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February  2019, 12(1): 159-176. doi: 10.3934/krm.2019007

## Elastic limit and vanishing external force for granular systems

 1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China 2 Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Fei Meng

Received  February 2017 Revised  February 2018 Published  July 2018

Fund Project: This work is supported by the National Natural Science Foundation of China(Grant No. 11531005); the Scientific Research Foundation of NUPT(Grant No. NY218091).

We consider two popular models derived from the theory of granular gases. The first model is the inelastic Boltzmann equation with a diffusion term representing the heat bath, the second model is obtained by a self-similar transformation for the inelastic Boltzmann equation in the homogeneous cooling problem. We prove that the steady states of the two models converge to a Maxwellian equilibrium or a Dirac distribution in the elastic limit and the vanishing external force, respectively. Our results show that the limits of the steady states depend on the ratio of external energy and dissipated energy due to inelastic collision. These results provide a partial answer to a question proposed by Gamba, Panferov and Villani (Comm. Math. Phys. 246,503-541. 2004).

Citation: Fei Meng, Xiao-Ping Yang. Elastic limit and vanishing external force for granular systems. Kinetic and Related Models, 2019, 12 (1) : 159-176. doi: 10.3934/krm.2019007
##### References:
 [1] D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, Math. Model. Numer. Anal., 35 (2001), 899-905.  doi: 10.1051/m2an:2001141. [2] M. Bisi, J. A. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for Granular gases: Diffusive Equilibria, J. Stat. Phys., 118 (2005), 301-331.  doi: 10.1007/s10955-004-8785-5. [3] M. Bisi, J. A. Carrillo and G. Toscani, Decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model, J. Stat. Phys., 124 (2006), 625-653.  doi: 10.1007/s10955-006-9035-9. [4] A. V. Bobylev, J. A. Carrillo and I. M. Gamba, On Some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Stat. Phys., 98 (2000), 743-773.  doi: 10.1023/A:1018627625800. [5] A. V. Bobylev and C. Cercignani, Moment equations for a granular material in a thermal bath, J. Stat. Phys., 106 (2002), 547-567.  doi: 10.1023/A:1013754205008. [6] A. V. Bobylev and C. Cercignani, Self-similar asympototics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys., 110 (2003), 333-375.  doi: 10.1023/A:1021031031038. [7] A. V. Bobylev, I. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Stat. Phys., 116 (2004), 1651-1682.  doi: 10.1023/B:JOSS.0000041751.11664.ea. [8] F. Bolley and J. A. Carrillo, Tanaka theorem for inelastic Maxwell models, Comm. Math. Phys., 276 (2007), 287-314.  doi: 10.1007/s00220-007-0336-x. [9] C. Cercignani, R. Illner and C. Stoica, On Diffusive equilibria in generalized kinetic theory, J. Stat. Phys., 105 (2001), 337-352. [10] I. M. Gamba, V. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541.  doi: 10.1007/s00220-004-1051-5. [11] S. Mischler, C. Mouhot and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres. Part Ⅰ: The Cauchy problem, J. Stat. Phys., 124 (2006), 655-702.  doi: 10.1007/s10955-006-9096-9. [12] S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. Part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8. [13] S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard sphere, Comm. Math. Phys., 288 (2009), 431-502.  doi: 10.1007/s00220-009-0773-9. [14] S. Mischler and C. Mouhot, Stability, Convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media, Discrete. Contin. Dyn. Syst., 24 (2009), 159-185.  doi: 10.3934/dcds.2009.24.159. [15] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1. [16] C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.

show all references

##### References:
 [1] D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, Math. Model. Numer. Anal., 35 (2001), 899-905.  doi: 10.1051/m2an:2001141. [2] M. Bisi, J. A. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for Granular gases: Diffusive Equilibria, J. Stat. Phys., 118 (2005), 301-331.  doi: 10.1007/s10955-004-8785-5. [3] M. Bisi, J. A. Carrillo and G. Toscani, Decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model, J. Stat. Phys., 124 (2006), 625-653.  doi: 10.1007/s10955-006-9035-9. [4] A. V. Bobylev, J. A. Carrillo and I. M. Gamba, On Some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Stat. Phys., 98 (2000), 743-773.  doi: 10.1023/A:1018627625800. [5] A. V. Bobylev and C. Cercignani, Moment equations for a granular material in a thermal bath, J. Stat. Phys., 106 (2002), 547-567.  doi: 10.1023/A:1013754205008. [6] A. V. Bobylev and C. Cercignani, Self-similar asympototics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys., 110 (2003), 333-375.  doi: 10.1023/A:1021031031038. [7] A. V. Bobylev, I. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Stat. Phys., 116 (2004), 1651-1682.  doi: 10.1023/B:JOSS.0000041751.11664.ea. [8] F. Bolley and J. A. Carrillo, Tanaka theorem for inelastic Maxwell models, Comm. Math. Phys., 276 (2007), 287-314.  doi: 10.1007/s00220-007-0336-x. [9] C. Cercignani, R. Illner and C. Stoica, On Diffusive equilibria in generalized kinetic theory, J. Stat. Phys., 105 (2001), 337-352. [10] I. M. Gamba, V. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541.  doi: 10.1007/s00220-004-1051-5. [11] S. Mischler, C. Mouhot and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres. Part Ⅰ: The Cauchy problem, J. Stat. Phys., 124 (2006), 655-702.  doi: 10.1007/s10955-006-9096-9. [12] S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. Part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8. [13] S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard sphere, Comm. Math. Phys., 288 (2009), 431-502.  doi: 10.1007/s00220-009-0773-9. [14] S. Mischler and C. Mouhot, Stability, Convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media, Discrete. Contin. Dyn. Syst., 24 (2009), 159-185.  doi: 10.3934/dcds.2009.24.159. [15] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1. [16] C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.
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