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

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Trails in Kinetic Theory

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 25))

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.

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Notes

  1. 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. 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. 3.

    Physically more realistic, in part because of the spontaneous loss of space homogeneity that has been observed in [58].

  4. 4.

    Namely, the initial condition is chosen with a lot of exponential moments in velocity, and close to a space homogeneous profile.

  5. 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].

References

  1. Almazán, L., Carrillo, J.A., Garzó, V., Salueña, C., Pöschel, T.: Supplementary online material (2012). https://iopscience.iop.org/article/10.1088/1367-2630/15/4/043044

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Araki, S., Tremaine, S.: The dynamics of dense particle disks. Icarus 65(1), 83–109 (1986)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Benedetto, D., Pulvirenti, M.: On the one-dimensional Boltzmann equation for granular flows. M2AN Math. Model. Numer. Anal. 35(5), 899–905 (2002)

    Google Scholar 

  10. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bobylev, A.V., Cercignani, C.: Moment equations for a granular material in a thermal bath. J. Statist. Phys. 1063–4, 547–567 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bolley, F., Carrillo, J.A.: Tanaka theorem for inelastic Maxwell models. Commun. Math. Phys. 276(2), 287–314 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bolley, F., Gentil, I., Guillin, A.: Uniform convergence to equilibrium for granular media. Arch. Ration. Mech. Anal. 208(2), 429–445 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  28. Bougie, J., Moon, S.J., Swift, J., Swinney, H.: Shocks in vertically oscillated granular layers. Phys. Rev. E 66, 051301 (2002)

    Article  Google Scholar 

  29. Brenier, Y., Grenier, E.: Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35(6), 2317–2328 (1998) (electronic)

    Google Scholar 

  30. Brey, J.J., Dufty, J.W., Santos, A.: Kinetic models for granular flow. J. Stat. Phys. 97, 281–322 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Brilliantov, N.V., Pöschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)

    Book  MATH  Google Scholar 

  32. Brilliantov, N.V., Salueña, C., Schwager, T., Pöschel, T.: Transient structures in a granular gas. Phys. Rev. Lett. 93, 134301 (2004)

    Article  Google Scholar 

  33. Carlen, E., Chow, S.-N., Grigo, A.: Dynamics and hydrodynamic limits of the inelastic Boltzmann equation. Nonlinearity 23(8), 1807–1849 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  39. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994)

    Google Scholar 

  40. Daerr, A.: Dynamique des Avalanches. PhD thesis, Université Denis Diderot Paris §7, 2000

    Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  44. Ernst, M.H., Brito, R.: Driven inelastic Maxwell models with high energy tails. Phys. Rev. E 65, 040301 (2002)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  47. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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  Google Scholar 

  50. Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. European Mathematical Society, Zürich (2014)

    Google Scholar 

  51. Gamba, I., Panferov, V., Villani, C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246(3), 503–541 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  54. Garzó, V.: Granular Gaseous Flows: A Kinetic Theory Approach to Granular Gaseous Flows. Springer, Berlin (2019)

    Book  MATH  Google Scholar 

  55. Goldhirsch, I.: Scales and kinetics of granular flows. Chaos 9(3), 659–672 (1999)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  57. Goldhirsch, I.: Rapid granular flows. Ann. Rev. Fluid Mech. 35, 267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  58. Goldhirsch, I., Zanetti, G.: Clustering instability in dissipative gases. Phys. Rev. Lett. 70(11), 1619–1622 (1993)

    Article  Google Scholar 

  59. Goldshtein, A., Shapiro, M.: Mechanics of collisional motion of granular materials. Part 1: general hydrodynamic equations. J. Fluid Mech. 282, 75 (1995)

    Google Scholar 

  60. Haff, P.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–30 (1983)

    Article  MATH  Google Scholar 

  61. Hill, S.A., Mazenko, G.F.: Granular clustering in a hydrodynamic simulation. Phys. Rev. E 67, 061302 (2003)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  63. Huang, F., Wang, Z.: Well posedness for pressureless flow. Commun. Math. Phys. 222(1), 117–146 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  64. Jabin, P.-E., Rey, T.: Hydrodynamic limit of granular gases to pressureless Euler in dimension 1. Q. Appl. Math. 75, 155–179 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  67. Johnson, C.G., Gray, J. M.N.T.: Granular jets and hydraulic jumps on an inclined plane. J. Fluid Mech. 675, 87–116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. Kang, M.-J., Kim, J.: Propagation of the mono-kinetic solution in the Cucker–Smale-type kinetic equations (2019). arXiv preprint 1909.07525

