## Electronic Journal of Probability

### Infinite energy solutions to inelastic homogeneous Boltzmann equations

#### 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

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

Rights

#### 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

#### 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, GÃ¶ttingen 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.