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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Arsenio, D., Saint-Raymond, L.: From the Vlasov–Maxwell–Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. arXiv:1604.01547
Cao, Y.: Regularity of Boltzmann equation with external fields in convex domains of diffuse reflection. submitted, arXiv:1812.09388
Cao, Y.: A note on two species collisional plasma in bounded domains, arXiv:1903.04935
Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer, New York, 1994
Chen, H., Kim, C., Li, Q.: Local Well-Posedness of Vlasov–Possion–Boltzmann System with Generalized Diffuse Boundary Condition, preprint
Desvillettes, L., Dolbeault, J.: On long time asymptotics of the Vlasov-Poisson-Boltzmann equation. Commun. Partial Differ. Equ. 16(2–3), 451–489 (1991)
DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321–366 (1989)
Devillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)
Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the Hydrodynamic limit. Ann. PDE 4(1), Art. 1, pp. 119, 2018
Esposito, R., Guo, Y., Kim, C., Marra, R.: Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law. Commun. Math. Phys. 323(1), 177–239 (2003)
Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order, Reprint of the, 1998th edn. Classics in Mathematics, Springer, Berlin (2001)
Glassey, R., Strauss, W.: Decay of the linearized Boltzmann–Vlasov system. Transp. Theor. Stat. 28(2), 35–156 (1999)
Gualdani, M.P., Mischler, S., Mouhot, C.: Factorization for non-symmetric operators and exponential H-theorem. Mem. Soc. Math. Fr. (N.S.) No. 153, pp. 137, 2017
Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Commun. Pure Appl. Math. 55(9), 1104–1135 (2002)
Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153(3), 593–630 (September 2003)
Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains. Invent. Math. 207(1), 115–290 (January 2017)
Guo, Y., Kim, C., Tonon, D., Trescases, A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Ration. Mech. Anal. 220(3), 1045–1093 (2016)
Guo, Y.: Decay and Continuity of Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 713–809 (2010)
Guo, Y., Jang, J.: Global Hilbert expansion for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 299(2), 469–501 (2010)
Guo, Y.: Regularity of the Vlasov equations in a half space. Indiana. Math. J. 43, 255–320 (1994)
Glassey, R.: The Cauchy Problems in Kinetic Theory. SIAM, Philadelphia (1996)
Hwang, H.-J., Velazquez, J.: Global existence for the Vlasov-Poisson system in bounded domains. Arch. Rat. Mech. Anal. 195(3), 763–796 (2010)
Jang, J., Masmoudi, N.: Derivation of Ohm's law from the kinetic equations. SIAM J. Math. Anal. 44(5), 3649–3669 (2012)
Kim, C.: Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308(3), 641–701 (2011)
Kim, C.: Boltzmann equation with a large external field. Comm. PDE. 39(8), 1393–1423 (2014)
Kim, C., Lee, D.: The Boltzmann equation with specular boundary condition in convex domains. Comm. Pure Appl. Math. 71(3), 411–504 (2018)
Kim, C., Lee, D.: Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains. Arch. Ration. Mech. Anal. 230(1), 49–123 (2018)
Kim, C., Lee, D.: Hölder Regularity Propagation for the Boltzmann equation Past an Obstacle, in preparation
Kim, C., Yun, S.: The boltzmann equation near a rotational local maxwellian. SIAM J. Math. Anal. 44(4), 2560–2598 (2012)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2001)
Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 157, 49–88 (1866)
Mischler, S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210, 447–466 (2000)
Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Springer, Berlin (2009)
Strain, R.: Asymptotic stability of the relativistic Boltzmann equation for the soft potentials. Commun. Math. Phys. 300(2), 529–597 (2010)
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
About this article
Cite this article
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