# American Institute of Mathematical Sciences

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 & 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

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##### 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
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