Professor Guo's research is concerned with the rigorous mathematical study of partial differential equations arising in various scientific applications. More specifically, he has been working on PDE arising in
the kinetic theory of statistical physics, especially in connection with the nonlinear stability of their steady states. Kinetic theory is concerned with the study of the dynamics of a large ensemble of 'particles'. Interestingly, such abstract 'particles' can be tiny
gas molecules, or enormous stars in a galaxy. The most fundamental equation in the kinetic theory for describing gas molecules is the celebrated Boltzmann equation. Many fundamental macroscopic fluid
equations, such as the Euler and Navier-Stokes equations, can be derived from the Boltzmann theory. He has been working on stability of Maxwellian states in the Boltzmann theory. In a kinetic theory of stars, collisions among stars are sufficiently rare to be ignored. Therefore, a galaxy or a globular cluster can be modeled as an
ensemble of particles, i.e., stars, which interact only by the gravitational field which they create collectively. The time evolution of a galaxy can then be described by the Vlasov theory. There are many well known steady state galaxy models. Professor Guo has been developing mathematical tools to analyze the dynamical stability of these steady galaxy models. Instabilities of equilibria in many physical and biological sciences has always attracted great attention. It is important, from a scientific point of view, to understand the rate, time scale, structure, pattern and dynamics of various instabilities in a fully nonlinear setting. Professor Guo has been working on developing general mathematical framework to prove and characterize such nonlinear instabilities.