Abstract
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients \({\alpha \in [0,1]}\) . In the physical regime of a small inelasticity (that is \({\alpha \in [\alpha_*,1)}\) for some constructive \({\alpha_* \in [0,1)}\)) we prove uniqueness of the self-similar profile for given values of the restitution coefficient \({\alpha \in [\alpha_*,1)}\) , the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation).
Moreover, if the initial datum lies in \({L^1_3}\) , and under some smallness condition on \({(1-\alpha_*)}\) depending on the mass, energy and \({L^1 _3}\) norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called homogeneous cooling state).
These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the “quasi-elastic self-similar temperature” and the rate of convergence towards self-similarity at first order in terms of (1−α), are obtained from our study.
These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.
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References
Abrahamsson F.: Strong L 1 convergence to equilibrium without entropy conditions for the Boltzmann equation. Comm. Part. Diff. Eqs. 24, 1501–1535 (1999)
Bisi M., Carrillo J.A., Toscani G.: Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria. J. Stat. Phys. 118(1–2), 301–331 (2005)
Bisi M., Carrillo J.A., Toscani G.: Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model. J. Stat. Phys. 124(2–4), 625–653 (2006)
Blake M.D.: A spectral bound for asymptotically norm-continuous semigroups. J. Op. Th. 45, 111–130 (2001)
Bobylev A.V., Carillo J.A., Gamba I.: On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000)
Baranger C., Mouhot C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Matem. Iberoam. 21, 819–841 (2005)
Bobylev A.V., Cercignani C., Toscani G.: Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Stat. Phys. 111, 403–417 (2003)
Bobylev A.V., Gamba I., Panferov V.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. J. Stat. Phys. 116, 1651–1682 (2004)
Brilliantov N.V., Pöschel T.: Kinetic Theory of Granular Gases. Oxford Graduate Texts. Oxford University Press, Oxford (2004)
Caglioti E., Villani C.: Homogeneous cooling states are not always good approximations to granular flows. Arch. Rat. Mech. Anal. 163, 329–343 (2002)
Carleman T.: Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60, 91–146 (1932)
Carleman, T.: Problèmes mathématiques dans la théorie cinétique des gaz. Uppsala: Almqvist and Wiksells Boktryckeri Ab, 1957
Cercignani, C.: Recent developments in the mechanics of granular materials. In: Fisica matematica e ingegneria delle strutture, Bologna: Pitagora Editrice, 1995, pp. 119–132
Csiszár I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 85–108 (1963)
Ernst M.H., Brito R.: Driven inelastic Maxwell molecules with high energy tails. Phys. Rev. E 65, 85–108 (2002)
Ernst M.H., Brito R.: Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails. J. Stat. Phys. 109, 407–432 (2002)
Gamba I., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246, 503–541 (2004)
Grad, H.: Asymptotic theory of the Boltzmann equation. II. In: Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962), Vol. I, New York: Academic Press, 1963, pp 26–59
Haff P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430 (1983)
Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. Math. Ann. 72, (1912), New York: Chelsea Publ., 1953
Kato T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1995)
Kullback S.: Information Theory and Statistics. John Wiley, New York (1959)
Mischler S., Mouhot C., Rodriguez Ricard M.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem. J. Stat. Phys. 124, 655–702 (2006)
Mischler S., Mouhot C.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I:Self-similar solutions and tail behavior. J. Stat. Phys. 124, 703–746 (2006)
Mischler, S., Mouhot, C.: Work in progress
Mischler S., Wennberg B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. Henri Poincaré, Analyse non linéaire 16, 467–501 (1999)
Mouhot C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 261, 629–672 (2006)
Mouhot C., Villani C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rat. Mech. Anal. 173, 169–212 (2004)
Nirenberg, L.: Topics in Nonlinear Functional Analysis. With a chapter by E. Zehnder. Notes by R. A. Artino. Lecture Notes, 1973–1974. New York: Courant Institute of Mathematical Sciences, New York University, 1974
Pulvirenti A., Wennberg B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183, 145–160 (1997)
Villani C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234, 455–490 (2003)
Yao P.F.: On the inversion of the Laplace transform of C 0 semigroups and its applications. SIAM J. Math. Anal. 26, 1331–1341 (1995)
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Mischler, S., Mouhot, C. Stability, Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres. Commun. Math. Phys. 288, 431–502 (2009). https://doi.org/10.1007/s00220-009-0773-9
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DOI: https://doi.org/10.1007/s00220-009-0773-9