This research project focuses on the modeling and mathematical analysis of nonlinear systems arising in physical and biological science as well as engineering applications. The models under investigation include hyperbolic balance laws for the investigation of nonlinear systems arising in continuum physics, mixtures of fluids for the study of fluid-particle and fluid-solid interaction as well as kinetic models for the investigation of the collective self-organization of agents. Specific themes include: (a) Structure and stability of solutions to hyperbolic balance laws and models of fluid mixtures. (b) Diffusion and hydrodynamic limits for collisional kinetic equations. (c) Large-data existence theory for polymeric fluids. (d) Self-organized dynamics: long-time behavior and stability of models capturing flocking behavior.
Mathematical models that directly address fluid-particle interaction and collective self-organization of agents are very relevant to a variety of problems: biological, and non-biological (social networks). Progress in this investigation will enhance our understanding of complex systems. Furthermore, a successful completion of the proposed activities will contribute to the development of new analytical and numerical techniques and to the design of high performance computational algorithms. The project includes a substantial educational component that reflects an ongoing commitment to education on all levels (undergraduate, graduate, and post-graduate education). Special attention is given to outreach activities, minorities, and to developing programs that foster and encourage training in the field of applied mathematics.
