Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2021 | OriginalPaper | Buchkapitel

Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods

verfasst von : José Antonio Carrillo, Jingwei Hu, Zheng Ma, Thomas Rey

Erschienen in: Trails in Kinetic Theory

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review Villani (J Stat Phys 124(2):781–822, 2006) by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks
Fußnoten
1
Because of that, the elastic collision operator is simply equal to 0 for a one-dimensional velocity space, the Boltzmann equation reducing only to the free transport equation.
 
2
Note that using a BBGKY approach [50] to derive (11) is not expected to succeed, because among other problems the macroscopic size of the particles composing a granular gas is incompatible with the Boltzmann-Grad scaling assumption.
 
3
Physically more realistic, in part because of the spontaneous loss of space homogeneity that has been observed in [58].
 
4
Namely, the initial condition is chosen with a lot of exponential moments in velocity, and close to a space homogeneous profile.
 
5
One can see the velocity scaling function ω as the inverse of the variance of the distribution f. This scaling is then a continuous “zoom” on the blowup, and can be used to develop numerical methods for solving the full granular gases equation, see [49].
 
Literatur
1.
2.
Almazán, L., Carrillo, J.A., Salueña, C., Garzó, V., Pöschel, T.: A numerical study of the Navier–Stokes transport coefficients for two-dimensional granular hydrodynamics. New J. Phys. 15(4), 043044 (2013)CrossRef
3.
Alonso, R., Cañizo, J.A., Gamba, I., Mouhot, C.: A new approach to the creation and propagation of exponential moments in the Boltzmann equation. Commun. Partial. Differ. Equ. 38(1), 155–169 (2013)MathSciNetMATHCrossRef
4.
Alonso, R., Lods, B.: Uniqueness and regularity of steady states of the Boltzmann equation for viscoelastic hard-spheres driven by a thermal bath. Commun. Math. Sci. 11(4), 851–906 (2013)MathSciNetMATHCrossRef
5.
Alonso, R., Lods, B.: Boltzmann model for viscoelastic particles: asymptotic behavior, pointwise lower bounds and regularity. Commun. Math. Phys. 331 2, 545–591 (2014)MathSciNetMATHCrossRef
6.
Alonso, R.J., Lods, B.: Free cooling and high-energy tails of granular gases with variable restitution coefficient. SIAM J. Math. Anal. 42(6), 2499–2538 (2010)MathSciNetMATHCrossRef
7.
Araki, S., Tremaine, S.: The dynamics of dense particle disks. Icarus 65(1), 83–109 (1986)CrossRef
8.
Benedetto, D., Caglioti, E., Carrillo, J.A., Pulvirenti, M.: A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91(5/6), 979–990 (1998)MathSciNetMATHCrossRef
9.
Benedetto, D., Pulvirenti, M.: On the one-dimensional Boltzmann equation for granular flows. M2AN Math. Model. Numer. Anal. 35(5), 899–905 (2002)
10.
Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994)
11.
Bisi, M., Cañizo, J.A., Lods, B.: Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath. SIAM J. Math. Anal. 43(6), 2640–2674 (2011)MathSciNetMATHCrossRef
12.
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)MathSciNetMATHCrossRef
13.
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)MathSciNetMATHCrossRef
14.
Bizon, C., Shattuck, M., Swift, J.B., Swinney, H.: Transport coefficients for granular media from molecular dynamics simulations. Phys. Rev. E 60, 4340 (1999)CrossRef
15.
Bobylev, A.V., Carrillo, J.A., Gamba, I.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys. 98(3), 743–773 (2000)MathSciNetMATHCrossRef
16.
