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Topics on Existence Theory of the Boltzmann Equation

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Rarefied Gas Flows Theory and Experiment

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 224))

Abstract

The basic object under study in kinetic theory is the so-called distribution function F(t, x, ξ). At a given instant t and a given place x in ordinary affine eudidean space (in three dimensions) it gives the (relative) probability density of finding a molecule with velocity ξ. We shall say that x belongs to position space and that ξ belongs to velocity space. Latin characters will be used for vectors of position space and Greek ones for vectors of velocity space. From the distribution function F we can deduce the number density of molecules defined as the number of molecules found, at a given instant in the unit volume of (position) space, namely

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References

  1. H. GRAD (1958), Principles of the Kinetic Theory of Gases In Handbuch der Physik, Vol XII, Thermodynamick der Gaze, p. 205–294.

    Google Scholar 

  2. C. CERCIGNANI (1972), On the Boltzmann Equation for Rigid Spheres. Transport Theory and Statistical Physics, 2, 3, p. 211–225, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. CARLEMAN (1957), Problémes mathématiques dans la théorie unitique des gaz. Publications mathématiques de l’Institut Mittag-Leffler.

    Google Scholar 

  4. C. CERCIGNANI (1969), Mathematical Methods in Kinetic Theory. Plemum Press, 1969.

    Google Scholar 

  5. H. GRAD (1963), Asymptotic Theory of the Boltzmann Equation. In LAURMANN Rarefied Gas Dynamics 1962, Vol 1, p. 26–59.

    Google Scholar 

  6. C. CERCIGNANI (1967), On the Boltzmann Equation with Cut-off Potentials, Phys. Fluids, 10, 10, p. 2097–2104, 1967.

    Article  MathSciNet  Google Scholar 

  7. H. DRANGE (1972), The linearized Boltzmann Collision Operator for Cut-off Potentials. Dept. Appl. Math. Univ. Berger Rept. 37.

    Google Scholar 

  8. M.N. KOGAN (1969) Rarefied Gas Dynamics. Plemum Press 1969.

    Google Scholar 

  9. R.G. BARANTSEV (1972), Some Problems of Gas-solid Surface Interaction. Kucheman Progress in aerospace Suonce Volume 13, p. 1–80. Pergamon 1972.

    Google Scholar 

  10. I. KUSCER (1971), Reciprocity in Scattering of Gas Molecules by Surfaces. Surface Science, 25, p. 225–237, 1971.

    Article  Google Scholar 

  11. C. CERCIGNANI, M. LAMPIS (1971), Kinetic Models for Gas Surface Interactions. Transport Theory and Statistical Physics, 1, 2, p. 101–114, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. CERCIGNANI, Scattering Kernels for Gas-surface Interactions. Transport Theory and Statistical Physics, 2, 1, p. 27–53, (1972).

    Article  MathSciNet  Google Scholar 

  13. J.S. DARROZES, J.P. GUIRAUD (1966), Generalisation formelle du theorème H en prèsence de parois. Comptes Rendus 262, p. 1368–1371, 1966.

    Google Scholar 

  14. K.M. CASE (1972), Uniqueness Theorems for the Linearized Boltzmann Equation. Physics Fluids, 15, 3, p. 377–379, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  15. J.P. GUIRAUD (1974), An H Theorem for a Gas of Rigid Spheres in a Bounded Domain. Communication to “Colloque international de CNRS n. 236 Théories cinétiques classiques et relativistes”, Paris, Juin 1974 p. 29–58.

    Google Scholar 

  16. J.H. CHONG, L. SIROVICH (1970), Structure of Three Dimensional Supersonic and Hypersonic Flow. Phys. Fluids 13, 8, p. 1990–1999, 1970.

    Article  MATH  Google Scholar 

  17. N. NARASIMHA (1968), Asymptotic Solutions for the Distribution Function in Non-equilibrium Flow. Part I, The Weak Shock. Journal Fluid Mechanics, 34, 1, p. 1–24, 1968.

    Article  MATH  Google Scholar 

  18. J.P. GUIRAUD (1970), Problème aux Limites Intérieur pour l’équation de Boltzmann linéaire. Journal de Mécanique, 9, 3, p. 443–490, 1970.

