institution d’accueil

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Contribution nette de l'UE
€ 940 540,34
Adresse
RUE MICHEL ANGE 3
75794 Paris
France

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Région
Ile-de-France Ile-de-France Paris
Type d’activité
Research Organisations
Liens
Coût total
€ 940 540,34

Bénéficiaires (1)