Inventiones mathematicae

, Volume 159, Issue 2, pp 245–316 | Cite as

On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation

  • L. Desvillettes
  • C. Villani


As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, started in [21], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t-∞). Our results hold conditionally to some strong but natural estimates of smoothness, decay at large velocities and strict positivity, which at the moment have only been established in certain particular cases. Among the most important steps in our proof are 1) quantitative variants of Boltzmann’s H-theorem, as proven in [52,60], based on symmetry features, hypercontractivity and information-theoretical tools; 2) a new, quantitative version of the instability of the hydrodynamic description for non-small Knudsen number; 3) some functional inequalities with geometrical content, in particular the Korn-type inequality which we established in [22]; and 4) the study of a system of coupled differential inequalities of second order, by a treatment inspired from [21]. We also briefly point out the particular role of conformal velocity fields, when they are allowed by the geometry of the problem.


Boltzmann Equation Entropy Production Differential Inequality Collision Operator Landau Equation 
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  1. 1.
    Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55, 30–70 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, 61–95 (2004)MathSciNetGoogle Scholar
  3. 3.
    Arkeryd, L.: Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 103, 151–167 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arkeryd, L., Nouri, A.: Boltzmann asymptotics with diffuse reflection boundary conditions. Monatsh. Math. 123, 285–298 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the Boltzmann and Landau operators. To appear in Rev. Mat. Iberoam.Google Scholar
  6. 6.
    Bardos, C., Golse, F., Levermore, D.: Fluid dynamical limits of kinetic equations, II: Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46, 667–753 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brush, S.: Kinetic Theory, Vol. 2: Irreversible Processes. Oxford: Pergamon Press 1966Google Scholar
  8. 8.
    Cáceres, M.-J., Carrillo, J.-A., Goudon, T.: Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Commun. Partial Differ. Equations 28, 969–989 (2003)CrossRefGoogle Scholar
  9. 9.
    Caflisch, R.: The Boltzmann equation with a soft potential. I. linear, spatially-homogeneous. II. nonlinear, spatially-periodic. Commun. Math. Phys. 74, 71–95, 97–109 (1980)MathSciNetGoogle Scholar
  10. 10.
    Carleman, T.: Sur la théorie de l’equation intégrodifférentielle de Boltzmann. Acta Math. 60, 369–424 (1932)Google Scholar
  11. 11.
    Carlen, E., Carvalho, M.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys. 67, 575–608 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Carlen, E., Carvalho, M.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74, 743–782 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Carlen, E., Carvalho, M., Loss, M.: Determination of the spectral gap for Kac’s master equation and related stochastic evolutions. Acta Math. 191, 1–54 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cercignani, C.: Ludwig Boltzmann, the man who trusted atoms. New York: Oxford University Press 1998Google Scholar
  15. 15.
    Cercignani, C.: Rarefied gas dynamics. From basic concepts to actual calculations. Cambridge: Cambridge University Press 2000Google Scholar
  16. 16.
    Degond, P., Pareschi, L., Russo, G. (eds.): Modeling and computational methods for kinetic equations. Birkhäuser 2003Google Scholar
  17. 17.
    Desvillettes, L.: Entropy dissipation rate and convergence in kinetic equations. Commun. Math. Phys. 123, 687–702 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Ration. Mech. Anal. 110, 73–91 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Desvillettes, L.: Convergence to equilibrium in various situations for the solution of the Boltzmann equation. In: Nonlinear kinetic theory and mathematical aspects of hyperbolic systems (Rapallo, 1992), pp. 101–114. River Edge, NJ: World Sci. Publishing 1992Google Scholar
  20. 20.
    Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. Commun. Partial Differ. Equations 25, 261–298 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Commun. Pure Appl. Math. 54, 1–42 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Desvillettes, L., Villani, C.: On a variant of Korn’s inequality arising in statistical mechanics. ESAIM, Control Optim. Calc. Var. 8, 603–619 (2002) (electronic). A tribute to J.L. LionsMathSciNetCrossRefGoogle Scholar
  23. 23.
    Diaconis, P., Saloff-Coste, L.: Bounds for Kac’s master equation. Commun. Math. Phys. 209, 729–755 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    DiPerna, R., Lions, P.-L.: On the Cauchy problem for the Boltzmann equation: Global existence and weak stability. Ann. Math. (2) 130, 312–366 (1989)Google Scholar
  25. 25.
    Fellner, K., Neumann, L., Schmeiser, C.: Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes. Monatsh. Math. 141, 289–299 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Filbet, F., Russo, G.: High order numerical methods for the space non-homogeneous Boltzmann equation. J. Comput. Phys. 186, 457–480 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Filbet, F.: Quelques résultats numériques sur l’équation de Boltzmann non homogène. Preprint 2004Google Scholar
