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Theory of existence and uniqueness for the nonlinear Maxwell-Boltzmann equation I

Published online by Cambridge University Press:  17 April 2009

Aleksander Glikson
Aleksander Glikson
Affiliation:
Department of Mathematics, University of New England, Armidale, New South Wales.
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Abstract

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A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 16 , Issue 3 , June 1977 , pp. 379 - 414
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Arkeryd, Leif, “On the Boltzmann equation. Part I: Existence”, Arch. Rational Mech. Anal. 45 (1972), 1–16.CrossRefGoogle Scholar
[2]Arkeryd, Leif, “On the Boltzmann equation. Part II: The full initial value problem”, Arch. Rational Mech. Anal. 45 (1972), 17–34.CrossRefGoogle Scholar
[3]Arkeryd, Leif, “An existence theorem for a modified space-inhomogeneous, nonlinear Boltzmann equation”, Bull. Amer. Math. Soc. 78 (1972), 610–614.CrossRefGoogle Scholar
[4]Arnold, V.I. and Avez, A., Ergodic problems of classical mechanics (W.A. Benjamin, New York, Amsterdam, 1968).Google Scholar
[5]Арсеньев, А.А. [A.A. Arsen'ev], ‘Задача Ноши для линеаризованного уравнения Больцмана” [The Cauchy problem for the linearized Boltzmann equation], Ž. Vyčisl. Mat. i Mat. Fiz. 5 (1965), 864–882; U.S.S.R. Computational Math, and Math. Phys. 5 (1965), 110–136.Google ScholarPubMed
[6]Cercignani, Carlo, Mathematical methods in kinetic theory (Plenum Press, New York, 1969).CrossRefGoogle Scholar
[7]Chapman, Sydney and Cowling, T.G., The mathematical theory of nonuniform gases, third edition (Cambridge University Press, Cambridge, 1970).Google Scholar
[8]Ferziger, J.H. and Kaper, H.G., Mathematical theory of transport processes in gases (North-Holland, Amsterdam, London, 1972).Google Scholar
[9]Glikson, Aleksander, “On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off”, Arch. Rational Mech. Anal. 45 (1972), 35–46.CrossRefGoogle Scholar
[10]Glikson, Aleksander, “On solution of the nonlinear Boltzmann equation with a cut-off in an unbounded domain”, Arch. Rational Mech. Anal. 47 (1972), 389–394.CrossRefGoogle Scholar
[11]Grad, Harold, “Principles of the kinetic theory of gases”, Thermodynamik der Gase, 205–294 (Handbuch der Physik, 12. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1958).Google Scholar
[12]Grad, Harold, “Asymptotic theory of the Boltzmann equation”, Phys. Fluids 6 (1963), 147–181.CrossRefGoogle Scholar
[13]Grad, Harold, “Asymptotic theory of the Boltzmann equation, II”, Rarefied gas dynamics, Volume I, 26–59 (Proc. Third Internat. Sympos. Rarefied Gas Dynamics, Paris, 1962. Academic Press, New York, London, 1963).Google Scholar
[14]Grad, Harold, “Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations”, Applications of nonlinear partial differential equations in mathematical physics, 154–183 (Proc. Sympos. Appl. Maths. Amer. Math. Soc., New York City, 1964, Volume 17. Amer. Math. Soc., Providence, Rhode Island, 1965).CrossRefGoogle Scholar
[15]Grad, Harold, “Solution of the Boltzmann equation in an unbounded domain”, Comm. Pure Appl. Math. 18 (1965), 345–354.CrossRefGoogle Scholar
[16]Guiraud, J.-P., “Gas dynamics from the point of view of kinetic theory”, Applied Mechanics., Proc. Thirteenth Internat. Congress Theoretical and Appl. Mech., Moscow, 1972, 104–123 (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
[17]Guiraud, J.-P., “An H theorem for a gas of rigid spheres in a bounded domain”, Colloques Internationaux C.N.R.S., No. 236, Théories cinétiques classiques et relativistes (1975), 29–58.Google Scholar
[18]Kamke, E., Differentialgleichungen reeller Funktionen (Chelsea, New York, 1947).Google Scholar
[19]*Kogan, M.N., Rarefied gas dynamics (Plenum Press, New York, 1969).CrossRefGoogle Scholar
[20]Krasnosel'skiĭ, M.A., The operator of translation along the trajectories of differential equations (Translations of Mathematical Monographs, 19. Amer. Math. Soc., Providence, Rhode Island, 1968).Google Scholar
[21]*Łunc, Michał, “Détermination de la fonction de distribution des vitesses moléculaires dugaz en mouvement stationnaire par la méthode demographique”, Arch. Mech. Stos. 9 (1957), 731–737.Google Scholar
[22]Morgenstern, Dietrich, “Analytical studies related to the Maxwell-Boltzmann equation”, J. Rational Mech. Anal. 4 (1955), 533–555.Google Scholar
[23]Повзнер, А.Я. [A.Ja. Povzner], “Об уравнении больцмана в кинетечесной теории газов” [On the Boltzmann equation of the kinetic theory of gases], Mat. Sb. N.S. 58 (1962), 65–86.Google ScholarPubMed
[24]Scharf, G., “Functional-analytic discussion of the linearized Boltzmann equation”, Helv. Phys. Acta 40 (1967), 929–945.Google Scholar
[25]Scharf, G., “Normal solutions of the linearized Boltzmann equation”, Helv. Phys. Acta 42 (1969), 5–22.Google Scholar
[26]Shizuta, Yasushi, “On the classical solutions of the Boltzmann equation”, Comm. Pure Appl. Math, (to appear).Google Scholar
[27]Shizuta, Yasushi, “The existence and approach to equilibrium of classical solutions of the Boltzmann equation”, Comm. Math. Phys. (to appear).Google Scholar
[28] Truesdell, C. and Muncaster, R.G., Book on Maxwell's theory, in preparation.Google Scholar
[29]Ukai, Seiji, “On the existence of global solutions of mixed problem for non-linear Boltzmann equation”, Proc. Japan Acad. 50 (1974), 179–184.Google Scholar
[30]Валландер, С.В. [S.V. Vallander], “Новые кинетические уравнения в теории одноатомных газов” [New kinetic equations in the theory of monatomic gases], Dokl. Akad. Nauk SSSR 131 (1960), 58–60; Soviet Physics Dokl. 5 (1960/1961), 269–271.Google ScholarPubMed
[31]Wild, E., “On Boltzmann's equation in the kinetic theory of gases”, Proc. Cambridge Philos. Soc. 47 (1951), 602–609.CrossRefGoogle Scholar
[32]*Willis, D.R., “A study of some nearly-free molecular flow problems” (PhD thesis, Princeton University, 1958); or Aero. Eng. Report No. 440, Princeton University, 1958.Google Scholar
[33]Willis, D. Roger, “On the flow of gases under nearly free molecular conditions”, AFOSR TN 58–1093, Report No. 442, AD 207 594 (1958).Google Scholar