Archive for Rational Mechanics and Analysis

, Volume 194, Issue 1, pp 253–282 | Cite as

Upper Maxwellian Bounds for the Spatially Homogeneous Boltzmann Equation

  • I. M. GambaEmail author
  • V. Panferov
  • C. Villani


For the spatially homogeneous Boltzmann equation with cutoff hard potentials, it is shown that solutions remain bounded from above uniformly in time by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.


Boltzmann Equation Comparison Principle Linear Boltzmann Equation Moment Inequality Collision Kernel 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arkeryd L.: On the Boltzmann equation. I. Existence II. The full initial value problem. Arch. Ration. Mech. Anal. 45, 1–34 (1972)zbMATHGoogle Scholar
  2. 2.
    Arkeryd L.: L estimates for the space-homogeneous Boltzmann equation. J. Stat. Phys. 31(2), 347–361 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arkeryd L., Esposito R., Pulvirenti M.: The Boltzmann equation for weakly inhomogeneous data. Comm. Math. Phys. 111(3), 393–407 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bellomo N., Toscani G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic stability. J. Math. Phys. 26(2), 334–338 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical Physics Reviews, Vol. 7, vol. 7 of Soviet Sci. Rev. Sect. C Math. Phys. Rev. Harwood Academic, Churchill, 1988, pp. 111–233Google Scholar
  6. 6.
    Bobylev A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88(5–6), 1183–1214 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bobylev A.V., Gamba I. M., Panferov V.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5–6), 1651–1682 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carleman T.: Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60, 91–146 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carleman, T.: Problèmes mathématiques dans la théoriecinétique des gaz. Publ. Sci. Inst. Mittag-Leffler, 2. Almqvist and Wiksell, Uppsala, 1957Google Scholar
  10. 10.
    Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  11. 11.
    Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78(3), 385–390 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Desvillettes L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Ration. Mech. Anal. 123(4), 387–404 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Di Blasio G.: Differentiability of spatially homogeneous solutions of the Boltzmann equation in the non Maxwellian case. Comm. Math. Phys. 38, 331–340 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    DiPerna R., Lions P.-L.: On the Cauchy problem for the Boltzmann equation: Global existence and weak stability. Ann. Math. 130, 321–366 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Elmroth T.: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Ration. Mech. Anal. 82(1), 1–12 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Evans L.C.: Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)Google Scholar
  17. 17.
    Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  18. 18.
    Gamba I.M., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys. 246(3), 503–541 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Vol. 1. Academic Press, New York, London, 1964. Translation of the second Russian edition, Moscow, 1958.Google Scholar
  20. 20.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd ed., Vol. 224 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1983Google Scholar
  21. 21.
    Goudon T.: Generalized invariant sets for the Boltzmann equation. Math. Models Methods Appl. Sci. 7(4), 457–476 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik, Vol.12 (Ed. S. Flügge). Springer, Berlin, 1958, 205–294Google Scholar
  23. 23.
    Hamdache K.: Existence in the large and asymptotic behaviour for the Boltzmann equation. Jpn. J. Appl. Math. 2(1), 1–15 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Illner R., Shinbrot M.: The Boltzmann equation: global existence for a rare gas in an infinite vacuum. Comm. Math. Phys. 95(2), 217–226 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kaniel S., Shinbrot M.: The Boltzmann equation. I. Uniqueness and local existence. Comm. Math. Phys. 58(1), 65–84 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Landau, L.D., Lifshitz, E. M.: Mechanics, 3rd edn. Course of theoretical physics, Vol. 1. Pergamon Press, Oxford, 1976. Translation of the third Russian edition, Moscow, 1973Google Scholar
  27. 27.
    Lanford, III, O.E.: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), 1–111. Springer, Berlin, 1975. Lecture Notes in Phys., Vol. 38Google Scholar
  28. 28.
    Lions P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. II. J. Math. Kyoto Univ. 34(2), 429–461 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mischler S., Perthame B.: Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian. SIAM J. Math. Anal. 28(5), 1015–1027 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mischler S., Wennberg B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(4), 467–501 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Povzner A.J.: On the Boltzmann equation in the kinetic theory of gases. Mat. Sb. (N.S.) 58(100), 65–86 (1962)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Pulvirenti A., Wennberg B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys. 183(1), 145–160 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vedenjapin, V.V.: On an inequality for convex functions, and on an estimate of the collision integral of the Boltzmann equation for a gas of elastic spheres. Dokl. Akad. Nauk SSSR 226(5) 997–1000, (1976). English translation in Soviet Math. Dokl. 17(1) (1976) 218–222Google Scholar
  34. 34.
    Villani, C.: A Review of Mathematical Topics in Collisional Kinetic Theory. Handbook of mathematical fluid dynamics, Vol. I. North-Holland, Amsterdam, 2002, 71–305Google Scholar
  35. 35.
    Wennberg B.: An example of nonuniqueness for solutions to the homogeneous Boltzmann equation. J. Stat. Phys. 95(1–2), 469–477 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  3. 3.UMPA, ENS LyonLyon Cedex 07France

Personalised recommendations