Bobylev, A.V., Carrillo, J.A., Gamba, I.: Erratum on: “On some properties of kinetic and hydrodynamic equations for inelastic interactions”. J. Statist. Phys. 103(5–6), 1137–1138 (2001)MathSciNetMATHCrossRef
17.
Bobylev, A.V., Cercignani, C.: Moment equations for a granular material in a thermal bath. J. Statist. Phys. 1063–4, 547–567 (2002)MathSciNetMATHCrossRef
18.
Bobylev, A.V., Cercignani, C., Gamba, I.: Generalized kinetic Maxwell type models of granular gases. In: Mathematical Models of Granular Matter. Lecture Notes in Math., vol. 1937, pp. 23–57. Springer, Berlin (2008)
19.
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(1–2), 403–417 (2003)MathSciNetMATHCrossRef
20.
Bobylev, A.V., Gamba, I., Panferov, V.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5), 1651–1682 (2004)MathSciNetMATHCrossRef
21.
22.
Bolley, F., Gentil, I., Guillin, A.: Uniform convergence to equilibrium for granular media. Arch. Ration. Mech. Anal. 208(2), 429–445 (2013)MathSciNetMATHCrossRef
23.
Bony, J.-M.: Solutions globales bornées pour les modèles discrets de l’équation de Boltzmann, en dimension 1 d’espace. In: Journées “Équations aux derivées partielles” (Saint Jean de Monts, 1987). École Polytechnique, Palaiseau, 1987. Exp. No. XVI, 10 pp
24.
Bouchut, F., Desvillettes, L.: A proof of the smoothing properties of the positive part of Boltzmann’s kernel. Rev. Mat. Iberoam. 14(1), 47–61 (1998)MathSciNetMATHCrossRef
25.
Bouchut, F., James, F.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Commun. Partial Differ. Equ. 24(11–12), 2173–2189 (1999)MathSciNetMATH
26.
Boudin, L.: A solution with bounded expansion rate to the model of viscous pressureless gases. SIAM J. Math. Anal. 32(1), 172–193 (2000)MathSciNetMATHCrossRef
27.
Bougie, J., Kreft, J., Swift, J.B., Swinney, H.: Onset of patterns in an oscillated granular layer: continuum and molecular dynamics simulations. Phys. Rev. E 71, 021301 (2005)CrossRef
28.
Bougie, J., Moon, S.J., Swift, J., Swinney, H.: Shocks in vertically oscillated granular layers. Phys. Rev. E 66, 051301 (2002)CrossRef
29.
Brenier, Y., Grenier, E.: Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35(6), 2317–2328 (1998) (electronic)
30.
31.
Brilliantov, N.V., Pöschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)MATHCrossRef
32.
Brilliantov, N.V., Salueña, C., Schwager, T., Pöschel, T.: Transient structures in a granular gas. Phys. Rev. Lett. 93, 134301 (2004)CrossRef
33.
Carlen, E., Chow, S.-N., Grigo, A.: Dynamics and hydrodynamic limits of the inelastic Boltzmann equation. Nonlinearity 23(8), 1807–1849 (2010)MathSciNetMATHCrossRef
34.
Carlen, E.A., Carrillo, J.A., Carvalho, M.C.: Strong convergence towards homogeneous cooling states for dissipative Maxwell models. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(5), 1675–1700 (2009)MathSciNetMATHCrossRef
35.
Carrillo, J.A., McCann, R.J., Villani, C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19(3), 971–1018 (2004)MathSciNetMATH
36.
Carrillo, J.A., Poëschel, T., Salueña, C.: Granular hydrodynamics and pattern formation in vertically oscillated granular disk layers. J. Fluid Mech. 597, 119–144 (2008)MathSciNetMATHCrossRef
37.
Carrillo, J.A., Salueña, C.: Modelling of shock waves and clustering in hydrodynamic simulations of granular gases. In: Modelling and Numerics of Kinetic Dissipative Systems, pp. 163–176. Nova Sci. Publ., Hauppauge (2006)
38.
Carrillo, J.A., Toscani, G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma (7) 6, 75–198 (2007)
39.
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994)
40.
Daerr, A.: Dynamique des Avalanches. PhD thesis, Université Denis Diderot Paris §7, 2000
41.