    MATH  MathSciNet  Google Scholar 

  19. J.M. SCHECHTER (1971), Principles of Functional Analysis. Academic Press, 1971.

    Google Scholar 

  20. F. RIESZ, B.Sz NAGY (1953), Leçon d’analyse fonctionelle. Académie des Sciences de Hongrie, 1953.

    Google Scholar 

  21. T. KATO (1966), Perturbation Theory for Linear Operators. Springer 1966.

    Google Scholar 

  22. N. DUNFORD, J.T. SCHWARZ, Linear Operators, Part 1, Part 2. Interscience 1958, 1963.

    Google Scholar 

  23. ZAANEN (19 ), Linear Analysis

    Google Scholar 

  24. J. L. LIONS, E. MAGENES (1968), Problèmes aux limites non homogènes et applications. Vol. 1, Dunod 1968.

    Google Scholar 

  25. C. CERCIGNANI (1968), Existence, Uniqueness and Convergence of the Solutions of Models in Kinetic Theory. Journal of Mathematical Physics, 9, 4, p. 633–639, 1968.

    Article  MATH  Google Scholar 

  26. C. CERCIGNANI (1967), Existence and Uniqueness in the Large for Boundary Value Problems in Kinetic Theory. Journal of Mathematical Physics, 8, 8, p. 1653–1656, 1974.

    Article  Google Scholar 

  27. C. RIGOLOT-TURBAT (1971), Problème de Kramers pour l’équation de Boltzmann en théorie cinétique des gaz. Comptes Rendus Ac. Sci. Paris, 273, A, p. 58–61, 1971.

    MathSciNet  Google Scholar 

  28. J.P. GUIRAUD (1972), Problème aux limites intérieur pour l’équation de Boltzmann en régime stationaire, faiblement non linéaire. Journal de Mécanique, 11, 2, p. 183–231, 1972.

    MATH  MathSciNet  Google Scholar 

  29. H. GRAD (1965), Asymptotic Equivalence of the Navier-Stokes and Non-linear Boltzmann Equations. Proceedings of Symposia in Applied Mathematics, Vol. XVII. Applications of Partial Differential Equations in Mathematical Physics. p. 154–183.

    Google Scholar 

  30. Y.P. PAO, Boundary Value Problems for the Linearized and Weakly Non-linear Boltzman Equation. Journal of Math. Physics, 8, 90. 1893–1898, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  31. C. TURBAT (1968), Equation de Boltzmann sur la droite, Comptes Rendus Ac.Sci., Paris 266, p. 258–260.

    MathSciNet  Google Scholar 

  32. C. TURBAT (1969), Equation de Boltzmann sur la droite, Comptes Rendus Ac.Sci., Paris 269, p. 481–484.

    MATH  MathSciNet  Google Scholar 

  33. C. RIGOLOT-TURBAT (1971), Equation de Boltzmann linéaire sur la droite, Comptes Rendus Ac.Sci., Paris 272, p. 697–700.

    MATH  MathSciNet  Google Scholar 

  34. C. RIGOLOT-TURBAT (1971), Equation de Boltzmann non linéaire sur la droite, Comptes Rendus Ac.Sci., Paris 272, p. 763–766.

    MATH  MathSciNet  Google Scholar 

  35. C. RIGOLOT-TURBAT (1976), Equation de Boltzmann linéaire, non statiounairee, sur la demi-droite, avec conditions de reflexion indépendantes du temps. Comptes Rendus 283, p. 683–686.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Universitè P. & M. Curie, Paris

    J. P. Guiraud

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  1. J. P. Guiraud
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Editor information

Editors and Affiliations

  1. Institute of Fundamental Technological Research Warsaw, Poland

    W. Fiszdon

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© 1981 Springer-Verlag Wien

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Guiraud, J.P. (1981). Topics on Existence Theory of the Boltzmann Equation. In: Fiszdon, W. (eds) Rarefied Gas Flows Theory and Experiment. International Centre for Mechanical Sciences, vol 224. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2898-5_2

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  • DOI: https://doi.org/10.1007/978-3-7091-2898-5_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81595-3

  • Online ISBN: 978-3-7091-2898-5

  • eBook Packages: Springer Book Archive

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