Séminaire EDP et Applications | Nancy - Institut Élie Cartan de Lorraine

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Dérivation d'un modèle d'écoulement compressible à bulles

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 22 novembre 2022 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Hélène Mathis (Université de Montpelliers) Résumé :

On s’intéresse à la modélisation d’un écoulement à bulles compressibles par une méthode d’homogénéisation.

A l’échelle microscopique, le fluide est décrit par le système de Navier-Stokes compressible, le comportement des bulles étant décrit individuellement.

L’originalité du modèle réside dans la prise en compte de la tension de surface aux interfaces.

Considérant que le nombre de bulles diverge, on propose une construction des quantités macroscopiques.

On présente le système d’équations macroscopiques associé qui contient non seulement une équation d’évolution de la fraction de volume mais également une équation cinétique sur la fonction de répartition des bulles.

Il s’agit d’un travail en collaboration avec M. Hillairet et N. Seguin.

We introduce a general framework of low regularity integrators which allows us to approximate the time dynamics of a large class of equations, including parabolic and hyperbolic problems, as well as dispersive equations, up to arbitrary high order on general domains. The structure of the local error of the new schemes is driven by nested commutators which in general require (much) lower regularity assumptions than classical methods do. Our main idea lies in embedding the central oscillations of the nonlinear PDE into the numerical discretisation. The latter is achieved by a novel decorated tree formalism inspired by singular SPDEs with Regularity Structures and allows us to control the nonlinear interactions in the system up to arbitrary high order on the infinite dimensional (continuous) as well as finite dimensional (discrete) level.

This is a joint work written in collaboration with Yvonne Alama Bronsard and Katharina Schratz.

Quelques résultats d’existence et de régularité des solutions de l'équation fractionnaire d’Hamilton-Jacobi

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 18 octobre 2022 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Somia ATMANI (Université Abou Bakr Belkaïd, Tlemcen) Résumé :

Abstract

Stabilization of 1D systems of PDEs

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 11 octobre 2022 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Amaury Hayat Résumé :

As part of control theory, stabilization consists in finding a way to make stable a trajectory of a system on which one has some means of action. In this talk, we will discuss recent advances in stabilization of PDEs, starting with one of the most natural approaches for nonlinear systems, quadratic Lyapunov functions, to more complex approaches such as Fredholm backstepping. Backstepping consists in finding a control operator such that the PDE system can be invertibly mapped to a simpler PDE system for which stability is known. Surprisingly powerful, this approach offers the possibility to deal with very general classes of systems. We will review the origin of the method and present new results that resolve a question opened in 2017 and illustrate it on the rapid stabilization of the linearized water-wave equations. Finally, if time allows we will talk about a completely different subject: teaching mathematics to an AI and we will consider two questions, can we train an AI to predict the solution of a mathematical problem? can we train an AI to prove a statement?

A morphelastic model

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 21 juin 2022 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Ulisse Stefanelli (University of Vienna) Résumé :

I will present some recent work in collaboration with Elisa Davoli (TU Wien) and Katerina Nik (University of Vienna) on a three-dimensional quasistatic morpholelastic model. The mechanical response of the body and its growth are modeled by the interplay of hyperelastic energy minimization and growth dynamics. An existence result is obtained by regularization and time-discretization, also taking advantage of an exponential-update scheme. Then, we allow the growth dynamics to depend on an additional scalar field modeling nutrient concentration, and formulate an optimal control problem. Eventually, we tackle the existence of coupled morphoelastic and nutrient solutions, when the latter is allowed to diffuse and interact with the growing body.

Control of parameter dependent systems: how to compute greedy, ensemble or averaged controls?

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 7 juin 2022 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Jérôme Lohéac (CRAN) Résumé :

In this talk I will provide an overview on the problem of controllability of parameter dependent systems. I will explore different control notions successfully developed through the last decade.
The aim of the control function is to steer the system to a state satisfying some properties prescribed either at some time instant T>0 or during a given time interval. These properties may be separated with respect to parameter values and can refer just to a single system itself (e.g. greedy control), or may consider solutions corresponding to the whole parameter range (e.g. ensemble control, averaged control). In the latter case control functions are designed as parameter invariant, implying a same control is to be applied to the system independently of a particular realization of the parameter, while in the first case controls vary along with the parameter. Beside the positive theoretical results, for each notion we provide a computational algorithm.

A varifold perspective on discrete surfaces

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 17 mai 2022 10:45-11:45 Lieu : Salle Döblin Oratrice ou orateur : Blanche Buet (Laboratoire de mathématiques d'Orsay) Résumé :

Joint work with: Gian Paolo Leonardi (Trento), Simon Masnou (Lyon) and Martin Rumpf (Bonn).

We propose a natural framework for the study of surfaces and their different discretizations based on varifolds. Varifolds have been introduced by Almgren to carry out the study of minimal surfaces. Though mainly used in the context of rectifiable sets, they turn out to be well suited to the study of discrete type objects as well.

While the structure of varifold is flexible enough to adapt to both regular and discrete objects, it allows to define variational notions of mean curvature and second fundamental form based on the divergence theorem. Thanks to a regularization of these weak formulations, we propose a notion of discrete curvature (actually a family of discrete curvatures associated with a regularization scale) relying only on the varifold structure. We prove nice convergence properties involving a natural growth assumption: the scale of regularization must be large with respect to the accuracy of the discretization. We performed numerical computations of mean curvature and Gaussian curvature on point clouds in R^3 to illustrate this approach.

Building on the explicit expression of approximate mean curvature we propose, we perform mean curvature flow of point cloud varifolds beyond the formation of singularities and we recover well-known soap films.

Remarques sur le Problème de Cauchy pour le laplacien et Contrôle lagrangien de l'équation d’Euler

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 10 mai 2022 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Otared Kavian (Université de Versailles) Résumé :

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