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Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system. (English) Zbl 1129.35023
Authors’ abstract: The time evolution of the distribution function for the charged particles in a dilute gas is governed by the Vlasov-Poisson-Boltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible Navier-Stokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian.

35F20General theory of first order nonlinear PDE
35Q35PDEs in connection with fluid mechanics
35Q60PDEs in connection with optics and electromagnetic theory
76N10Compressible fluids, general
82D10Plasmas (statistical mechanics)
Full Text: DOI
[1] Boltzmann L. (1964) (translated by Stephen G. Brush): Lectures on Gas Theory. New York, Dover Publications Inc
[2] Carleman T. (1932) Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60, 91--142 · doi:10.1007/BF02398270
[3] Cercignani C. (1988) The Boltzmann equation and its applications Applied Mathematical Sciences 67. New York, Springer-Verlag · Zbl 0646.76001
[4] Cercignani C., Illner R., Pulvirenti M. (1994) The Mathematical Theory of Dilute Gases Applied Mathematical Sciences 106. New York, Springer-Verlag · Zbl 0813.76001
[5] Deckelnick K. (1992) Decay estimates for the compressible Navier--Stokes equations in unbounded domains. Math. Z. 209, 115--130 · Zbl 0752.35048 · doi:10.1007/BF02570825
[6] Desvillettes L., Dolbeault J. (1991) On long time asymptotics of the Vlasov--Poisson--Boltzmann equation. Comm. Partial Differ. Eqs. 16 (2--3): 451--489 · Zbl 0737.35127 · doi:10.1080/03605309108820765
[7] Duan R., Yang T., Zhu C.-J. (2005) Boltzmann equation with external force in infinite vacuum. J. Math. Phys. 46(5): 053307 · Zbl 1110.82034 · doi:10.1063/1.1899985
[8] Glassey R.: The Cauchy Problem in Kinetic Theory. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1996 · Zbl 0858.76001
[9] Glassey R., Strauss W.-A. (1999) Decay of the linearized Boltzmann--Vlasov system. Transport Theory Statist. Phys. 28(2): 135--156 · Zbl 0983.82018 · doi:10.1080/00411459908205653
[10] Glassey R., Strauss W.-A. (1999) Perturbation of essential spectra of evolution operators and the Vlasov--Poisson--Boltzmann system. Discrete Contin. Dynam. Systems 5(3): 457--472 · Zbl 0951.35102 · doi:10.3934/dcds.1999.5.457
[11] Glassey R., Schaeffer J., Zheng Y.-X. (1996) Steady states of the Vlasov--Poisson--Fokker--Planck system. J. Math. Anal. Appl. 202(3): 1058--1075 · Zbl 0867.35026 · doi:10.1006/jmaa.1996.0360
[12] Golse F., Perthame B., Sulem C. (1986) On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rational Mech. Anal. 103, 81--96 · Zbl 0668.76089 · doi:10.1007/BF00292921
[13] Grad H.: Asymptotic Theory of the Boltzmann Equation II, Rarefied Gas Dynamics. J. A. Laurmann, ed., Vol. 1, New York: Academic Press 1963, pp. 26--59
[14] Guo Y. (2002) The Vlasov--Poisson--Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9): 1104--1135 · Zbl 1027.82035 · doi:10.1002/cpa.10040
[15] Guo Y. (2003) The Vlasov--Maxwell--Boltzmann system near Maxwellians. Invent. Math. 153(3): 593--630 · Zbl 1029.82034 · doi:10.1007/s00222-003-0301-z
[16] Guo Y. (2001) The Vlasov--Poisson--Boltzmann system near vacuum. Commun. Math. Phys. 218(2): 293--313 · Zbl 0981.35057 · doi:10.1007/s002200100391
[17] Guo Y. (2004) The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4): 1081--1094 · Zbl 1065.35090 · doi:10.1512/iumj.2004.53.2574
[18] Hilbert D., (1953) Grundzuge einer Allgemeinen Theorie der Linearen Integralgleichungen (in German). New York N.-Y, Chelsea Publishing Company · Zbl 0050.10201
[19] Huang F.-M., Xin Z.-P., Yang T.: Contact discontinuity with general perturbations for gas motions. Preprint 2004, available at http://www.cityu.edu.hk/rcms/publication/preprint18.pdf
[20] Kawashima S., Matsumura A. (1985) Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1): 97--127 · Zbl 0624.76095 · doi:10.1007/BF01212358
[21] Li H.-L., Matsumura A.: Asymptotic behavior of compressible Navier--Stokes--Poisson system. Preprint
[22] Lions P.-L.: On kinetic equations. In: Proceedings of the International Congress of Mathematicians, Kyoto: Math. Soc. Japan, 1991, pp.1173--1185 · Zbl 0806.35143
[23] Liu T.-P., Yang T., Yu S.-H. (2004) Energy method for the Boltzmann equation. Physica D 188(3-4): 178--192 · Zbl 1098.82618 · doi:10.1016/j.physd.2003.07.011
[24] Liu T.-P., Yang T., Yu S.-H., Zhao H.-J. (2006) Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch. Rational Mech. Anal. 181(2): 333--371 · Zbl 1095.76024 · doi:10.1007/s00205-005-0414-1
[25] Liu T.-P., Yu S.-H. (2004) Boltzmann equation: Micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1): 133--179 · Zbl 1092.82034 · doi:10.1007/s00220-003-1030-2
[26] Matsumura A., Nishida T. (1983) Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89(4): 445--464 · Zbl 0543.76099 · doi:10.1007/BF01214738
[27] Mischler S. (2000) On the initial boundary value problem for the Vlasov--Poisson--Boltzmann system. Commun. Math. Phys. 210(2): 447--466 · Zbl 0983.45007 · doi:10.1007/s002200050787
[28] Ukai S., Yang T., Zhao H.-J. (2005) Global solutions to the Boltzmann equation with external forces. Analysis and Applications 3(2): 157--193 · Zbl 1152.76464 · doi:10.1142/S0219530505000522
[29] Ukai S., Yang T., Zhao H.-J. (2006) Convergence rate to stationary solutions for Boltzmann equation with external force. To appear in Chin. Ann. Math. Ser. B. 27(4): 363--378 · Zbl 1151.76571 · doi:10.1007/s11401-005-0199-4
[30] Yang T., Yu H.-J., Zhao H.-J.: Cauchy problem for the Vlasov--Poisson--Boltzmann system. Arch. Rational Mech. Anal. 182(3), (2006) · Zbl 1104.76086
[31] Yang T., Zhao H.-J. (2006) A new energy method for the Boltzmann equation. J. Math. Phys. 47(5): 053301 · Zbl 1111.82048 · doi:10.1063/1.2195528
[32] Yang T., Zhao H.-J. (2005) A half-space problem for the Boltzmann equation with specular reflection boundary condition. Commun. Math. Phys. 255(3): 683--726 · Zbl 1075.35060 · doi:10.1007/s00220-004-1278-1