Dufty, J.W.: Nonequilibrium Statistical Mechanics and Hydrodynamics for a Granular Fluid. In: Cichocki, E., Napiorkowski, M., Piasekcki, J. (eds.) Second Warsaw School on Statistical Physics, no. June 2007, p. 64. Warsaw University Press, Warsaw (2008)
42.
Weinan, E., Rykov, Y.G., Sinai, Y.G.: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun. Math. Phys. 177(2), 349–380 (1996)MathSciNetMATHCrossRef
43.
Ellis, R.S., Pinsky, M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54(9), 125–156 (1975)MathSciNetMATH
44.
Ernst, M.H., Brito, R.: Driven inelastic Maxwell models with high energy tails. Phys. Rev. E 65, 040301 (2002)CrossRef
45.
Esteban, M., Perthame, B.: On the modified Enskog equation for elastic and inelastic collisions. Models with spin. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(3–4), 289–308 (1991)MathSciNetMATH
46.
Faraday, M.: On a peculiar class of acoustical figure and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Philos. Trans. R. Soc. Lond. 121, 299 (1831)
47.
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)MathSciNetMATHCrossRef
48.
Filbet, F., Pareschi, L., Toscani, G.: Accurate numerical methods for the collisional motion of (heated) granular flows. J. Comput. Phys. 202(1), 216–235 (2005)MathSciNetMATHCrossRef
49.
Filbet, F., Rey, T.: A rescaling velocity method for dissipative kinetic equations. Applications to granular media. J. Comput. Phys. 248, 177–199 (2013)MATH
50.
Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. European Mathematical Society, Zürich (2014)
51.
Gamba, I., Panferov, V., Villani, C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246(3), 503–541 (2004)MathSciNetMATHCrossRef
52.
Gamba, I.M., Tharkabhushanam, S.H.: Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. J. Comput. Phys. 228, 2012–2036 (2009)MathSciNetMATHCrossRef
53.
García de Soria, M.I., Maynar, P., Mischler, S., Mouhot, C., Rey, T., Trizac, E.: Towards an H-theorem for granular gases. J. Stat. Mech: Theory Exp. 2015(11), P11009 (2015)
54.
Garzó, V.: Granular Gaseous Flows: A Kinetic Theory Approach to Granular Gaseous Flows. Springer, Berlin (2019)MATHCrossRef
55.
56.
Goldhirsch, I.: Probing the boundaries of hydrodynamics. In: Pöschel, T., Luding, S. (eds.) Granular Gases. Lecture Notes in Physics, vol. 564, pp. 79–99. Springer, Berlin (2001)CrossRef
57.
58.
Goldhirsch, I., Zanetti, G.: Clustering instability in dissipative gases. Phys. Rev. Lett. 70(11), 1619–1622 (1993)CrossRef
59.
Goldshtein, A., Shapiro, M.: Mechanics of collisional motion of granular materials. Part 1: general hydrodynamic equations. J. Fluid Mech. 282, 75 (1995)
60.
Haff, P.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–30 (1983)MATHCrossRef
61.
Hill, S.A., Mazenko, G.F.: Granular clustering in a hydrodynamic simulation. Phys. Rev. E 67, 061302 (2003)CrossRef
62.
Hu, J., Ma, Z.: A fast spectral method for the inelastic Boltzmann collision operator and application to heated granular gases. J. Comput. Phys. 385, 119–134 (2019)MathSciNetMATHCrossRef
63.
64.
Jabin, P.-E., Rey, T.: Hydrodynamic limit of granular gases to pressureless Euler in dimension 1. Q. Appl. Math. 75, 155–179 (2017)MathSciNetMATHCrossRef
65.
Jenkins, J., Richman, M.W.: Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Ration. Mech. Anal. 87, 355 (1985)MathSciNetMATHCrossRef
66.
Jenkins, J., Richman, M.W.: Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485 (1985)MATHCrossRef
67.
68.
Kang, M.-J., Kim, J.: Propagation of the mono-kinetic solution in the Cucker–Smale-type kinetic equations (2019). arXiv preprint 1909.07525
69.
Kang, M.-J., Vasseur, A.F.: Asymptotic analysis of Vlasov-type equations under strong local alignment regime. Math. Models Methods Appl. Sci. 25(11), 2153–2173 (2015)MathSciNetMATHCrossRef
