CAREER: Nonlocal partial differential equations in collisional kinetic theory

University Of Texas At Austin, Austin TX

Investigators

Abstract

This research project is concerned with important physical phenomena driven by collision and diffusion of particles, whose mathematical description is based on partial differential equations of kinetic type. It is driven by applications in gas dynamics and plasma physics as well by mathematical interests in analysis, partial differential equations and mathematical physics. Kinetic equations are used to describe evolution of interacting particles, such as gas molecules, ions and electrons in a plasma. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936 Lev Landau derived from the Boltzmann equation a new mathematical model for motion of plasma. This latter equation was named the Landau equation. Despite the fact that many mathematicians and physicists have been working on these equations, many important questions are still unanswered due to their mathematical complexity. The proposed research will fundamentally contribute to this field by bridging the gap between mathematical analysis and physics and enable further mathematical understanding of physical phenomena. The work will be enhanced by collaborations at the national and international levels and will help strengthen inter-institutional ties. The research program will be integrated with educational and outreach activities designed to (i) broaden the students' understanding of different research areas, (ii) provide the students with a modern skill set that is essential for different career paths, and (iii) promote existing connections between local institutions in the DC Metro area and help establish new avenues for collaboration and training of high quality scientific workforce. The proposed research will advance the knowledge in global properties of solutions to collisional kinetic mathematical models, improving our understanding of non-linear dynamics, entropy, equilibrium, and regularizing effects. The corresponding equations contain integro-differential operators that are highly nonlinear, singular and with degenerating coefficients. Integro-differential equations of this type have received increased attention recently: well-posedness and regularity theory are being developed since many new applications have emerged, among which conformal geometry, stochastic control and image processing. The project aims at advancing knowledge in the theories of kinetic equations, nonlocal integro-differential operators and degenerate differential operators.