The Kompaneets Equation, Anomalous Diffusion, and Brownian Entanglement

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This research project is devoted to mathematical study of questions arising in the investigation of three natural phenomena; these studies are united by the use of related mathematical tools. The first set of questions concerns mathematically understanding behavior of photons in low-density plasmas. The second set of questions concerns the behavior of particles in fluid flow. In certain situations important for applications, particles diffuse in unusual ways. This project investigates such "anomalous" diffusive behavior on intermediate (as opposed to long) time scales. The third set of questions is motivated by mathematically quantifying the entanglement of long polymer molecules; it aims to quantify the winding of trajectories of the related random walks. Each of these research projects involves graduate students and postdoctoral associates, who will be broadly trained through the course of their work. This research will use tools from applied analysis, partial differential equations, and probability. The first set of questions concerns the long-time behavior of the Kompaneets equation, and focuses on understanding the formation of a Bose-Einstein condensate. The second set of questions studies the small-noise limit of passive scalar transport and other related models arising in the context of fluids. While the long-time behavior of these equations is well known, here interest is in the small-noise intermediate-time regime where an anomalous diffusive effect is observed, and the limiting behavior is described by a time changed Brownian motion and time-fractional equations. The third set of questions studies the long-time "winding" or "entanglement" of Brownian trajectories on compact Riemannian manifolds with boundary. This is closely related to heat kernel estimates on covering spaces, and is motivated by the study of the entanglement of long polymer molecules.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.