Dans cet exposé, nous présenterons plusieurs résultats concernant un problème d’optimisation en écologie spatiale et qui peut se formuler ainsi: comment, au sein d’un domaine, répartir les ressources accessibles à une population afin de garantir que cette dernière soit de taille maximale? Nous nous concentrerons sur les propriétés qualitatives de ce problème. Nous mettrons en évidence, entre autre, des propriétés de type concentration/fragmentation des ressources: vaut-il mieux répartir le plus possible les ressources ou, au contraire, les concentrer en un unique endroit? Contrairement à plusieurs critères mieux connus (comme la capacité de survie), où la concentration de ressources est toujours favorable, et ce indépendamment de la vitesse de déplacement des individus, pour la taille de la population, nous montrons que, plus cette vitesse de déplacement est faible, plus la fragmentation est un atout. La première partie de l’exposé sera essentiellement descriptive, et nous donnerons des éléments de preuve dans la seconde. Les différents travaux qui seront présentés ont été réalisés en collaboration avec G. Nadin, Y. Privat et D. Ruiz-Balet.
First, we consider a system of two wave equations coupled by velocities in one-dimensional space with one boundary fractional damping and we prove that the energy of our system decays polynomially with different rates. Second, we investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping of Kelvin-Voigt type and we prove that the energy of our system decays polynomially with rate 1/t. Finally, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients and we establish the exponential stability of the solution if and only if the two waves have the same speed of propagation. In case of different speed propagation, we prove that the energy of our system decays polynomially with rate 1/t.
The lake equations arise as a geophysical model for the description of shallow water. The system is introduced as a 2D model for the vertically averaged horizontal component of a 3D incompressible fluid. A lake is characterised by a 2D domain and a non-negative topography function. The 2D velocity satisfies an anelastic constraint rather than a divergence-free condition. The equations are degenerate if the topography may vanish. More precisely, velocity and vorticity are then related through degenerate elliptic problems. In this talk, we discuss the stability of the lake equations for singular geometries and degenerated topographies. Specifically, we prove stability results for two scenarios: First, motivated by natural phenomena such as flooding or erosion we consider a sequence of lakes with an island that disappears. In addition, we highlight crucial differences to the incompressible 2D Euler equations (flat topography). Second, we address the stability of the equations for a sequence of lakes for which an island appears in the limit, e.g. due to a decreasing level of water. This is joint work with C. Lacave and E. Miot.
In the last years, measure solutions to PDE, in particular those modeling populations, have drawn much attention. The talk will be devoted to the presentation of a recent, unusual result in this field, that we obtained with Pierre Gabriel. First, I will expose some wellposedness and asymptotic results for two famous population equations in the L^p and measure frameworks, and explain the critical case that interested us. Then, I will define the notion of solution we used, and if needed, recall some basic definitions about semigroups. Moving to the proof itself, I will present the main steps of the proof of the wellposedness of the problem, that relies on a duality relation used to build a solution expressed as a semigroup acting on an initial measure. Then, I will go a little more into details of the demonstration of the asymptotic behaviour. In particular, I will exhibit how we used Harris' ergodic theorem to obtain a uniform exponential convergence in (weighted) total variation norm toward an oscillating measure.
The equations of motion for compressible (barotropic) fluids have the structure of a simple conservative dynamical system when expressed in Lagrangian variables. This can be exposed interpreting the Lagrangian flow as a curve of vector-valued L2 functions, and the internal energy of the fluid as a functional on the same space. Particle methods are a natural discretization strategy in this setting, since in this case the flow is discretized using piecewise constant functions on a given partition of the domain, but they require some form of regularization to define the internal energy of the fluid. In this talk I will describe a particle method in which the internal energy is replaced by its Moreau-Yosida regularization in the L2 space, which can be efficiently computed as a semi-discrete optimal transport problem. I will also show how the convexity of the energy in the Eulerian variables can be exploited in the non-convex Lagrangian setting to prove quantitative convergence estimates towards smooth solution of this problem, and how this result generalizes to dissipative porous media flow.
Le but de l'optique anidolique, aussi appelée optique non imageante, est de construire des composants optiques qui transportent l'énergie lumineuse d'une source de lumière vers une cible de lumière donnée. La modélisation de ce type de problèmes inverses conduit dans certains cas à des équations de type Monge-Ampère. Dans cet exposé, je montrerai comment de telles équations peuvent être résolues numériquement à l'aide d'une discrétisation géométrique particulière appelée semi-discrète. Je présenterai aussi des applications en optique anidolique avec la construction de miroirs et de lentilles.
Les champignons endomycorhiziens forment des communautés mutualistes qui aident les plantes à accroître leur système racinaire et donc leur biomasse. Depuis plusieurs décennies, ces champignons sont utilisés comme engrais vert. Cependant quel est l'impact de ces champignons commerciaux sur les communautés sauvages? Afin de comprendre ces interactions j'ai développé en collaboration avec M. Martignoni, R. Tyson et M. Hart (Univ British Columbia) un nouveau modèle mutualiste basé sur des équations aux dérivées partielles. Dans cette exposé, je vous présenterai des critères analytiques d'existence et de stabilité des solutions stationnaires pour lesquels la coexistence apparaît. Ensuite je m'intéresserai à l'invasion spatiale d'une communauté par une autre en montrant l'existence de solutions de type front progressif pour le système et en caractérisant leur vitesse de propagation.
In this talk we consider the Cauchy problem for the 2D Euler equations for incompressible inviscid fluids. It is well-known that given a smooth initial datum, the Cauchy problem is well-posed and in particular the energy is conserved and the vorticity is transported by the flow of the velocity. When we consider weak solutions this might not be the case anymore. We will review some recent results obtained in collaboration with Gianluca Crippa and Gennaro Ciampa where we extend those properties to the case of irregular vorticities. In particular, under low integrability assumptions on the vorticity we show that certain approximations important from a physical and a numerical point of view converge to solutions satisfying those properties.
