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

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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

Abstract
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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 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: http://www.math.uvic.ca/ agueh/Publications.html.
[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: http://www.math.uvic.ca/ agueh/Publications.html.
[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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