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

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.
Citation: Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic and 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: , (). 

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, Basel, 2005.

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641.

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media, M2AN Math. Model. Numer. Anal., 33 (1999), 439-441. doi: 10.1051/m2an:1999118.

[5]

D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal., 35 (2001), 899-905. doi: 10.1051/m2an:2001141.

[6]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for multidimensional aggregation model, Comm. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.

[7]

J.-M. Bony, Existence globale et diffusion en théorie cinétique discrète, in Advances in Kinetic Theory and Continuum Mechanics, Springer-Verlag, Berlin, 1991, 81-90.

[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), 971-1018. doi: 10.4171/RMI/376.

[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), 217-263. doi: 10.1007/s00205-005-0386-1.

[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), 229-271. doi: 10.1215/00127094-2010-211.

[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), 1-16. doi: 10.1007/BF00376253.

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[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), 1409-1413. doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.

[14]

T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964. doi: 10.1080/03605300701318955.

[15]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics, 1048, Springer-Verlag, Berlin, 1984, 60-110. doi: 10.1007/BFb0071878.

[16]

G. Toscani, One-dimensional kinetic models for granular flows, RAIRO Modél. Math. Anal. Numér., 34 (2000), 1277-1292. doi: 10.1051/m2an:2000127.

show all references

References:
[1]

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

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, Basel, 2005.

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641.

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media, M2AN Math. Model. Numer. Anal., 33 (1999), 439-441. doi: 10.1051/m2an:1999118.

[5]

D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal., 35 (2001), 899-905. doi: 10.1051/m2an:2001141.

[6]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for multidimensional aggregation model, Comm. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.

[7]

J.-M. Bony, Existence globale et diffusion en théorie cinétique discrète, in Advances in Kinetic Theory and Continuum Mechanics, Springer-Verlag, Berlin, 1991, 81-90.

[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), 971-1018. doi: 10.4171/RMI/376.

[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), 217-263. doi: 10.1007/s00205-005-0386-1.

[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), 229-271. doi: 10.1215/00127094-2010-211.

[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), 1-16. doi: 10.1007/BF00376253.

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[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), 1409-1413. doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.

[14]

T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964. doi: 10.1080/03605300701318955.

[15]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics, 1048, Springer-Verlag, Berlin, 1984, 60-110. doi: 10.1007/BFb0071878.

[16]

G. Toscani, One-dimensional kinetic models for granular flows, RAIRO Modél. Math. Anal. Numér., 34 (2000), 1277-1292. doi: 10.1051/m2an:2000127.

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