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Initial-Boundary value problems for the Boltzmann equation: Global existence of weak solutions

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Abstract

For Ω an open set of ℝ3 bounded or not, we consider initial-boundary value problems for the Boltzmann equation. For general gas-surface interaction laws and for hard potentials, we prove a global existence result for weak solutions. The proof uses the regularization of the collision operator and the renormalization method for the regularized problem. By using weak compactness in L1 and averaged stability ofQ(f,f), we prove the existence of weak solutions of our problem.

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References

  1. Arkeryd, L. On the Boltzmann equation. Part I. Existence. Arch. Rational Mech. Anal.45, 1–16 (1972).

    Google Scholar 

  2. Arkeryd, L. On the Boltzmann equation. Part. II. The full initial value problem. Arch. Rational Mech. Anal.45, 17–34 (1972).

    Google Scholar 

  3. Arkeryd, L. Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rational. Mech. Anal.77, 11–23 (1981).

    Google Scholar 

  4. Arkeryd, L. Loeb solutions of the Boltzmann equation. Arch. Rational Mech. Anal.86, 85–97 (1984).

    Google Scholar 

  5. Asano, K. Local solutions to the initial and initial boundary value problems for the Boltzmann equation with an external force, J. Math. Kyoto Univ.24, 225–238 (1984).

    Google Scholar 

  6. Asano, K.On the initial boundary value problem of the nonlinear Boltzmann equation in an exterior domain. To appear.

  7. Caflisch, R. The Boltzmann equation with a soft potential. Part I. Comm. Math. Phys.74, 71–95 (1980).

    Google Scholar 

  8. Carleman, T. Problèmes mathématiques dans la théorie cinétique des gaz. Almquist Wiksell, Uppsala (1957).

    Google Scholar 

  9. Cercignani, C. Theory and application of the Boltzmann equation. Scottish Academic Press (1975).

  10. Cercignani, C. Mathematical methods in kinetic theory. Plenum Press, New York (1969).

    Google Scholar 

  11. Cessenat, M. Théorèmes de trace L p pour des espaces de fonctions de la neutronique. Note C.R. Acad. Sci. Paris, Sér. 1,299, 831–834 (1984).

    Google Scholar 

  12. Cessenat, M. Théorèmes de traces pour les espaces de fonctions de la neutronique. Note C.R. Acad. Sci. Paris, Sér. 1,300, 89–92 (1985).

    Google Scholar 

  13. Darozes, J. S. &Guiraud, J. P. Généralisation formelle du théorème H en présence de parais. Note C.R. Acad. Sci. Paris,262A, 368–1371 (1966).

    Google Scholar 

  14. Dautray, R. &Lions, J. L. Analyse mathématique et calcul numérique pour les sciences et les techniques. Tôme 3, chap. 21, Masson (1985).

  15. DiPerna, R. &Lions, P. L. On the Cauchy problem for Boltzmann equation. Global existence and weak stability. Ann. of Math.130, 321–366 (1989).

    Google Scholar 

  16. DiPerna, R. &Lions, P. L. To appear.

  17. Elmroth, T. Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rational Mech. Anal.82, 1–12 (1983).

    Google Scholar 

  18. Elmroth, T. On the H-function and convergence towards equilibrium for a space — homogeneous molecular density. SIAM J. Appl. Math.44, 150–159 (1984).

    Google Scholar 

  19. Elmroth, T. Loeb solutions of the Boltzmann equation with initial boundary values and external forces. Preprint 1984-23. Chalmers Univ. of Technology Göteborg.

  20. Gatignol, R. Théorie cinétique des gaz à répartition discrète de vitesses. Lecture Notes in Physics36, Springer-Verlag (1975).

  21. Godunov, S. K. &Sultangazin, U. M. On discrete models of kinetic Boltzmann equation. Russian Math Surveys26, 1–56 (1971).

    Google Scholar 

  22. Golse, F., Perthame, B. &Sentis, R. Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport. Note C.R. Acad. Sci. Paris301, 341–344 (1985).

    Google Scholar 

  23. Golse, F., Lions, P. L., Perthame, B. &Sentis, R. Regularity of the moments of the solution of a transport equation. J. Funct. Anal.76, 110–125 (1988).

    Google Scholar 

  24. Grad, H. Asymptotic theory of the Boltzmann equation, II. in Rarefied Gas Dynamics. Vol. 1, Ed.Laurmann, J., 26–59, Acad. Press (1963).

