Some Problems on Hypocoercivity for Kinetic Equations

The hypocoercivity theory, which is closely related but different to the hypoellipticity theory, has become one of the main focuses in the study of problems from mathematical physics. The main feature of this theory is that the coupling of a degenerate diffusion operator and a conservative operator may give the dissipation in all variables, and the convergence to the equilibrium state which lies in a subspace smaller than the kernel of the diffusion operator. Breakthroughs have been made and substantial results have been obtained recently, especially by Villani and his collaborators, on problems in bounded domains or a torus. However, many challenging problems still remain unsolved. This project will focus on problems on the hypocoercivity theory for kinetic equations in the whole space and try to obtain the optimal convergence rates of solutions to the equilibrium state in some new function spaces. One of the main differences between problems in a bounded domain and those in the whole space is that only algebraic convergence rates are expected in the whole space rather than (almost) exponential decay in the bounded domain.