Mathematical Analysis of Moment Systems Derived from Kinetic Equations

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Partial differential equations (PDEs) are indispensable tools in modern science. Many physical phenomena such as fluid motion, sound propagation, and heat transfer can all be modeled by certain types of PDEs. Theoretical investigation of PDEs is crucial for us to understand the fundamental properties or behaviors of complex systems.My research concentrates on analysis of PDEs. The main topic of this research program is to study models related to a particular type of PDEs called kinetic equations. Kinetic equations are classical tools to model the evolution of density distributions of particles in large systems. Traditional application areas of kinetic equations include gas dynamics and transport phenomena in nuclear reactors, semiconductors, and plasmas. More recent years have also seen emerging applications of kinetic theory to other branches of science, such as biology, behavioral science, and social networks.In practice when there are too many particles involved in a system, it is often difficult to directly simulate kinetic equations. A typical way to resolve this difficulty is to construct models to approximate the original kinetic equation. In this proposal I will study a family of approximations called moment systems. These systems reduce the full particle distribution to only finitely many averaged quantities related to the distribution function. My main goal in this program is to construct and analyze proper moment systems which can give high accuracy when approximating the underlying kinetic equations. In addition, these moment systems will serve as extensions of classical fluid equations such as the Navier-Stokes system. Hence they will be appropriate models in the regimes for which classical fluid equations are no longer valid. This program will facilitate modeling and computations for capturing multiscale phenomena.