The Vlasov-Poisson-Landau System in a Periodic Box

The classical Vlasov-Poisson-Landau system describes dynamics of a collisional plasma interacting with its own electrostatic …eld as well as its grazing collisions. Such grazing collisions are modeled by the famous Landau (Fokker-Planck) collision kernel, proposed by Landau in 1936. We construct global unique solutions to such a system for initial data which have small weighted H norms, but can have large H(k 3) norms with high velocity moments. Our construction is based on accumulative study on the Landau kernel in the past decade [G1] [SG1-3], with four extra ingredients to overcome the speci…c mathematical di¢ culties present in the Vlasov-Poisson-Landau system: a new exponential weight of electric potential to cancel the growth of the velocity, a new velocity weight to capture the weak velocity di¤usion in the Landau kernel, a decay of the electric …eld to close the energy estimate, and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude.