• Previous Article
    Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model
  • KRM Home
  • This Issue
  • Next Article
    Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations
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 & 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.  Google Scholar

[2]

M. BisiJ. 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.  Google Scholar

[3]

M. BisiJ. 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.  Google Scholar

[4]

A. V. BobylevJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. V. BobylevI. 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.  Google Scholar

[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.  Google Scholar

[9]

C. CercignaniR. Illner and C. Stoica, On Diffusive equilibria in generalized kinetic theory, J. Stat. Phys., 105 (2001), 337-352.   Google Scholar

[10]

I. M. GambaV. 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.  Google Scholar

[11]

S. MischlerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.  Google Scholar

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.  Google Scholar

[2]

M. BisiJ. 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.  Google Scholar

[3]

M. BisiJ. 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.  Google Scholar

[4]

A. V. BobylevJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. V. BobylevI. 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.  Google Scholar

[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.  Google Scholar

[9]

C. CercignaniR. Illner and C. Stoica, On Diffusive equilibria in generalized kinetic theory, J. Stat. Phys., 105 (2001), 337-352.   Google Scholar

[10]

I. M. GambaV. 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.  Google Scholar

[11]

S. MischlerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.  Google Scholar

[1]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[2]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[3]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029

[4]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[5]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[6]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[7]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454

[8]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009

[9]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[10]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[11]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[12]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[13]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[14]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[15]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[16]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[17]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[18]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[19]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

[20]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (225)
  • HTML views (250)
  • Cited by (0)

Other articles
by authors

[Back to Top]