The asymptotic structure of gravity in the asymptotically flat context will be described at spatial infinity by Hamiltonian methods. The first lecture will provide the main ideas, give the Poisson bracket algebra of the BMS charges and propose a supertranslation-invariant definition of the angular momentum.  The second lecture will discuss the connection with null infinity and provide in particular a justification of Strominger’s matching conditions of the fields between past null infinity and future null infinity.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

The asymptotic structure of gravity in the asymptotically flat context will be described at spatial infinity by Hamiltonian methods. The first lecture will provide the main ideas, give the Poisson bracket algebra of the BMS charges and propose a supertranslation-invariant definition of the angular momentum.  The second lecture will discuss the connection with null infinity and provide in particular a justification of Strominger’s matching conditions of the fields between past null infinity and future null infinity.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications1ère lecture d’un cours intitulé : « Une introduction aux singularités sur critiques ».Les lectures suivantes auront lieu les 14, 15 et 19 mars de 10h30 à 12h30 dans l’Amphithéâtre Léon Motchane :Lecture 2 : Explosion de type front.Lecture 3 : Sur les solutions auto similaires d’Euler compressible.Lecture 4 : Stabilité du mécanisme d’implosion.

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications 

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.