American Institute of Mathematical Sciences

Advanced
June  2015, 8(2): 201-214. doi: 10.3934/krm.2015.8.201

Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds

Martial Agueh 1, , Guillaume Carlier 2, and  Reinhard Illner 1,

1. 

Department of Mathematics and Statistics, University of Victoria, PO BOX 1700 STN CSC, Victoria, BC, V8W 2Y2, Canada, Canada
2. 

CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16

Received  November 2014 Revised  January 2015 Published  March 2015

We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solutions. Furthermore, using an additional assumption on the interaction kernel and a ``potential for interaction'', we prove a global entropy estimate in the one-dimensional case.
Keywords: global in time estimates, Kinetic granular media, asymptotic behavior, entropy bounds..
Mathematics Subject Classification: Primary: 82C21, 82C22, 82C7.
Citation: Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic & Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201
References:
[1]

M. Agueh, Local existence of weak solutions to kinetic models of granular media,, 2014. Available from: , ().   Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics, (2005).   Google Scholar

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Model. Math. Anal. Numer., 31 (1997), 615.   Google Scholar

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media,, M2AN Math. Model. Numer. Anal., 33 (1999), 439.  doi: 10.1051/m2an:1999118.  Google Scholar

[5]

D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows,, M2AN Math. Model. Numer. Anal., 35 (2001), 899.  doi: 10.1051/m2an:2001141.  Google Scholar

[6]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for multidimensional aggregation model,, Comm. Pure Appl. Math., 64 (2011), 45.  doi: 10.1002/cpa.20334.  Google Scholar

[7]

J.-M. Bony, Existence globale et diffusion en théorie cinétique discrète,, in Advances in Kinetic Theory and Continuum Mechanics, (1991), 81.   Google Scholar

[8]

J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates,, Rev. Matemàtica Iberoamericana, 19 (2003), 971.  doi: 10.4171/RMI/376.  Google Scholar

[9]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Ration. Mech. Anal., 179 (2006), 217.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[10]

J. A. Carrillo, M. DiFrancesco, A. Figalli, L. Laurent and D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations,, Duke Math. J., 156 (2011), 229.  doi: 10.1215/00127094-2010-211.  Google Scholar

[11]

C. Cercignani and R. Illner, Global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions,, Arch. Ration. Mech. Anal., 134 (1996), 1.  doi: 10.1007/BF00376253.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

R. Illner and G. Rein, Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case,, Math. Methods Appl. Sci., 19 (1996), 1409.  doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.  Google Scholar

[14]

T. Laurent, Local and global existence for an aggregation equation,, Comm. Partial Differential Equations, 32 (2007), 1941.  doi: 10.1080/03605300701318955.  Google Scholar

[15]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, in Kinetic Theories and the Boltzmann Equation, (1048), 60.  doi: 10.1007/BFb0071878.  Google Scholar

[16]

G. Toscani, One-dimensional kinetic models for granular flows,, RAIRO Modél. Math. Anal. Numér., 34 (2000), 1277.  doi: 10.1051/m2an:2000127.  Google Scholar

show all references

References:
[1]

M. Agueh, Local existence of weak solutions to kinetic models of granular media,, 2014. Available from: , ().   Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics, (2005).   Google Scholar

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Model. Math. Anal. Numer., 31 (1997), 615.   Google Scholar

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media,, M2AN Math. Model. Numer. Anal., 33 (1999), 439.  doi: 10.1051/m2an:1999118.  Google Scholar

[5]

D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows,, M2AN Math. Model. Numer. Anal., 35 (2001), 899.  doi: 10.1051/m2an:2001141.  Google Scholar

[6]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for multidimensional aggregation model,, Comm. Pure Appl. Math., 64 (2011), 45.  doi: 10.1002/cpa.20334.  Google Scholar

[7]

J.-M. Bony, Existence globale et diffusion en théorie cinétique discrète,, in Advances in Kinetic Theory and Continuum Mechanics, (1991), 81.   Google Scholar

[8]

J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates,, Rev. Matemàtica Iberoamericana, 19 (2003), 971.  doi: 10.4171/RMI/376.  Google Scholar

[9]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Ration. Mech. Anal., 179 (2006), 217.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[10]

J. A. Carrillo, M. DiFrancesco, A. Figalli, L. Laurent and D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations,, Duke Math. J., 156 (2011), 229.  doi: 10.1215/00127094-2010-211.  Google Scholar

[11]

C. Cercignani and R. Illner, Global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions,, Arch. Ration. Mech. Anal., 134 (1996), 1.  doi: 10.1007/BF00376253.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

R. Illner and G. Rein, Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case,, Math. Methods Appl. Sci., 19 (1996), 1409.  doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.  Google Scholar

[14]

T. Laurent, Local and global existence for an aggregation equation,, Comm. Partial Differential Equations, 32 (2007), 1941.  doi: 10.1080/03605300701318955.  Google Scholar

[15]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, in Kinetic Theories and the Boltzmann Equation, (1048), 60.  doi: 10.1007/BFb0071878.  Google Scholar

[16]

G. Toscani, One-dimensional kinetic models for granular flows,, RAIRO Modél. Math. Anal. Numér., 34 (2000), 1277.  doi: 10.1051/m2an:2000127.  Google Scholar

[1]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[2]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[3]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[4]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[5]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[6]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018
[7]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[8]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[9]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082
[10]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[11]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[12]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377
[13]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[14]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
[15]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[16]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[17]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[18]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[19]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[20]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences