Paper detail

Decay of the Vlasov-Poisson-Boltzmann system

We establish the time decay rates of the solutions to the Cauchy problem for the two-species Vlasov-Poisson-Boltzmann system near Maxwellians via a refined pure energy method. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov-Poisson-Boltzmann system in which the electric field decays at the optimal algebraic rate or the Vlasov-Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis, and a reformulation of the problem which well displays the cancelation property of the two-species system.