This talk is concerned with the wellposedness of Cauchy problem for Vlasov-Poisson-Boltzmann system and inertial Kuramoto-Sakaguchi equation around equilibriums. For the whole range of the one-species Vlasov-Poisson-Boltzmann system with cutoff soft potentials, we introduce a new time-velocity weighted energy method and based on optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables and obtain a unique global smooth solution for small initial perturbation. For the Kuramoto-Sakaguchi equation with inertia, in a perturbative framework around a Maxwellian type equilibrium, we establish the global-in-time existence and uniqueness of strong solutions for large initial data when the noise strength is large enough, and obtain exponential decay for solutions towards the equilibrium. These are joint works with Professors Seung-Yeal Ha, Huijiang Zhao et. al.