Christopher Henderson
University of Arizona
Title
A new approach to regularity, well-posedness, and blow-up in the Boltzmann equation
Abstract
When we imagine a gas at the microscopic scale, we envision a huge number of individual particles flying around and colliding with one another. The sheer number of particles makes this mathematically intractable. The Boltzmann equation provides a way to bypass this complexity through a kinetic partial differential equation. Due to its nonlocal and nonlinear structure, however, it is notoriously difficult to analyze.
In this talk, I will give an overview of a new approach to analyzing the Boltzmann equation by drawing on and generalizing ideas from parabolic equations (which govern systems like heat flow). This approach leads to a novel blow-up criterion; that is, a condition that guarantees the continued existence of a solution. We then use this to significantly expand the class of initial data for which solutions exist and for which we understand the long-time trend to equilibrium.