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]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943
[2]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[3]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[4]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[5]

Dohyun Kim. Asymptotic behavior of a second-order swarm sphere model and its kinetic limit. Kinetic & Related Models, 2020, 13 (2) : 401-434. doi: 10.3934/krm.2020014
[6]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[7]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651
[8]

Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058
[9]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
[10]

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

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47
[12]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509
[13]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[14]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[15]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
[16]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[17]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[18]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[19]

Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159
[20]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences