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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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The Vlasov-Poisson-Landau system in a periodic box
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by Yan Guo
J. Amer. Math. Soc. 25 (2012), 759-812
Published electronically: October 25, 2011


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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Bibliographic 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 759-812
  • MSC (2010): Primary 35-XX
  • DOI:
  • MathSciNet review: 2904573