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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, Birkhäuser, Basel, 2005.  Google Scholar

[3]

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

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media, M2AN Math. Model. Numer. Anal., 33 (1999), 439-441. 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), 899-905. 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), 45-83. 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, Springer-Verlag, Berlin, 1991, 81-90. 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), 971-1018. 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), 217-263. 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), 229-271. 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), 1-16. doi: 10.1007/BF00376253.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 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), 1409-1413. 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), 1941-1964. doi: 10.1080/03605300701318955.  Google Scholar

[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.  Google Scholar

[16]

G. Toscani, One-dimensional kinetic models for granular flows, RAIRO Modél. Math. Anal. Numér., 34 (2000), 1277-1292. 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, Birkhäuser, Basel, 2005.  Google Scholar

[3]

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

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media, M2AN Math. Model. Numer. Anal., 33 (1999), 439-441. 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), 899-905. 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), 45-83. 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, Springer-Verlag, Berlin, 1991, 81-90. 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), 971-1018. 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), 217-263. 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), 229-271. 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), 1-16. doi: 10.1007/BF00376253.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 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), 1409-1413. 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), 1941-1964. doi: 10.1080/03605300701318955.  Google Scholar

[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.  Google Scholar

[16]

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

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