The Vlasov-Poisson-Boltzmann System for Soft Potentials
  An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov-Poisson-Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov-Poisson-Boltzmann system for angular cutoff soft potentials with $-2\leq \gamma<0$. The main idea is to introduce a time dependent weight function in the velocity variable to capture the singularity of the cross-section at zero relative velocity. 
Renjun Duan, Tong Yang, Huijiang Zhao
math.AP