Global Strong Solutions of the Vlasov–Poisson–Boltzmann System in Bounded Domains

Abstract

When dilute charged particles are confined in a bounded domain, boundary effects are crucial in the global dynamics. We construct a unique global-in-time solution to the Vlasov–Poisson–Boltzmann system in convex domains with the diffuse boundary condition. The construction is based on an L2-L framework with a novel nonlinear-normed energy estimate of a distribution function in some weighted W1,p-spaces and C2,δ-estimates of the self-consistent electric potential. Moreover we prove an exponential convergence of the distribution function toward the global Maxwellian.

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Acknowledgements

The authors thank Hongxu Chen for finding errors in the original manuscript and fixing them. They also thank the referee(s) for useful comments which helped us to improve the clarity of presentation. The authors thank Yan Guo for his interest and discussions. They also thank Clément Mouhot, Lello Esposito, Rossana Marra, Misha Feldman, Hyung Ju Hwang, and Stéphane Mischler for their interest in this project. C.K. especially thanks James Callen (Center for Plasma Theory and Computation) for discussions on the several relevant kinetic models. The authors also thank the kind hospitality ofMFO at Oberwolfach, ICERM, KAIST-CMC,math/applied math departments of Brown, Cambridge, Princeton, USC (during a summer school organized by Juhi Jang), UMN, UIC, UT-Austin, POSTECH, NTU, Lyon 1, and Paris-Dauphine during this research.

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Correspondence to Chanwoo Kim.

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The research is supported in part by National Science Foundation under Grant No. 1501031, Wisconsin Alumni Research Foundation, and the Herchel Smith foundation of the University of Cambridge.

Communicated by N. Masmoudi

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Cao, Y., Kim, C. & Lee, D. Global Strong Solutions of the Vlasov–Poisson–Boltzmann System in Bounded Domains. Arch Rational Mech Anal 233, 1027–1130 (2019). https://doi.org/10.1007/s00205-019-01374-9

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