Abstract
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.
Similar content being viewed by others
References
Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55, 30–70 (2002)
Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, 61–95 (2004)
Arkeryd, L.: Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 103, 151–167 (1988)
Arkeryd, L., Nouri, A.: Boltzmann asymptotics with diffuse reflection boundary conditions. Monatsh. Math. 123, 285–298 (1997)
Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the Boltzmann and Landau operators. To appear in Rev. Mat. Iberoam.
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)
Brush, S.: Kinetic Theory, Vol. 2: Irreversible Processes. Oxford: Pergamon Press 1966
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)
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)
Carleman, T.: Sur la théorie de l’equation intégrodifférentielle de Boltzmann. Acta Math. 60, 369–424 (1932)
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)
Carlen, E., Carvalho, M.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74, 743–782 (1994)
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)
Cercignani, C.: Ludwig Boltzmann, the man who trusted atoms. New York: Oxford University Press 1998
Cercignani, C.: Rarefied gas dynamics. From basic concepts to actual calculations. Cambridge: Cambridge University Press 2000
Degond, P., Pareschi, L., Russo, G. (eds.): Modeling and computational methods for kinetic equations. Birkhäuser 2003
Desvillettes, L.: Entropy dissipation rate and convergence in kinetic equations. Commun. Math. Phys. 123, 687–702 (1989)
Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Ration. Mech. Anal. 110, 73–91 (1990)
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 1992
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)
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)
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. Lions
Diaconis, P., Saloff-Coste, L.: Bounds for Kac’s master equation. Commun. Math. Phys. 209, 729–755 (2000)
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)
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)
Filbet, F., Russo, G.: High order numerical methods for the space non-homogeneous Boltzmann equation. J. Comput. Phys. 186, 457–480 (2003)
Filbet, F.: Quelques résultats numériques sur l’équation de Boltzmann non homogène. Preprint 2004
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)
Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation: Convergence proof. Invent. Math. 155, 81–161 (2004)
Grad, H.: On Boltzmann’s H-theorem. J. Soc. Indust. Appl. Math. 13, 259–277 (1965)
Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391–434 (2002)
Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169, 305–353 (2003)
Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153, 593–630 (2003)
Guo, Y.: Private communication (December 2003)
Gustafsson, T.: Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 92, 23–57 (1986)
Helffer, B., Nier, F.: Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians. Preprint 03-25, Université de Rennes (2003). Available at http://name.math.univ-rennes1.fr/francis.nier/recherche/liste.html
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)
Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29, 288–304 (2001)
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 1956
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)
Lu, X.: Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior. SIAM J. Math. Anal. 30, 1151–1174 (1999)
McKean, H. J.: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)
Mouhot, C.: Quantitative lower bounds for the full Boltzmann equation. Preprint 2003
Mouhot, C., Villani, C.: Regularity theory for the spatially homogeneous Boltzmann equation with cutoff. Arch. Ration. Mech. Anal., in press
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)
Poincaré, H.: Le mécanisme et l’expérience. Revue de Métaphysique et de Morale I, 534–537 (1893)
Pulvirenti, A., Wennberg, B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183, 145–160 (1997)
Shizuta, Y., Asano, K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Japan Acad., Ser. A 53, 3–5 (1977)
Spohn, H.: Large scale dynamics of interacting particles. Texts and Monographs in Physics. Berlin: Springer 1991
Sznitman, A.-S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX–1989, pp. 165–251. Berlin: Springer 1991
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)
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)
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)
Truesdell, C., Muncaster, R.: Fundamentals of Maxwell’s kinetic theory of a simple monoatomic gas. New York: Academic Press 1980
Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974)
Villani, C.: Fisher information bounds for Boltzmann’s collision operator. J. Math. Pures Appl. 77, 821–837 (1998)
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. 2000/2001
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 2002
Villani, C.: On the Boltzmann equation with singular kernel. Unpublished notes
Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234, 455–490 (2003)
Villani, C.: Topics in Optimal Transportation, vol. 58 of Graduate Series in Mathematics. Providence: American Mathematical Society 2003
Wennberg, B.: Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonlinear Anal. 20, 935–964 (1993)
Wennberg, B.: Stability and exponential convergence for the Boltzmann equation. Arch. Ration. Mech. Anal. 130, 103–144 (1995)
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Desvillettes, L., Villani, C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. math. 159, 245–316 (2005). https://doi.org/10.1007/s00222-004-0389-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-004-0389-9