Collisions in Plasma: The Landau Equation and Related Models

University Of Texas At Austin, Austin TX

Investigators

Abstract

Phenomena in gas dynamics and plasma physics are driven by collisions and diffusion of particles. The mathematical modeling of these effects lies at the intersection of applied mathematics and statistical physics, and it involves the study of kinetic equations, a class of nonlinear partial differential equations that captures the behavior of large number of particles in terms of the particle density. This project concentrates on two specific kinetic equations: the first describes the time evolution of the particle distribution when particles interact through binary collisions that can occur at very large microscopical distances, the second arises in the statistical description of charged quantum particles. The intent is to validate the fidelity of these models to the physical systems by studying the qualitative properties of their solutions, to ensure that a temporary breakdown of the models does not occur. The project will provide training opportunities for graduate students and postdoctoral researchers. The project aims at expanding the mathematical understanding of the dynamics of collisions in dilute gases and plasma by analyzing two kinetic partial differential equations. The first is the Landau equation, which describes the time evolution of the particle distribution when particles interact through binary collisions of grazing type. The second is the Landau-Fermi-Dirac equation, which arises in the statistical description of charged quantum particles. Both equations present challenges due to nonlinear terms, nonlocal features, and degenerate coefficients. The first part of the project deals with the qualitative properties of solutions, such as global well-posedness and finite time blow-up, characterization of the long-time behavior, and connection with macroscopic fluid equations. The second part concerns the validity of the Landau-Fermi-Dirac approximation as a correction to the Boltzmann equation in the grazing regime. The techniques employed include a novel combination of classical kinetic theory and recent theories developed for nonlinear nonlocal integro-differential equations and degenerate nonlocal differential operators. 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.