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The Vlasov-Poisson-Landau system in a periodic box. (English) Zbl 1251.35167
Summary: The classical Vlasov-Poisson-Landau system describes the dynamics of a collisional plasma interacting with its own electrostatic field as well as its grazing collisions. Such grazing collisions are modeled by the famous Landau (Fokker-Planck) collision kernel, proposed by L. Landau in 1936 [Phys. Z. Sowjetunion 10, 154–164 (1936; Zbl 0015.38202)]. We construct global unique solutions to such a system for initial data which have small weighted \( H^{2}\) norms, but can have large high derivatives with high velocity moments. Our construction is based on the accumulative study of the Landau kernel in the past decade, with four extra ingredients to overcome the specific mathematical difficulties 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 diffusion in the Landau kernel, a decay of the electric field to close the energy estimate, and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude.

MSC:
35Q83 Vlasov equations
35Q84 Fokker-Planck equations
35C05 Solutions to PDEs in closed form
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