On the uniqueness of probability measure solutions to Liouville's equation of Hamiltonian PDEs.

paru sous le titre : On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEsInternational audienceIn this paper, we give a uniqueness result to a transport equation fulfilled by probability measure on a infinite dimensional Hilbert space. Main arguments are based on projective aspects and a probabilistic representation of the solutions. It extends the work of Maniglia, which concerns the finite dimensional case and the work of Ammari and Nier, for a wider class of velocity field

Dérivation des équations de Schrödinger non linéaires par une méthode des caractéristiques en dimension infinie

In this thesis, we justify the mean field approximation in a general framework for bosonic systems. The derivation of mean field dynamics is known for some specific quantum states. Therefore it is natural to expect the extension of these results for a general family of normal states. The mean field approximation for bosons consists in replacing the many-body quantum problem by a non linear one, so-called Hartree problem, when the number of particles tends to infinity. We establish a general result for bosons confined or not, interacting through a singular potential. The method used is based on Wigner measures. Our contribution consists in extending the characteristics method when the velocity field associated to the Hartree equation is subcritical or critical. It complements the work of Ammari and Nier and provides a result for critical potential for the Hartree equation. We also focus on bosonic systems interacting through a multi-body potential and we prove the mean field approximation under a strong assumption on this potential. All these results essentially rely on the flexibility of Wigner measures and we can give an alternative proof of the variational mean field approximation.Dans cette thèse, nous aborderons l'approximation de champ moyen pour des particules bosoniques. Pour un certain nombre d'états quantiques, la dérivation de la limite de champ moyen est connue, et il semble naturel d'étendre ces travaux à un cadre général d'états quantiques quelconques. L'approximation de champ moyen consiste à remplacer le problème à N corps quantique par un problème non linéaire, dit de Hartree, quand le nombre de particules est grand. Nous prouverons un résultat général pour un système de particules, confinées ou non, interagissant au travers d'un potentiel singulier. La méthode utilisée repose sur les mesures de Wigner. Notre contribution consiste en l'extension de la méthode des caractéristiques au cadre de champ de vitesse singulier associé à l'équation de Hartree. Cela complète les travaux d'Ammari et Nier et permet de prouver des résultats pour des potentiels critiques pour les équations de Hartree. En particulier, on s'intéressera à un système de bosons interagissant au travers d'un potentiel à plusieurs corps et nous démontrerons l'approximation de champ moyen sous une hypothèse de compacité forte sur ce dernier. Les résultats s’appuient en grande partie sur la flexibilité des mesures de Wigner, ce qui permet également de proposer une preuve alternative à l'approximation de champ moyen dans un cadre variationnel

Mean field limit for Bosons with compact kernels interactions by Wigner measures transportation

We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results in the subject.Comment: 27 pages. arXiv admin note: text overlap with arXiv:1111.5918 by other author

On the mean field approximation of many-boson dynamics

We show under general assumptions that the mean-field approximation for quantum many-boson systems is correct. Our contribution unifies and improves on most of the known results. The proof uses general properties of quantization in infinite dimensional spaces, phase-space analysis and measure transportation techniques

Database of pleistocene periglacial featuresin France: description of the online version

A database of Pleistocene periglacial features in France has been compiled from a review of academic literature and reports of rescue archaeology, the analysis of aerial photographs and new field surveys. Polygons, soil stripes, ice-wedge pseudomorphs, sand wedges and composite wedge pseudomorphs are included in the database together with their geographic coordinates, geological context, description and associated references. It is hoped that this database, which aim is to be integrated in broader studies, will stimulate further work on past permafrost reconstruction and will favour greater understanding of the climatic events that lead to the formation of the periglacial features. The database is available online on the AFEQ-CNF INQUA website (https://afeqeng.hypotheses.org/487). A folder that contains photographs and sketches of the features is also available on request.Une base de données des structures périglaciaires pléistocènes de France a été créée à partir d’une revue de la littérature scientifique, de rapports d’archéologie préventive, de l’analyse de photographies aériennes et de nouvelles prospections de terrain. Les polygones, les sols striés, les pseudomorphoses de coin de glace, les coins de sable et les pseudomorphoses de coin composite ont été répertoriés dans la base de données avec leurs coordonnées géographiques, le contexte géologique, leur description et les références bibliographiques associées. Nous espérons que cette base de données, dont le but est d’être intégrée dans des études plus larges, stimulera de prochains travaux sur la reconstitution du pergélisol pléistocène et favorisera une plus grande compréhension des événements climatiques qui ont conduit à la formation de ces structures périglaciaires. La base de données est disponible en ligne sur le site de l’AFEQ-CNF INQUA (https://afeqeng.hypotheses.org/487). Un dossier contenant les photographies et dessins des structures périglaciaires est également disponible sur demande