  25. Grad, H. Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equation. Proc. Symp. Appl. Math.17, 154–183 (1965).

    Google Scholar 

  26. Guiraud, J. P. An H-theorem for a gas of rigid spheres in a bounded domain. Colloque Int. C.N.R.S.236, 29–58 (1975).

    Google Scholar 

  27. Guiraud, J. P. Topics on existence theory of the Boltzmann equation. Rarefied Gas flows, Theory and Experiment. Ed.Fiszdon, W., Springer (1981).

  28. Hamdache, K. Problèmes aux Limites pour l'équation de Boltzmann: Existence globale de solution. Comm. Part. Diff. Eqs.13, (1988).

  29. Hamdache, K. Théorèmes de compacité par compensation et application en théorie cinétique des gaz. C.R. Acad. Sci. Paris302, 151–154 (1986).

    Google Scholar 

  30. Hamdache, K. Estimations uniformes des solutions de l'équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck. Note C.R. Acad. Sci. Paris302, 187–190 (1986).

    Google Scholar 

  31. Hamdache, K. Sur l'existence globale et le comportement asymptotique de quelques solutions de l'équation de Boltzmann. Thèse de Doctorat d'Etat. Université Pierre et Marie Curie, Paris (1986).

    Google Scholar 

  32. Kaniel, S. &Shinbrot, M. The Boltzmann equation. I, uniqueness and local existence. Commun. Math. Phys.58, 65–84 (1978).

    Google Scholar 

  33. Kogan, M. N. Rarefied Gas Dynamics. Plenum Press, New York (1969).

    Google Scholar 

  34. Monaco, R. Introduction à la théorie et aux applications de l'interaction gaz-paroi en théorie cinétique des gaz. Libreria editrice universitaria Levrotto & Bella (1986).

  35. Morgenstern, D. General existence and uniqueness proof for spatially homogeneous solution of the Maxwell-Boltzmann equation in the case of Maxwellien molecules. Proc. Nat. Acad. Sci. U.S.A.40, 719–721 (1954).

    Google Scholar 

  36. Morgenstern, D. Analytical studies related to the Maxwell-Boltzmann equation. J. Rational Mech. Anal.4, 533–555 (1955).

    Google Scholar 

  37. Povzner, A. JA. About the Boltzmann equation in kinetic gas theory. Mat. Sborn.58 (100), 65–86 (1962).

    Google Scholar 

  38. Shinbrot, M. The Boltzmann equation. Flaw of cloud of gas past a body. Transport Theory and Stat. Physics.15, 317–322 (1986).

    Google Scholar 

  39. Sultangazin, U. M. Discrete nonlinear models of the Boltzmann equation. Nauka, Moscow (1987).

    Google Scholar 

  40. Toscani, G. &Protopopescu, V. Existence globale pour un problème mixte associé à l'équation de Boltzmann non linéaire. Note C.R. Acad. Sci. Paris, Sér. 1,302, 255–258 (1986).

    Google Scholar 

  41. Truesdell, C. &Muncaster, R. G. Fundamentals of Maxwell's kinetic theory of simple monoatomic gas. Academic Press. New York (1980).

    Google Scholar 

  42. Ukaï, S. &Asano, K. Steady solutions of the Boltzmann equation for a gas flow past an obstacle, I. Existence. Arch. Rational Mech. Anal.84, 249–291 (1983).

    Google Scholar 

  43. Ukaï, S. &Asano, K. Steady solution of the Boltzmann equation for a gas flow past an obstacle. II. Stability. Publ. RIMS. Kyoto Univ.22, 1035–1062 (1986).

    Google Scholar 

  44. Ukaï, S. On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation. Proc. Japan Acad.50, 179–184 (1974).

    Google Scholar 

  45. Ukaï, S. Les solutions globales de l'équation de Boltzmann dans l'espace tout entier et le demi-espace. C.R. Acad. Sci. Paris,282A, 317–320 (1976).

    Google Scholar 

  46. Ukaï, S. Solutions of the Boltzmann equation. InPatterns and Waves. InQualitative analysis of differential equations, 37–96 (1986).

  47. Voigt, J. Functional analytic treatment of the initial boundary value problem of collisionless gas. Habilitationsschrift. Universität München (1980).

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  1. Chemin de la Hunière, CNRS - ENSTA/GHN Centre de l'Yvette, 91120, Palaiseau, France

    K. Hamdache

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  1. K. Hamdache
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Communicated by C. M.Dafermos

Dedicated to the Memory of Ronald DiPerna

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Hamdache, K. Initial-Boundary value problems for the Boltzmann equation: Global existence of weak solutions. Arch. Rational Mech. Anal. 119, 309–353 (1992). https://doi.org/10.1007/BF01837113

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