This project is concerned with the investigation of physically important phenomena driven by gravity and diffusion, based on partial differential equations (PDE) approaches. In particular, such diverse phenomena as collapse of stars or generation of vortices at the interface between two fluids give rise to similar mathematical models. These model equations (Euler-Poisson, Navier-Stokes systems, Fokker-Planck equations, kinetic transport and Boltzmann equations) are widely used to describe the motion of compressible fluids and gases. They have rich applications in mathematical sciences and engineering and pose formidable mathematical challenges. This project aims to advance knowledge in this fundamental area of mathematics and to influence other domains of mathematics. Results will provide some evidence to other disciplines such as astrophysics, plasma physics, aerodynamics, and computational physics, chemistry, and biology, and may lead to scientific and technological advances. The research project will be integrated with educational and outreach activities such as student research projects, one-to-one mentoring activities, course development, summer schools for undergraduate and graduate students and interdisciplinary conferences. The project will strengthen the Applied Mathematics program at University of California-Riverside by means of integrated research, education and outreach activities. Through summer schools and conferences, it will be providing excellent learning opportunities for students with one of the goals being to broaden participation of underrepresented groups.
In particular, the following topics will be studied: (i) gravitational collapses of the Euler-Poisson system, the dynamics of rotating stars, relativity and radiation, (ii) Rayleigh-Taylor instability and the counterpart stability of two-fluid compressible Navier-Stokes system with or without surface tension, (iii) absorbing, elastically and inelastically reflecting boundary collisions for the kinetic Fokker-Planck equations, (iv) multi-scale dynamics from the kinetic transport and Boltzmann equations in the presence of boundary: theory and computation. The goal is to find a mathematical framework where these physical phenomena can be captured and to develop an appropriate mathematical theory by using PDE methods.