Derivation of the non linear Schrödinger equations by the characteristics method in a infinite dimensional space

Dans cette thèse, nous aborderons l'approximation de champ moyen pour des particules bosoniques. Pour un certain nombre d'états quantiques, la dérivation de la limite de champ moyen est connue, et il semble naturel d'étendre ces travaux à un cadre général d'états quantiques quelconques. L'approximation de champ moyen consiste à remplacer le problème à N corps quantique par un problème non linéaire, dit de Hartree, quand le nombre de particules est grand. Nous prouverons un résultat général pour un système de particules, confinées ou non, interagissant au travers d'un potentiel singulier. La méthode utilisée repose sur les mesures de Wigner. Notre contribution consiste en l'extension de la méthode des caractéristiques au cadre de champ de vitesse singulier associé à l'équation de Hartree. Cela complète les travaux d'Ammari et Nier et permet de prouver des résultats pour des potentiels critiques pour les équations de Hartree. En particulier, on s'intéressera à un système de bosons interagissant au travers d'un potentiel à plusieurs corps et nous démontrerons l'approximation de champ moyen sous une hypothèse de compacité forte sur ce dernier. Les résultats s’appuient en grande partie sur la flexibilité des mesures de Wigner, ce qui permet également de proposer une preuve alternative à l'approximation de champ moyen dans un cadre variationnel.In this thesis, we justify the mean field approximation in a general framework for bosonic systems. The derivation of mean field dynamics is known for some specific quantum states. Therefore it is natural to expect the extension of these results for a general family of normal states. The mean field approximation for bosons consists in replacing the many-body quantum problem by a non linear one, so-called Hartree problem, when the number of particles tends to infinity. We establish a general result for bosons confined or not, interacting through a singular potential. The method used is based on Wigner measures. Our contribution consists in extending the characteristics method when the velocity field associated to the Hartree equation is subcritical or critical. It complements the work of Ammari and Nier and provides a result for critical potential for the Hartree equation. We also focus on bosonic systems interacting through a multi-body potential and we prove the mean field approximation under a strong assumption on this potential. All these results essentially rely on the flexibility of Wigner measures and we can give an alternative proof of the variational mean field approximation

On well-posedness and uniqueness for general hierarchy equations of Gross-Pitaevskii and Hartree type

International audienceGross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to an initial value problem, respectively the Gross-Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations and secondly solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is $U(1)$-invariant and prove a general principle which can be stated formally as follows: (i) Uniqueness for weak solutions of an initial value problem implies the uniqueness of solutions for the related hierarchy equation. (ii) Existence of solutions for the initial value problem implies existence of solutions for the related hierarchy equation. In particular, several new well-posedness results as well as a counterexample to uniqueness for the Gross-Pitaevskii hierarchy equation are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces

On well-posedness and uniqueness for general hierarchy equations of Gross-Pitaevskii and Hartree type

International audienceGross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to an initial value problem, respectively the Gross-Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations and secondly solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is $U(1)$-invariant and prove a general principle which can be stated formally as follows: (i) Uniqueness for weak solutions of an initial value problem implies the uniqueness of solutions for the related hierarchy equation. (ii) Existence of solutions for the initial value problem implies existence of solutions for the related hierarchy equation. In particular, several new well-posedness results as well as a counterexample to uniqueness for the Gross-Pitaevskii hierarchy equation are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces