La deuxième séance de conférences Master sera l'occasion d'écouter Florian Loiseau (M2 MFA), Manon Simonot (M2 IS) et Matthieu Trotreau (M2 IS) qui nous parleront de leur expérience de stage de M1.
Plus précisément,
- Florian Loiseau parlera de sa "Formation en théorie des nombres" (stage réalisé à l'Université de Lille),
- Manon Simonot présentera la "Conception d'une application web comme outil clinique d'aide à l'analyse et au suivi des troubles de la marche" (stage réalisé au LMJL UMR 6629 Nantes Université),
- Matthieu Trotreau exposera sur "Les déterminants d’une carrière hospitalo-universitaire, sous la perspective du genre" (stage réalisé au Laboratoire de Psychologie de Nantes Université (LPPL - UR 4638)).
Dans cet exposé j'aborderai un de mes résultats récents sur la limite de l’équation de Schrödinger discrète sur le réseau $h\mathbb{Z}^d$ pour d = 1, 2 quand la taille de la grille h tend vers 0. En particulier, j'esquisserai comment obtenir des taux de convergence explicites dans des espaces de Sobolev arbitraires de la solution discrète vers celle de l’équation continue pour tout temps. La démonstration repose sur deux ingrédients principaux : le contrôle de l’évolution des normes de Sobolev discrètes de la solution, ainsi que des estimations bilinéaires vérifiées par l’interpolation de Shannon. L’estimation obtenue est dite compatible, au sens où plus de régularité sur la condition initiale permet d’obtenir une meilleure convergence en h. Une large portion de l'exposé sera dédiée à des rappels sur le cas continu, notamment sur l'intérêt des estimations de dispersion pour NLS.
systèmes dynamiques, d'EDPs et d'analyse numérique
Date de début de l'actualité
27-09-2023 14:00
Date de fin de l'actualité
29-09-2023 12:00
Rencontre ANR - salle Eole
Wednesday 27 :
2pm -> 3pm : Frédéric Rousset (Université Paris-Saclay), Interaction of a solitary water wave with a slowly varying bottom
3.30pm -> 4.30pm : Beatrice Langella(SISSA),Time periodic solutions of resonant Klein-Gordon equations on the 3d sphere
Thursday 28 :
9:30am->10:30am : Francisco Torres de Lizaur(University of Seville), Quasiperiodic solutions and invariant manifolds in the Euler equation of hydrodynamics
11am -> 12am : Fernando Casas (Universitat Jaume I), A short introduction to splitting methods for differential equations
2pm->3pm : Yoann Le Hénaff (Université Rennes 1), Grid-free Weighted Particle method applied to the Vlasov-Poisson equation
3:30pm->4:30pm : Qi Zhou (NanKai University), Quantitative structured almost reducibility and its applications
Friday 29:
9:30am->10:30am : Michela Procesi (Università Roma Tre), Almost-periodic solutions for the NLS with convolution potential
11am -> 12am : Zhiyan Zhao (Université Côte d’Azur), Symplectic Normal Forms and Classification of Growth of Sobolev norms
Et les résumés :
Frédéric Rousset : Interaction of a solitary water wave with a slowly varying bottom
In this talk we will discuss how a solution of the water wave system which behaves like a pure solitary wave at infinity is influenced by a slowly varying monotonously increasing or decreasing bottom. The main framework will be the water wave system with strong surface tension but we shall also use a Whitham type equation to illustrate the ideas. Joint work with Maria Eugenia Martinez (Lyon) and Claudio Munoz (Santiago).
Beatrice Langella : Time periodic solutions of resonant Klein-Gordon equations on the 3d sphere
In this talk I will focus on a class of completely resonant Klein-Gordon equations on the 3 dimensional sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, which arise as toy models in General Relativity. I will show that these equations admit small amplitude, time periodic solutions. Their existence is obtained by a variational Lyapunov-Schmidt decomposition, which reduces the problem to the search of Mountain Pass critical points of a restricted Euler-Lagrange action functional.
In order to gain compactness of the gradient of such a functional and smoothness of the critical points, a key point is to implement Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on $\mathbb{S}^3$. Based on a joint work with Massimiliano Berti and Diego Silimbani.
Francisco Torres de Lizaur : Quasiperiodic solutions and invariant manifolds in the Euler equation of hydrodynamics
I will review recent results on existence and dynamics of finite dimensional invariant manifolds of the Euler equation. These are families of divergence-free vector fields, parametrized by some manifold N, with the property that the solutions of the Euler equation with initial condition in the family exist and remain there for all time, defining a finite-dimensional ODE on N. For example if N is a torus and the ODE is a linear flow, these give families of periodic or quasiperiodic solutions of the Euler equation. This is joint work with A. Enciso and D. Peralta-Salas.
Fernando Casas : A short introduction to splitting methods for differential equations
Splitting methods constitute a powerful tool for the numerical integration of differential equations, either arising directly from dynamical systems or from partial differential equations of evolution previously discretized in space. Efficient high-order schemes have been designed that provide accurate solutions whilst preserving some of the most salient qualitative features of the system. In this talk we present an overview of this type of schemes and some of their applications. We will also consider a new class of reversible splitting methods and analyze their preservation properties when they are applied to the integration of unitary problems.
Yoann Le Hénaff : Grid-free Weighted Particle method applied to the Vlasov-Poisson equation
In this talk, a particle method for the collisionless Vlasov-Poisson system will be presented. It is based on the exact representation of an approximate electric field and, thanks to its grid-free nature, circumvents some of the usual drawbacks of PIC-like methods. Convergence and numerical aspects will be presented.
Qi Zhou : Quantitative structured almost reducibility and its applications
We propose a method called quantitative structured almost reducibility for analytic quasiperiodic $SL(2,\R)$-cocycles, which allows us to deal with infinitely many normal frequency resonances. Also we give will its dynamical and spectral applications.
Michela Procesi : Almost-periodic solutions for the NLS with convolution potential
I shall discuss some recent results, in collaboration with Livia Corsi and Guido Gentile proving the existence of maximal invariant tori for the NLS with a convolution potential of arbitrary regularity.
This is the first result where the regularity of the potential is not compensated by a corresponding smoothing in the non-linearity.
Zhiyan Zhao : Symplectic Normal Forms and Classification of Growth of Sobolev norms
For the Hamiltonian PDEs quantized by a quadratic polynomial with time-dependent coefficients, we give a complete classification for the long-time behaviors of the solution in Sobolev space, through Metaplectic representation and Schrödinger representation. Every behavior of Sobolev norm is characterized by a certain symplectic normal form by means of reducibility.
Imène Bouafia (M2 IS) : Statistiques Bayésiennes et contraintes Stratigraphiques. Stage réalisé au LMJL UMR 6629 Nantes Université.
Marie Compain (M2 MACS) : Simulations numériques d’une solution de référence pour les phénomènes viscothermiques dans les instruments de musique à vent. Stage réalisé à Inria Bordeaux.
Florian Loiseau (M2 MFA) : Formation en théorie des nombres. Stage réalisé à l'Université de Lille.
Manon Simonot (M2 IS) : Conception d'une application web comme outil clinique d'aide à l'analyse et au suivi des troubles de la marche. Stage réalisé au LMJL UMR 6629 Nantes Université.
Matthieu Trotreau (M2 IS) : Les déterminants d’une carrière hospitalo-universitaire, sous la perspective du genre. Stage réalisé au Laboratoire de Psychologie de Nantes Université (LPPL - UR 4638).
