ReferencesLaure Saint-Raymond, Hydrodynamic limits of the Boltzmann equation. (ISBN: 978-3540928461).

Course Description

This course will focus on the mathematics of kinetic theory, and the connections to equations in fluid mechanics.

The kinetic theory describes systems of interacting particles at the mesoscopic scale, that lies between microscopic dynamics given by Newton’s laws and the macroscopic hydrody- namic limits. One of the most important equations in this regime is the Boltzmann equation in gas dynamics. The importance and challenges of the development of mathematics theory on kinetic equations have been addressed in the famous Hilbert’s sixth problem, where he said Boltzmann’s work on the principle of mechanics suggests the problem of devel- oping mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua.

In this course, we will study mathematical theories for Boltzmann equations, as well as the rigorous derivation of hydrodynamic limits, including incompressible and compressible Euler and Navier-Stokes equations. Some powerful analytical tools will be discussed, including averaging lemma, Hilbert expansion and Chapman-Enskog expansion, DiPerna-Lions theory, relative entropy method, and more.

Amazingly, the mathematical framework of gas dynamics and mathematical tools developed for Boltzmann equation can be applied to new models arise from physics, biology, and social sciences. If time permits, we will discuss some applications in plasma physics and swarming dynamics.

Lecture notes

The lecture notes will be updated frequently. Please point out any typos or mistakes.