Diffusion in Kinetic Equations

Kinetic equations model the evolution of densities of a large system of interactive particles. They may be used, for example, to study the evolution of a gas or a plasma. The Principal Investigator (PI) is interested in the study of the Boltzmann and Landau equations, for systems of particles that repel each other by power-law potentials. These equations exhibit a regularization effect. An outstanding open problem is to understand if a singularity could emerge from the natural flow of the equation, or if the regularization effects actually dominate the evolution and keep the solutions smooth. The PI mentors graduate students and postdocs in research on the topics of this project. This project aims at developing tools in the analysis of nonlocal equations, parabolic equations and hypoelliptic theory targeted to their applications in kinetic equations. The Boltzmann collision operator acts as a nonlinear diffusive operator of fractional order. It can be studied in the framework of parabolic integro-differential equations. The Landau equation is a model from statistical mechanics used to describe the dynamics of plasma. It can be obtained as a limit case of the Boltzmann equation when grazing collisions prevail. It is a second order, nonlinear, parabolic equation. The project connects different areas of mathematics and mathematical physics, relating recent progress in nonlinear integro-differential equations with the classical Boltzmann equation from statistical mechanics. Kinetic equations involve a nonlinear diffusive operator with respect to velocity, combined with a transport equation with respect to space. The regularization effect in all variables requires ideas from hypoelliptic theory. For the Boltzmann equation in the case of very soft potentials, as well as for the Landau equation with Coulomb potentials, the diffusive part of the equations is not strong enough to prevent the solution from blowing up in theory. In that case, new ideas are needed to properly understand the regularization effects of the equation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.