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  70. Kawai, T., Shida, K.: An inelastic collision model for the evolution of “planetary rings”. J. Phys. Soc. Jpn 59(1), 381–388 (1990)

    Article  MathSciNet  Google Scholar 

  71. Li, H., Toscani, G.: Long-time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal. 172(3), 407–428 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  72. McNamara, S., Young, W.R.: Inelastic collapse in two dimensions. Phys. Rev. E 50, R28–R31 (1993)

    Article  Google Scholar 

  73. Melo, F., Umbanhowar, P., Swinney, H.L.: Hexagons, kinks, and disorder in oscillated granular layers. Phys. Rev. Lett. 75, 3838–3841 (1995)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  81. Pöschel, T., Brilliantov, N., Schwager, T.: Transient clusters in granular gases. J. Phys. Condens. Matter 17, 2705–2713 (2005)

    Article  MATH  Google Scholar 

  82. Rericha, E., Bizon, C., Shattuck, M., Swinney, H.: Shocks in supersonic sand. Phys. Rev. Lett. 88, 1 (2002)

    Google Scholar 

  83. Rey, T.: Blow up analysis for anomalous granular gases. SIAM J. Math. Anal. 44(3), 1544–1561 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  84. Rey, T.: A spectral study of the linearized Boltzmann equation for diffusively excited granular media (2013). Preprint arXiv 1310.7234

    Google Scholar 

  85. Toscani, G.: One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal. 34(6), 1277–1291 (2000)

    Google Scholar 

  86. Toscani, G.: Kinetic and hydrodynamic models of nearly elastic granular flows. Monatsh. Math. 142(1), 179–192 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  87. Tristani, I.: Boltzmann equation for granular media with thermal forces in a weakly inhomogeneous setting. J. Funct. Anal. 270(5), 1922–1970 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  88. Tsimring, L.S., Aranson, I.S.: Localized and cellular patterns in a vibrated granular layer. Phys. Rev. Lett. 79, 213–216 (1997)

    Article  Google Scholar 

  89. Umbanhowar, P.B., Melo, F., Swinney, H.L.: Localized excitations in a vertically vibrated granular layer. Nature 382, 793–796 (1996)

    Article  Google Scholar 

  90. Umbanhowar, P.B., Melo, F., Swinney, H.L.: Periodic, aperiodic, and transient patterns in vibrated granular layers. Phys. A 249, 1–9 (1998)

    Article  Google Scholar 

  91. Villani, C.: Mathematics of granular materials. J. Stat. Phys. 124(2), 781–822 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  92. Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202, 184 (2009)

    Google Scholar 

  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)

    Google Scholar 

  94. Wu, Z.: L 1 and BV-type stability of the inelastic Boltzmann equation near vacuum. Continuum Mech. Thermodyn. 22(3), 239–249 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  95. Wu, Z.: On the inelastic Enskog equation near vacuum. J. Math. Phys. 51(3), 033508 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

JAC acknowledges support by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363), by the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/P031587/1, and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). JH was partially funded by NSF grant DMS-1620250 and NSF CAREER grant DMS-1654152. TR was partially funded by Labex CEMPI (ANR-11-LABX-0007-01) and ANR Project MoHyCon (ANR-17-CE40-0027-01).

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  1. Mathematical Institute, University of Oxford, Oxford, UK

    José Antonio Carrillo

  2. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Jingwei Hu & Zheng Ma

  3. Univ. Lille, CNRS, UMR 8524, Inria – Laboratoire Paul Painlevé, Lille, France

    Thomas Rey

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    Sara Merino-Aceituno

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Carrillo, J.A., Hu, J., Ma, Z., Rey, T. (2021). Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods. In: Albi, G., Merino-Aceituno, S., Nota, A., Zanella, M. (eds) Trails in Kinetic Theory. SEMA SIMAI Springer Series, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-67104-4_1

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