Strong Convergence Towards Homogeneous Cooling States for Dissipative Maxwell Models
Annales de l'I.H.P. Analyse non linéaire, Volume 26 (2009) no. 5, p. 1675-1700
@article{AIHPC_2009__26_5_1675_0,
     author = {Carlen, Eric and Carrillo, Jos\'e A. and Carvalho, Maria C.},
     title = {Strong Convergence Towards Homogeneous Cooling States for Dissipative Maxwell Models},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {26},
     number = {5},
     year = {2009},
     pages = {1675-1700},
     doi = {10.1016/j.anihpc.2008.10.005},
     zbl = {1175.82046},
     mrnumber = {2566705},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2009__26_5_1675_0}
}
Carlen, Eric A.; Carrillo, José A.; Carvalho, Maria C. Strong Convergence Towards Homogeneous Cooling States for Dissipative Maxwell Models. Annales de l'I.H.P. Analyse non linéaire, Volume 26 (2009) no. 5, pp. 1675-1700. doi : 10.1016/j.anihpc.2008.10.005. http://www.numdam.org/item/AIHPC_2009__26_5_1675_0/

[1] Bisi M., Carrillo J. A., Toscani G., Contractive Metrics for a Boltzmann Equation for Granular Gases: Diffusive Equilibria, J. Statist. Phys. 118 (2005) 301-331. | MR 2122557 | Zbl 1085.82008

[2] Bisi M., Carrillo J. A., Toscani G., Decay Rates in Probability Metrics Towards Homogeneous Cooling States for the Inelastic Maxwell Model, J. Statist. Phys. 124 (2006) 625-653. | MR 2264621 | Zbl 1135.82028

[3] Bobylev A. V., Fourier Transform Method in the Theory of the Boltzmann Equation for Maxwellian Molecules, Dokl. Akad. Nauk USSR 225 (1975) 1041-1044. | MR 398379 | Zbl 0361.76077

[4] Bobylev A. V., The Theory of the Nonlinear Spatially Uniform Boltzmann Equation for Maxwell Molecules, Sov. Sci. Rev. C. Math. Phys. 7 (1988) 111-233. | MR 1128328 | Zbl 0850.76619

[5] Bobylev A. V., Carrillo J. A., Gamba I., On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions, J. Statist. Phys. 98 (2000) 743-773, Erratum on:, J. Statist. Phys. 103 (2001) 1137-1138. | MR 1749231 | Zbl 1126.82323

[6] Bobylev A. V., Cercignani C., Self-Similar Asymptotics for the Boltzmann Equation With Inelastic and Elastic Interactions, J. Statist. Phys. 110 (2003) 333-375. | MR 1966332 | Zbl 1134.82324

[7] Bobylev A. V., Cercignani C., Gamba I., Generalized Kinetic Maxwell Models of Granular Gases, in: Capriz G., Giovine P., Mariano P. M. (Eds.), Mathematical Models of Granular Matter, Lecture Notes in Mathematics, vol. 1937, Springer, 2008. | MR 2436467

[8] A.V. Bobylev, C. Cercignani, I. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Comm. Math. Phys., in press. | Zbl pre05655529

[9] Bobylev A. V., Cercignani C., Toscani G., Proof of an Asymptotic Property of Self-Similar Solutions of the Boltzmann Equation for Granular Materials, J. Statist. Phys. 111 (2003) 403-417. | MR 1964277 | Zbl 1119.82318

[10] Bolley F., Carrillo J. A., Tanaka Theorem for Inelastic Maxwell Models, Comm. Math. Phys. 276 (2007) 287-314. | MR 2346391 | Zbl 1136.82033

[11] Carlen E. A., Carvalho M. C., Strict Entropy Production Bounds and Stability of the Rate of Convergence to Equilibrium for the Boltzmann Equation, J. Statist. Phys. 67 (1992) 575-608. | MR 1171145 | Zbl 0899.76317

