Séminaire Jeunes chercheurs (Richard MEDINA RODRIGUEZ, jeudi 9 janvier 2025)

La prochaine séance du séminaire Jeunes chercheurs du CEREMADE aura lieu le jeudi 9 janvier 2025 à 17h en salle A707. Nous aurons le plaisir d'écouter Richard MEDINA RODRIGUEZ (CEREMADE), qui nous parlera de

Brief introduction to the Boltzmann equation: well-posedness and long-time behavior in the torus

Abstract
The Boltzmann equation models a system of moving particles which interact through collisions, that we will assume in this talk to be elastic and instantaneous. In this talk we will present the theory of well-posedness and long-time behavior for this equation in the torus. 
First we will introduce the model and the physical relevance of the different operators, then we will present the expected equilibrium and the linearized equation around such equilibrium.
We will introduce then the concept of hypocoercivity for the linearized Boltzmann operator in order to obtain a constructive decay estimate in a suitable functional space. 
Using the Duhamel formula we will relate the previous functional space with a weighted $L^\infty$ space, which is where we can properly control the non-linear Boltzmann collision operator. 
We will conclude by constructing solutions for the fully non-linear Boltzmann equation in the perturbative regime by providing a construcitve rate of decay to equilibrium.