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The Vlasov-Poisson-Landau system in a periodic box

Author: Yan Guo
Journal: J. Amer. Math. Soc. 25 (2012), 759-812
MSC (2010): Primary 35-XX
Published electronically: October 25, 2011
MathSciNet review: 2904573
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Abstract: 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 Landau in 1936. 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.

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Additional Information

Yan Guo
Affiliation: Division of Applied Mathematics, Brown University, Box F, Providence, Rhode Island 02912

Received by editor(s): March 19, 2011
Received by editor(s) in revised form: June 27, 2011, and September 3, 2011
Published electronically: October 25, 2011
Additional Notes: This research is supported in part by NSF grant #0905255 and FRG grants as well as a Chinese NSF grant #10828103.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.