Dans cet exposé, nous considérerons l'équation de Landau spatialement homogène dans le cas des potentiels mous. Dans ce cadre, l'existence de solutions faibles est un résultat connu. Nous nous intéresserons à l'obtention de bornes $L^p$ pour $1<$ $p<\infty$ à partir d'une inégalité dite de $\varepsilon$-Poincaré. Nous verrons ensuite comment la méthode de De Giorgi permet de déduire...
In this talk, we consider the one-dimensional nonlinear Schrödinger equation with a trapping potential that exhibits a growth of $|x|^s (s > 1)$ at infinity. Our main focus is on the construction and invariance of Gibbs measures associated with the equation. We determine conditions on the nonlinearity and the growth of potential that dictate whether Gibbs measures can be normalized, and we...
We consider the variational wave equation with a stochastic forcing which appears in modeling liquid crystals. In this talk, we study the following results: the existence of local-in-time regular solutions, the finite-time blow-up with a high probability, and the existence of global martingale weak solutions.
Les observations expérimentales montrent que la viscosité, constante pour certains fluides tels que l'eau et le miel, dépend du taux de déformation dans de nombreux autres cas. On parle alors de fluide complexe ou non newtonien. Une manière simple de modéliser un fluide complexe est de postuler une relation entre taux de déformation et viscosité, dont les paramètres sont ajustés pour coller au...
Les cellules et leur environnement constituent une matière active à l’origine de dynamiques complexes, par exemple lors du développement embryonnaire, de la croissance d’une tumeur ou d’un processus de cicatrisation. Dans une approche interdisciplinaire, combinant mathématiques et biophysique, nous nous intéressons à la modélisation mathématique du mouvement collectif de cellules. Chaque...
We focus on the study of solitary waves for two deep water wave models: the Whitham equation and the Zakharov water wave system. Specifically, we analyze the behavior of a solitary wave when it encounters a change in the environment, for example, when the bottom of the domain containing the fluid is altered.
Zakharov water waves arises as a free surface model for an irrotational and...
In this talk we present a new variant of mathematical prediction-correction model for crowd motion. As the first step (prediction) is somehow classical and performed via a transport equation, we shall focus on the second step (correction) which relies on a minimum flow problem. We discuss some duality results and how to use solve the minimum flow problem via a primal-dual algorithm. We provide...
In the context of proving the semiclassical mean-field limit from the N-body Schrödinger equation to the Hartree-Fock and Vlasov equations, a crucial component is obtaining inequalities uniform in the Planck constant and the number of particles. These inequalities are the analogue of the estimates obtained in the corresponding kinetic models of classical statistical mechanics.
Hence, in...
Le comportement en temps grand des solutions d'équations de réaction-diffusion est généralement dicté par des solutions particulières de type front progressif. Ces fronts permettent entre autres de décrire des phénomènes d'invasion en dynamique des populations et en biologie. En général, l'existence de plusieurs états d'équilibre conduit à un profil de propagation multiple passant par...
In recent years, there has been a spike in interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke’s, Brinkman, or Darcy’s law. While these velocity-pressure relations have been studied in the literature, little emphasis has been placed on the fine relationship between them. In this talk, I will address this question...
The aim of this presentation is to introduce a new hyperbolic model to describe the breaking wave phenomenon. The breaking wave model is obtained by depth averaging the Large Eddy Simulation (LES) equations. In the derivation, the small-scale turbulence is modeled by a turbulent viscosity, while the large scales are considered by an additional variable called enstrophy. Typically, the...
