Workshop - Problèmes mathématiques et modélisation en théorie cinétique

Amit Einav

Local Cauchy Theory for the Boltzmann-Nordheim Equation

One of the most influential equations in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in a classical dilute gas. The irrefutable appearance of Quantum Mechanics, however, required a modification to this celebrated kinetic equation, resulting in the Boltzmann-Nordheim equation. In this presentation we will discuss a newly found local Cauchy Theory for a general solution to the spatially homogeneous bosonic Boltzmann-Nordheim equation in any dimension $d \geq 3$, a case of much interest due to the so-called Bose-Einstein condensation.

The methods used to achieve this theory are similar to those available for the classical Boltzmann equation, yet are entangled with L^infty control that dominates the difference between the classical and quantum kinetic equation. Time permitting we will discuss some details about the existence of a global solution to the equation.

This is a joint work with Marc Briant.




ANR CNRS Rennes 1 Rennes 2 Nantes INSA Rennes INRIA ENSRennes UBO UBS Angers UBL