Abstract
The dynamics of dilute electrons can be modelled by the fundamental Vlasov–Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [17, 19] by obtaining new entropy estimates for this physical model.
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References
Boltzmann L. (1964) (translated by Stephen G. Brush): Lectures on Gas Theory. Dover Publications, Inc. New York
Cercignani C. The Boltzmann Equation and its Applications. Applied Mathematical Sciences, 67. Springer-Verlag, New York, xii+455 pp, 1988
Cercignani, C., Illner, R., Pulvirenti, M. The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer-Verlag, New York, viii+347 pp, 1994
Desvillettes L., Dolbeault J. (1991) On long time asymptotics of the Vlasov-Poisson-Boltzmann equation. Comm. Partial Differential Equations 16, 451–489
Diperna R.J., Lions P.-L. (1989) Global weak solutions ofVlasov-Maxwell systems. Comm. Pure Appl. Math. 42, 729–757
Glassey, R.T. The Cauchy problem in kinetic theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, xii+241 pp, 1996
Glassey R.T., Strauss W.A. (1999) Decay of the linearized Boltzmann-Vlasov system. Transport Theory Statist. Phys. 28, 135–156
Golse F., Perthame B., Sulem C. (1986) On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Ration Mech. Anal. 103, 81–96
Grad H. (1963) Asymptotic theory of the boltzmann equation II. Rarefied Gas Dynamics (Laurmann, J.A. Ed.). Vol. 1, Academic Press, New York, pp. 26–59
Guo Y. (2002) The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55, 1104–1135
Guo Y. (2003) The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153, 593–630
Guo Y. (2001) The Vlasov-Poisson-Boltzmann system near vacuum. Comm. Math. Phys. 218, 293–313
Kawashima S., Matsumura A. (1985) Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys. 101, 97–127
Lions, P.-L. On kinetic equations. Proceedings of the International Congress of Mathematicians Kyoto, 1173–1185, Math. Soc. Japan, 1991
Liu T.-P., Yang T., Yu S.-H. (2004) Energy method for the Boltzmann equation. Physica D 188, 178–192
Liu T.-P., Yang, T., Yu, S.-H., Zhao, H.-J. Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch. Ration Mech. Anal. DOI: 10.1007/s00205-005-0414-1
Liu T.-P., Yu S.-H. (2004) Boltzmann equation: Micro-macro decompositions and positivity of shock profiles. Comm. Math. Phys. 246, 133–179
Mischler S. (2000) On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Comm. Math. Phys. 210, 447–466
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Communicated by C.M. Dafermos
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Yang, T., Yu, H. & Zhao, H. Cauchy Problem for the Vlasov–Poisson–Boltzmann System. Arch Rational Mech Anal 182, 415–470 (2006). https://doi.org/10.1007/s00205-006-0009-5
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DOI: https://doi.org/10.1007/s00205-006-0009-5