An introduction to kinetic equations

DATES: 8-12 April 2019 (5 sessions)
TIME: 10:00 - 12:00 (a total of 10 hours)
LOCATION: BCAM Seminar room

The main goal of the lecture will be to give an overview of the problem studied in the field of kinetic equations. Those PDEs appear naturally in the modelling of systems composed by a large number of objects which are all subjected to the same physical laws. The panorama of objects that can be considered with those equations is very large: one can think about the electrons in a plasma or the stars in a galaxy (Vlasov equation), the particle of dust in suspension in a gas (Fokker-Plank equation), or the molecules of a gas (Boltzmann equation). We will particularly focus on the rigorous physical derivation of the equations, the well definition of their solutions, the large time asymptotic when it can be established, and the derivation of simpler equations.

CONTENT:

We will start by introducing the relevant function to study a system of many particles. In order to understand better that point of view, we will first consider the linear Vlasov equation which is nothing less than the simple transport equation describing a finite mass of particles subjected to an external force. Those equations have some remarkable property that are very useful in the study of the real non linear case. We will also see that they have a large number of equilibrium. For the force associated with the harmonic potential, we will see that when the spatial density stays constant, the distribution function do not most of the time.

We will then turn to study the real (non linear) Vlasov equation which describes a system of particles which are all linked by an interaction force. We will prove the existence of solution in the regular case and give some example of the system described by this kind of equations. We will explain how this equation can be obtained as the limit equation of the density of a system of N particle when N goes to infinity (mean field limit). We will end by a few words about the recent progress in the study of its large time behaviour.

As a first step in collisional kinetic equations, we will then come to the fokker-Planck equation. As before, we will explain how it can be obtained rigorously by a mean field limit in a more probabilistic meaning. This will lead us to talk about the propagation of chaos and explain why the action of Brownian motion is traduced by a Laplacian in the evolution equation. The large time asymptotic will be established in the spatially homogeneous case. We will say a few words to explain how the proof can be adapted in the general case by hypocohercivity. Depending on the time left we may say a few words about the diffusion asymptotics which allows us to define more simple evolution equations.

We will end by a short introduction to Boltzmann equation. We will explain its formal derivation, show its irreversibility thanks to the famous H-theorem. This will lead us to get very easily some expectations about its large time asymptotic. Depending on the time remaining, we might explain how it is possible to establish rigorous results in the spatially homogeneous case. Whatever, we will end by presenting some different aspects of those equations which have been studied in the last decades.


OBJECTIVES:
Get familiar with the techniques of Carleman estimates and monotonicity formulas.

PREREQUISITES:
Basic knowledge in functional analysis: distributions, Lp spaces. Most of the specific technics to study those PDE will be explained during the lecture. Some probability knowledge will be necessary for the lecture on Wednesday.


*Registration is free, but mandatory before April 5th: So as to inscribe go to https://bit.ly/2R51v2F and fill the registration form. Student grants are available. Please, let us know if you need support for travel and accommodation expenses when you fill the form.