Electronic Journal of Probability

Infinite energy solutions to inelastic homogeneous Boltzmann equations

Federico Bassetti, Lucia Ladelli, and Daniel Matthes

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Abstract

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a class of spatially homogeneous kinetic equations in space dimensions $d\ge2$. Our results cover in particular Bobylev's model for inelastic Maxwell molecules.  First, we show under certain conditions on the collision kernel, that there exists an index $\alpha\in(0,2)$  such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $\alpha$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation.  Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $\alpha$-stable distribution  converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of projections  of randomly rotated  i.i.d. random vectors.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 89, 34 pp.

Dates
Accepted: 8 September 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067195

Digital Object Identifier
doi:10.1214/EJP.v20-3531

Mathematical Reviews number (MathSciNet)
MR3399825

Zentralblatt MATH identifier
1345.60060

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K40: Other physical applications of random processes 82C40: Kinetic theory of gases

Keywords
Central Limit Theorems Inelastic Boltzmann Equation Infinite Energy Solutions Normal Domain of Attraction Multidimensional Stable Laws

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bassetti, Federico; Ladelli, Lucia; Matthes, Daniel. Infinite energy solutions to inelastic homogeneous Boltzmann equations. Electron. J. Probab. 20 (2015), paper no. 89, 34 pp. doi:10.1214/EJP.v20-3531. https://projecteuclid.org/euclid.ejp/1465067195