70.
Kawai, T., Shida, K.: An inelastic collision model for the evolution of “planetary rings”. J. Phys. Soc. Jpn 59(1), 381–388 (1990)MathSciNetCrossRef
71.
Li, H., Toscani, G.: Long-time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal. 172(3), 407–428 (2004)MathSciNetMATHCrossRef
72.
McNamara, S., Young, W.R.: Inelastic collapse in two dimensions. Phys. Rev. E 50, R28–R31 (1993)CrossRef
73.
Melo, F., Umbanhowar, P., Swinney, H.L.: Hexagons, kinks, and disorder in oscillated granular layers. Phys. Rev. Lett. 75, 3838–3841 (1995)CrossRef
74.
Mischler, S., Mouhot, C.: Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior. J. Statist. Phys. 124(2), 703–746 (2006)MathSciNetMATH
75.
Mischler, S., Mouhot, C.: Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres. Commun. Math. Phys. 288(2), 431–502 (2009)MathSciNetMATHCrossRef
76.
Mischler, S., Mouhot, C.: Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete Contin. Dyn. Syst. 24(1), 159–185 (2009)MathSciNetMATHCrossRef
77.
Mischler, S., Mouhot, C., Ricard, M.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I: the Cauchy problem. J. Statist. Phys. 124(2), 655–702 (2006)MathSciNetMATH
78.
Mischler, S., Mouhot, C., Wennberg, B.: A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. Probab. Theory Relat. Fields 161(1–2), 1–59 (2015)MathSciNetMATHCrossRef
79.
Naldi, G., Pareschi, L., Toscani, G.: Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit. ESAIM: Math. Model. Numer. Anal. 37, 73–90 (2003)MathSciNetMATHCrossRef
80.
Oleinik, O.: On Cauchy’s problem for nonlinear equations in a class of discontinuous functions. Doklady Akad. Nauk SSSR (N.S.) 95, 451–454 (1954)
81.
Pöschel, T., Brilliantov, N., Schwager, T.: Transient clusters in granular gases. J. Phys. Condens. Matter 17, 2705–2713 (2005)MATHCrossRef
82.
Rericha, E., Bizon, C., Shattuck, M., Swinney, H.: Shocks in supersonic sand. Phys. Rev. Lett. 88, 1 (2002)
83.
84.
Rey, T.: A spectral study of the linearized Boltzmann equation for diffusively excited granular media (2013). Preprint arXiv 1310.7234
85.
Toscani, G.: One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal. 34(6), 1277–1291 (2000)
86.
87.
Tristani, I.: Boltzmann equation for granular media with thermal forces in a weakly inhomogeneous setting. J. Funct. Anal. 270(5), 1922–1970 (2016)MathSciNetMATHCrossRef
88.
Tsimring, L.S., Aranson, I.S.: Localized and cellular patterns in a vibrated granular layer. Phys. Rev. Lett. 79, 213–216 (1997)CrossRef
89.
Umbanhowar, P.B., Melo, F., Swinney, H.L.: Localized excitations in a vertically vibrated granular layer. Nature 382, 793–796 (1996)CrossRef
90.
Umbanhowar, P.B., Melo, F., Swinney, H.L.: Periodic, aperiodic, and transient patterns in vibrated granular layers. Phys. A 249, 1–9 (1998)CrossRef
91.
92.
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202, 184 (2009)
93.
Womersley, R.S.: Efficient spherical designs with good geometric properties. In: Contemporary Computational Mathematics—a Celebration of the 80th Birthday of Ian Sloan, pp. 1243–1285. Springer, Cham (2018)
94.
Wu, Z.: L 1 and BV-type stability of the inelastic Boltzmann equation near vacuum. Continuum Mech. Thermodyn. 22(3), 239–249 (2009)MathSciNetMATHCrossRef
95.
Metadaten
Titel
Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods
verfasst von
José Antonio Carrillo
Jingwei Hu
Zheng Ma
Thomas Rey
Copyright-Jahr
2021
Buch
Trails in Kinetic Theory

Print ISBN: 978-3-030-67103-7

Electronic ISBN: 978-3-030-67104-4

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-3-030-67104-4_1