Quantum hydrodynamic (QHD) systems arise in the effective description of phenomena where quantistic behavior can be seen also at a macroscopic scale. This is the case for instance in Bose-Einstein condensation, superfluidity or in the modeling of semiconductor devices. Standard results for global existence of finite energy weak solutions to the QHD system often exploit the analogy with a nonlinear Schrödinger equation; by using the Madelung transform it is possible to define a solution to the QHD by considering the momenta (mass and current density) associated to a wave function. In particular this argument requires the initial data to be determined by a given wave function. This usual approach hence shows the existence of solutions but can not be used to study their stability properties in a general framework. In this talk I will present some recent developments that overcome those difficulties for the one dimensional QHD system. First of all I will provide an existence result for a large class of initial data, without requiring them to be generated by a wave function. Furthermore, I will prove a stability result for weak solutions. This exploits a novel functional which formally controls the L^2 norm of the chemical potential, weighted with the particle density. This is a joint work with P. Marcati and H. Zheng.
Le but de l'exposé sera de présenter une preuve alternative de la stabilité asymptotique d'équilibres spatialement homogènes pour des perturbations localisées d'équations de Vlasov posées dans l'espace entier. La preuve originale due à Bedrossian-Mouhot-Masmoudi, est inspirée de la preuve du Landau damping pour des solutions périodiques et utilise les propriétés dispersives du transport libre en Fourier. On présentera une approche basée sur la méthode des caractéristiques et une étude des propriétés dispersives du linéarisé dans l'espace physique. (collaboration avec D. Han-Kwan (Polytechnique) et T. Nguyen (Penn-State))
Nicolas BESSET, IF, Grenoble Jean François BOUGRON, IF, Grenoble Dorin BUCUR, LAMA, Chambéry Emmanuelle CREPEAU, LJK, Grenoble Rita JUODAGALVYTE, LaMuse, Saint-Etienne Florian PATOUT, LAMA, Chambéry Arnaud MUNCH, LMPB, Clermont-Ferrand, Tran Duc Minh PHAN, LMBP, Clermont-Ferrand Laure SAINT-RAYMOND, UMPA, ENS-Lyon Filippo SANTAMBROGIO, ICJ, Lyon Simon SANTOSO, LJK, Grenoble Raphael WINTER, UMPA, ENS Lyon
The talk is concerned with geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains of a Riemannian manifold. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary value problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case where the first nonzero Neumann eigenvalue is not simple. In the second part of the talk, we focus on product manifolds with euclidean factors, and we classify the subdomains where the associated overdetermined boundary value problem has a solution. This is joint work with Moustapha Fall (AIMS Senegal).
Dans cet exposé nous nous intéresserons aux effets régularisants de l’équation de Kolmogorov sur l’espace de Wasserstein. Telle équation décrit la dynamique du semi-groupe généré par la solution d’une équation différentielle stochastique de type McKean-Vlasov (i.e. dont la dynamique dépend de la loi). Nous verrons comment de tels effets permettent de retrouver des résultats d’unicité faible et forte ainsi que des phénomènes de propagation du chaos pour des équations à coefficients peu réguliers.
In this talk, I will first recall a few standard results on predator-prey systems with or without Allee effect on the prey. Then I will present a brake-driven gene drive reversal model (spatialized population genetics) and show the link with the first part. Thanks to this link, a co-extinction result will be rigorously established and a co-invasion result will be partially proved, partially illustrated numerically. This is an interdisciplinary joint work with Vincent Calvez and Florence Débarre.
En 2005-2007 Burdzy, Caffarelli et Lin, Van den Berg ont conjecturé, dans des contextes différents, que la somme (ou le maximum) des valeurs propres fondamentales du Laplacien-Dirichlet associées à des cellules disjointes d'un domaine planaire est asymptotiquement minimale pour une structure en nid d'abeilles, quand le nombre de cellules devient très grand. Je vais discuter l'histoire de cette conjecture en détaillant les arguments de Fejes Toth et Hales sur le problème du nid d'abeilles classique, et je vais démontrer la conjecture (du maximum) pour les valeurs propres du Laplacien-Robin. Les résultats présentés ont été obtenus avec I. Fragala, B. Velichkov et G. Verzini.
We consider a particle constrained in a graph structure and excited by an external controlling field. Its dynamics is modeled by the bilinear Schrödinger equation i∂t ψ = −∆ψ + u(t)Bψ in the Hilbert space L2(G , C) where G is the graph. The Laplacian −∆ is equipped with self-adjoint boundary conditions. The action of the field is represented by the bounded symmetric operator B and by the control function u ∈ L2((0,T),R) with T > 0, which accounts its intensity. The exact controllability of the bilinear Schrödinger equation on bounded intervals was widely studied in literature. Nevertheless, the bilinear Schrödinger equation on graphs is in general a more delicate matter and it was only studied on compact networks. Up to our knowledge, the controllability on infinite graphs is still an open problem. The main reason can be found on the dispersive phenomena characterizing the equation (not considering the difficulties already appearing on compact graphs). A peculiarity of the Schrödinger equation is the loss of localization of the wave packets during the evolution, the dispersion. This effect can be measured by L ∞ -time decay. In this talk, we present the bilinear Schrödinger equation on infinite graphs. In par- ticular, we show the existence of suitable subspaces of L 2 (G , C) where the equation is well-posed. In such spaces, we define assumptions on the structure of the graph and on the control field such that the global exact controllability is guaranteed. The result leads to the so-called “energetic controllability”.