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References

  • Aaronson, Jon; Denker, Manfred. Characteristic functions of random variables attracted to $1$-stable laws. Ann. Probab. 26 (1998), no. 1, 399–415.
  • Alsmeyer, Gerold; Meiners, Matthias. Fixed points of the smoothing transform: two-sided solutions. Probab. Theory Related Fields 155 (2013), no. 1-2, 165–199.
  • Araujo, Aloisio; Giné, Evarist. The central limit theorem for real and Banach valued random variables. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-Chichester-Brisbane, 1980. xiv+233 pp. ISBN: 0-471-05304-X.
  • Bassetti, Federico; Ladelli, Lucia. Self-similar solutions in one-dimensional kinetic models: a probabilistic view. Ann. Appl. Probab. 22 (2012), no. 5, 1928–1961.
  • Bassetti, Federico; Ladelli, Lucia; Matthes, Daniel. Central limit theorem for a class of one-dimensional kinetic equations. Probab. Theory Related Fields 150 (2011), no. 1-2, 77–109.
  • Bassetti, Federico; Ladelli, Lucia; Regazzini, Eugenio. Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model. J. Stat. Phys. 133 (2008), no. 4, 683–710.
  • Bassetti, Federico; Ladelli, Lucia; Toscani, Giuseppe. Kinetic models with randomly perturbed binary collisions. J. Stat. Phys. 142 (2011), no. 4, 686–709.
  • Bassetti, Federico; Matthes, Daniel. Multi-dimensional smoothing transformations: existence, regularity and stability of fixed points. Stochastic Process. Appl. 124 (2014), no. 1, 154–198.
  • Bhattacharya, R. N. Speed of convergence of the $n$-fold convolution of a probability measure on a compact group. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 1–10.
  • Billingsley, Patrick. Probability and measure. Third edition. Wiley Series in Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xiv+593 pp. ISBN: 0-471-00710-2.
  • 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 (2006), no. 2-4, 625–653.
  • Bobylev, A. V. The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical physics reviews, Vol. 7, 111–233, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, 1988.
  • Bobylev, A. V.; Carrillo, J. A.; Gamba, I. M. On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys. 98 (2000), no. 3-4, 743–773.
  • Bobylev, A. V.; Cercignani, C. Self-similar solutions of the Boltzmann equation and their applications. J. Statist. Phys. 106 (2002), no. 5-6, 1039–1071.
  • Bobylev, A. V.; Cercignani, C. Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Statist. Phys. 110 (2003), no. 1-2, 333–375.
  • Bobylev, A. V.; Cercignani, C.; Gamba, I. M. Generalized kinetic Maxwell type models of granular gases. Mathematical models of granular matter, 23–57, Lecture Notes in Math., 1937, Springer, Berlin, 2008.
  • Bobylev, A. V.; Cercignani, C.; Toscani, G. Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Statist. Phys. 111 (2003), no. 1-2, 403–417.
  • Bobylev, A. V.; Cercignani, C.; Gamba, I. M. On the self-similar asymptotics for generalized nonlinear kinetic Maxwell models. Comm. Math. Phys. 291 (2009), no. 3, 599–644.
  • Bolley, F.; Carrillo, J. A. Tanaka theorem for inelastic Maxwell models. Comm. Math. Phys. 276 (2007), no. 2, 287–314.
  • Cannone, Marco; Karch, Grzegorz. Infinite energy solutions to the homogeneous Boltzmann equation. Comm. Pure Appl. Math. 63 (2010), no. 6, 747–778.
  • Carlen, Eric; Gabetta, Ester; Regazzini, Eugenio. On the rate of explosion for infinite energy solutions of the spatially homogeneous Boltzmann equation. J. Stat. Phys. 129 (2007), no. 4, 699–723.
  • Carlen, Eric A.; Carrillo, José A.; Carvalho, Maria C. Strong convergence towards homogeneous cooling states for dissipative Maxwell models. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, 1675–1700.
  • Carrillo, J. A.; Toscani, G. Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma (7) 6 (2007), 75–198.
  • Chow, Yuan Shih; Teicher, Henry. Probability theory. Independence, interchangeability, martingales. Third edition. Springer Texts in Statistics. Springer-Verlag, New York, 1997. xxii+488 pp. ISBN: 0-387-98228-0.
  • Dolera, Emanuele; Gabetta, Ester; Regazzini, Eugenio. Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem. Ann. Appl. Probab. 19 (2009), no. 1, 186–209.
  • Dolera, Emanuele; Regazzini, Eugenio. The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation. Ann. Appl. Probab. 20 (2010), no. 2, 430–461.
  • bysame, phProbabilistic representation for the solution of the homogeneous boltzmann equation for maxwellian molecules., (2011) ARXIV1103.4738.
  • Dolera, Emanuele; Regazzini, Eugenio. Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions. Probab. Theory Related Fields 160 (2014), no. 1-2, 315–389.
  • Durrett, Richard; Liggett, Thomas M. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275–301.
  • Ernst, M. H.; Brito, R. Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails. Special issue dedicated to J. Robert Dorfman on the occasion of his sixty-fifth birthday. J. Statist. Phys. 109 (2002), no. 3-4, 407–432.
  • Fortini, S.; Ladelli, L.; Regazzini, E. A central limit problem for partially exchangeable random variables. Teor. Veroyatnost. i Primenen. 41 (1996), no. 2, 353–379; translation in Theory Probab. Appl. 41 (1996), no. 2, 224–246 (1997)
  • Fristedt, Bert; Gray, Lawrence. A modern approach to probability theory. Probability and its Applications. Birkhauser Boston, Inc., Boston, MA, 1997. xx+756 pp. ISBN: 0-8176-3807-5.
  • Gabetta, Ester; Regazzini, Eugenio. Central limit theorem for the solutions of the Kac equation. Ann. Appl. Probab. 18 (2008), no. 6, 2320–2336.
  • Gabetta, Ester; Regazzini, Eugenio. Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics. Probab. Theory Related Fields 146 (2010), no. 3-4, 451–480.
  • Gabetta, G.; Toscani, G.; Wennberg, B. Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Statist. Phys. 81 (1995), no. 5-6, 901–934.
  • A. Hurwitz, Ueber die erzeugung der invarianten durch integration, Nach. Gesell. Wissen, Göttingen Math-Phys Klasse. (1897), 71–90.
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1.
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2.
  • McKean, H. P., Jr. Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rational Mech. Anal. 21 1966 343–367.
  • McKean, H. P., Jr. An exponential formula for solving Boltmann's equation for a Maxwellian gas. J. Combinatorial Theory 2 1967 358–382.
  • Meerschaert, Mark M.; Scheffler, Hans-Peter. Limit distributions for sums of independent random vectors. Heavy tails in theory and practice. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, 2001. xvi+484 pp. ISBN: 0-471-35629-8.
  • Mischler, S.; Mouhot, C. Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres. Comm. Math. Phys. 288 (2009), no. 2, 431–502.
  • Perversi, Eleonora; Regazzini, Eugenio. Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations. J. Stat. Phys. 159 (2015), no. 4, 823–852.
  • Pulvirenti, Ada; Toscani, Giuseppe. Asymptotic properties of the inelastic Kac model. J. Statist. Phys. 114 (2004), no. 5-6, 1453–1480.
  • Tanaka, Hiroshi. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 1, 67–105.
  • Toscani, G.; Villani, C. Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys. 94 (1999), no. 3-4, 619–637.
  • Villani, Cédric. A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, 71–305, North-Holland, Amsterdam, 2002.