[12] Carlen E. A., Gabetta E., Toscani G., Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas, Comm. Math. Phys. 305 (1999) 521-546. | MR 1669689 | Zbl 0927.76088

[13] Carrillo J. A., Toscani G., Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations, Riv. Mat. Univ. Parma 6 (2007) 75-198. | MR 2355628 | Zbl 1142.82018

[14] Csiszar I., Information-Type Measures of Difference of Probability Distributions and Indirect Observations, Stud. Sci. Math. Hung. 2 (1967) 299-318. | MR 219345 | Zbl 0157.25802

[15] Desvillettes L., About the Use of the Fourier Transform for the Boltzmann Equation, Riv. Mat. Univ. Parma 7 (2003) 1-99. | MR 2052786 | Zbl 1072.35513

[16] Desvillettes L., Mouhot C., Large Time Behavior of the a Priori Bounds for the Solutions to the Spatially Homogeneous Boltzmann Equations With Soft Potentials, Asymptotic Anal. 54 (2007) 235-245. | MR 2359877 | Zbl 1141.35337

[17] Ernst M. H., Brito R., High Energy Tails for Inelastic Maxwell Models, Europhys. Lett. 58 (2002) 182-187.

[18] Ernst M. H., Brito R., Scaling Solutions of Inelastic Boltzmann Equation With Over-Populated High Energy Tails, J. Statist. Phys. 109 (2002) 407-432. | MR 1942001 | Zbl 1015.82030

[19] Gamba I., Panferov V., Villani C., On the Boltzmann Equation for Diffusively Excited Granular Media, Comm. Math. Phys. 246 (2004) 503-541. | MR 2053942 | Zbl 1106.82031

[20] Gross L., Logarithmic Sobolev Inequalities, Amer. J. Math. 97 (1975) 1061-1083. | MR 420249 | Zbl 0318.46049

[21] Kullback S., Leibler R. A., On Information and Sufficiency, Ann. Math. Statist. 22 (1951) 79-86. | MR 39968 | Zbl 0042.38403

[22] Lions P. L., Toscani G., A Strenghtened Central Limit Theorem for Smooth Densities, J. Funct. Anal. 128 (1995) 148-167. | MR 1322646 | Zbl 0822.60018

[23] Mischler S., Mouhot C., Rodríguez Ricard M., Cooling Process for Inelastic Boltzmann Equations for Hard Spheres, Part I: the Cauchy Problem, J. Statist. Phys. 124 (2006) 655-702. | MR 2264622 | Zbl 1135.82325

[24] Mischler S., Mouhot C., Cooling Process for Inelastic Boltzmann Equations for Hard Spheres, Part II: Self-Similar Solution and Tail Behavior, J. Statist. Phys. 124 (2006) 655-702. | MR 2264622 | Zbl 1135.82325

[25] Mischler S., Mouhot C., Stability, Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres, preprint, arXiv: math.AP/0701449. | MR 2500990

[26] Mischler S., Mouhot C., Stability, Convergence to the Steady State and Elastic Limit for the Boltzmann Equation for Diffusively Excited Inelastic Hard Spheres, preprint, arXiv: 0712.0124v1 [math.AP].

[27] Toscani G., Sur L'inégalité Logarithmique De Sobolev, C. R. Acad. Sci. Paris, Sér. 1 324 (1997) 689-694. | MR 1447044 | Zbl 0905.46018

[28] Villani C., Fisher Information Estimates for Boltzmann's Collision Operator, J. Math. Pures Appl. 77 (1998) 821-837. | MR 1646804 | Zbl 0918.60093

[29] Villani C., Mathematics of Granular Materials, J. Statist. Phys. 124 (2006) 781-822. | MR 2264625 | Zbl 1134.82040

[30] Zhang X., Regularity and Long Time Behavior of the Boltzmann Equation for Granular Gases, J. Math. Anal. Appl. 324 (2006) 650-662. | MR 2262498 | Zbl 1104.35008