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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 |
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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