Seminar | November 13 | 3:40-5 p.m. | 740 Evans Hall
Nima Moini, UC Berkeley
The state of a system in the low density limit should be described (at the statistical level) by the kinetic density, i.e. by the probability of finding a particle with position x and velocity v at time t. This density is expected to evolve under both the effects of transport and binary elastic collisions, which are expressed in the Boltzmann equation. The Cauchy problem for this equation is still one of the most important open problems, a new concept appearing in the 1989 paper by DiPerna and Lions is the notions of renormalized solutions of transport equation, is the most recent breakthrough, they provide a proof of the global existence of weak solutions via compactness arguments without any a priori estimates on the derivatives. The regularity and uniqueness of these solutions are still open problems. On the other hand, in terms of connection with the physical problem of interacting bodies (liquid, gas, etc), it is necessary to study the qualitative behavior of system of particles with short range potentials, for example particles with short range binary interactions like hard spheres undergo elastic collisions or smooth, monotonic, compactly supported potentials. The point of interest is to show that as the number of the particles increases, behavior of the system will actually converge to the kind of evolution that is being described by the Boltzmann equation. In this first part of this talk, I will present a rigorous derivation of the Boltzmann equation as the low density limit of system of hard spheres based on works of Saint-Raymond, Cercignani, Gerasimenko and Petrina. As for using the DiPerna-Lions theory in this context, the first step would be to understand the counterpart of renormalization at the level of the microscopic dynamics.