Together with Luis Silvestre (University of Chicago), we wrote a survey article about a series of works we recently completed about the Boltzmann equation. See hal and arxiv.
The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an outstanding open problem. In the present article, we review a program focused on the type of particle interactions known as non-cutoff. It is dedicated to the derivation of a priori estimates in C∞, depending only on physically meaningful conditions. We prove that the solution will stay uniformly smooth provided that its mass, energy and entropy densities remain bounded, and away from vacuum.