  28. 28.
    Gallay, T., Wayne, C.: Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ2. Arch. Ration. Mech. Anal. 163, 209–258 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation: Convergence proof. Invent. Math. 155, 81–161 (2004)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Grad, H.: On Boltzmann’s H-theorem. J. Soc. Indust. Appl. Math. 13, 259–277 (1965)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391–434 (2002)CrossRefGoogle Scholar
  32. 32.
    Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169, 305–353 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153, 593–630 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Guo, Y.: Private communication (December 2003)Google Scholar
  35. 35.
    Gustafsson, T.: Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 92, 23–57 (1986)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Helffer, B., Nier, F.: Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians. Preprint 03-25, Université de Rennes (2003). Available at Scholar
  37. 37.
    Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential. Arch. Ration. Mech. Anal. 171, 151–218 (2004)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29, 288–304 (2001)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. Berkeley, Los Angeles: University of California Press 1956Google Scholar
  40. 40.
    Lions, P.-L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 158, 173–193, 195–211 (2001)CrossRefGoogle Scholar
  41. 41.
    Lu, X.: Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior. SIAM J. Math. Anal. 30, 1151–1174 (1999)MathSciNetCrossRefGoogle Scholar
  42. 42.
    McKean, H. J.: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Mouhot, C.: Quantitative lower bounds for the full Boltzmann equation. Preprint 2003Google Scholar
  44. 44.
    Mouhot, C., Villani, C.: Regularity theory for the spatially homogeneous Boltzmann equation with cutoff. Arch. Ration. Mech. Anal., in pressGoogle Scholar
  45. 45.
    Pitteri, M.: On the asymptotic behaviour of Boltzmann’s H function in the kinetic theory of gases. Rend. Sc. Fis. Mat. e Nat. 67, 248–251 (1979)MathSciNetGoogle Scholar
  46. 46.
    Poincaré, H.: Le mécanisme et l’expérience. Revue de Métaphysique et de Morale I, 534–537 (1893)Google Scholar
  47. 47.
    Pulvirenti, A., Wennberg, B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183, 145–160 (1997)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Shizuta, Y., Asano, K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Japan Acad., Ser. A 53, 3–5 (1977)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Spohn, H.: Large scale dynamics of interacting particles. Texts and Monographs in Physics. Berlin: Springer 1991Google Scholar
  50. 50.
    Sznitman, A.-S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX–1989, pp. 165–251. Berlin: Springer 1991Google Scholar
  51. 51.
    Toscani, G.: H-theorem and asymptotic trend to equilibrium of the solution for a rarefied gas in the vacuum. Arch. Ration. Mech. Anal. 100, 1–12 (1987)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Toscani, G., Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203, 667–706 (1999)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Toscani, G., Villani, C.: On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Stat. Phys. 98, 1279–1309 (2000)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Truesdell, C., Muncaster, R.: Fundamentals of Maxwell’s kinetic theory of a simple monoatomic gas. New York: Academic Press 1980Google Scholar
  55. 55.
    Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Villani, C.: Fisher information bounds for Boltzmann’s collision operator. J. Math. Pures Appl. 77, 821–837 (1998)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Villani, C.: Limites hydrodynamiques de l’équation de Boltzmann (d’après C. Bardos, F. Golse, C. D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond). Astérisque 282 (2002), Exp. No. 893, ix, 365–405. Séminaire Bourbaki, Vol. 2000/2001Google Scholar
  58. 58.
    Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, Vol. I, pp. 71–305. Amsterdam: North-Holland 2002Google Scholar
  59. 59.
    Villani, C.: On the Boltzmann equation with singular kernel. Unpublished notesGoogle Scholar
  60. 60.
    Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234, 455–490 (2003)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Villani, C.: Topics in Optimal Transportation, vol. 58 of Graduate Series in Mathematics. Providence: American Mathematical Society 2003Google Scholar
  62. 62.
    Wennberg, B.: Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonlinear Anal. 20, 935–964 (1993)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Wennberg, B.: Stability and exponential convergence for the Boltzmann equation. Arch. Ration. Mech. Anal. 130, 103–144 (1995)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag 2004

Authors and Affiliations

  1. 1.CMLAENS CachanCachanFrance
  2. 2.UMPAENS LyonLyonFrance

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