Semi-groupes stochastiques
Décomposition des difféomorphismes du tore en applications déviant la verticale (avec un appendice en collaboration avec Jean-Marc Gambaudo)
Une version feuilletée équivariante du théorème de translation de Brouwer
The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism of the plane, every point belongs to a Brouwer line: a proper topological embedding C of , disjoint from its image and separating (C) and (C). Suppose that commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines....
Étude Topologique des Applications Déviant la Verticale. / Patrice Le Calvez
Propriétés générales des applications déviant la verticale
Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes ?
Soit un homéomorphisme du plan qui préserve l’orientation et qui a un point périodique de période . Nous montrons qu’il existe un point fixe tel que le nombre d’enlacement de et ne soit pas nul. En d’autres termes, le nombre de rotation de l’orbite de dans l’anneau est un élément non nul de . Ceci donne une réponse positive à une question posée par John Franks.
Propriétés dynamiques des régions d'instabilité
Dynamique des homéomorphismes du plan au voisinage d'un point fixe
Multivalued Lyapunov functions for homeomorphisms of the 2-torus
Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,ℝ) is a cohomology class which is positive on the rotation set of f. We prove the existence of a smooth Lyapunov function of f whose derivative lifts a non-vanishing smooth closed form on 𝕋² whose cohomology class is κ.
Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data
Analyse statistique d'une formation limoneuse en vue de préciser son origine
Foreword
A rainbow inverse problem
We consider the radiative transfer equation (RTE) with reflection in a three-dimensional domain, infinite in two dimensions, and prove an existence result. Then, we study the inverse problem of retrieving the optical parameters from boundary measurements, with help of existing results by Choulli and Stefanov. This theoretical analysis is the framework of an attempt to model the color of the skin. For this purpose, a code has been developed to solve...
Mathematical and numerical modeling of early atherosclerotic lesions
This article is devoted to the construction of a mathematical model describing the early formation of atherosclerotic lesions. The early stage of atherosclerosis is an inflammatory process that starts with the penetration of low density lipoproteins in the intima and with their oxidation. This phenomenon is closely linked to the local blood flow dynamics. Extending a previous work [5] that was mainly restricted to a one-dimensional setting, we couple...
Diffusion models of multicomponent mixtures in the lung
In this work, we are interested in two different diffusion models for multicomponent mixtures. We numerically recover experimental results underlining the inadequacy of the usual Fick diffusion model, and the importance of using the Maxwell-Stefan model in various situations. This model nonlinearly couples the mole fractions and the fluxes of each component of the mixture. We then consider a subregion of the lower part of the lung, in which we compare...
Mucus dynamics subject to air and wall motion
This study presents a numerical investigation of basic interactions between respiratory mucus motion, air circulation and epithelium ciliated cells vibration. One focuses on identification of meaningful rheological parameters, physiological and numerical simulation dimensioning. These preliminary results are crucial before the study of more general configurations of respiratory mucus motion. The numerical study presented in this work aims at providing...
Influence of the spray retroaction on the airflow
In this work, we investigate the influence of a spray evolving in the air, in the respiration framework. We consider two kinds of situations: a moving spray in a motionless fluid, and motionless particles in a Poiseuille flow. We observe that the spray retroaction may not be neglected in some situations which can really happen, for instance, when one considers rather big particles, as it is possible for polluting particles and even for some therapeutic...
Asymptotic analysis of blood flow in stented arteries: time dependency and direct simulations
This work aims to extend in two distinct directions results recently obtained in [10]. In a first step we focus on the possible extension of our results to the time dependent case. Whereas in the second part some preliminary numerical simulations aim to give orders of magnitudes in terms of numerical costs of direct 3D simulations. We consider, in the first part, the time dependent rough problem for a simplified heat equation in a straight channel that...
Numerical study of the stopping of aura during migraine
This work is devoted to the study of migraine with aura in the human brain. Following [6], we class migraine as a propagation of a wave of depolarization through the cells. The mathematical model used, based on a reaction-diffusion equation, is briefly presented. The equation is considered in a duct containing a bend, in order to model one of the numerous circumvolutions of the brain. For a wide set of parameters, one can establish the existence...
High-order WENO scheme for polymerization-type equations
Polymerization of proteins is a biochemical process involved in different diseases. Mathematically, it is generally modeled by aggregation-fragmentation-type equations. In this paper we consider a general polymerization model and propose a high-order numerical scheme to investigate the behavior of the solution. An important property of the equation is the mass conservation. The WENO scheme is built to preserve the total mass of proteins along time....
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