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Dive into the research topics where Patrick Gérard is active.

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Featured researches published by Patrick Gérard.


Communications on Pure and Applied Mathematics | 1997

HOMOGENIZATION LIMITS AND WIGNER TRANSFORMS

Patrick Gérard; Peter A. Markowich; Norbert J. Mauser; Frédéric Poupaud

We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure. n n n nThe very general theory is illustrated by typical examples like (semi)classical limits of Schrodinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation.


Communications in Partial Differential Equations | 1991

Microlocal defect measures

Patrick Gérard

In order to study weak continuity of quadratic forms on spaces of L2 solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.A systematic use of these measures leads in particular to a compensated compactness theorem, generalizing MURATTARTARs compensated compactness to variable coefficients and GOLSELIONSPERTHAMESENTISs averaging lemma. We also obtain results on homogenization for differential operators of order I with oscillating coefficients.


American Journal of Mathematics | 1999

High frequency approximation of solutions to critical nonlinear wave equations

Hajer Bahouri; Patrick Gérard

This work is devoted to the description of bounded energy sequences of solutions to the equation (1) □u + |u|4 = 0 in [inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=01i /], up to remainder terms small in energy norm and in every Strichartz norm. The proof relies on scattering theory for (1) and on a structure theorem for bounded energy sequences of solutions to the linear wave equation. In particular, we infer the existence of an a priori estimate of Strichartz norms of solutions to (1) in terms of their energy.


Comptes Rendus De L Academie Des Sciences Serie I-mathematique | 1997

Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes

Nicolas Burq; Patrick Gérard

Resume On demontre que la condition de controle geometrique de C. Bardos, G. Lebeau et J. Rauch est une condition necessaire et suffisante pour la eontrolabilite exacte des ondes avec conditions de Dirichlet sur le bord.


Duke Mathematical Journal | 2007

Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

Nicolas Burq; Patrick Gérard; N. Tzvetkov

We give estimates for the


Journal D Analyse Mathematique | 2000

Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg

Hajer Bahouri; Patrick Gérard; Chao-Jiang Xu

L^p


Journal of Nonlinear Mathematical Physics | 2003

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

Nicolas Burq; Patrick Gérard; Nikolay Tzvetkov

norm (


Journal de Mathématiques Pures et Appliquées | 2001

Profile decomposition for the wave equation outside a convex obstacle

Isabelle Gallagher; Patrick Gérard

2leq p leq +infty


Partial differential equations and mathematical physics | 1996

A microlocal version of concentration-compactness

Patrick Gérard

) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.


Archive | 1997

Concentration effects in critical nonlinear wave equation and scattering theory

Hajer Bahouri; Patrick Gérard

In this paper, we prove dispersive and Strichartz inequalities on the Heisenberg group. The proof involves the analysis of Besov-type spaces on the Heisenberg group.

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Nicolas Burq

University of Paris-Sud

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Frédéric Poupaud

University of Nice Sophia Antipolis

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Nikolay Tzvetkov

Institut Universitaire de France

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Norbert J. Mauser

Technical University of Berlin

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Norbert J. Mauser

Technical University of Berlin

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