# American Institute of Mathematical Sciences

December  2013, 6(4): 801-808. doi: 10.3934/krm.2013.6.801

## On self-similar solutions to the homogeneous Boltzmann equation

 1 Université Paris-Est-Marne-la-Vallée, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050 CNRS, 5 boulevard Descartes, Cité Descartes Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France 2 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Received  July 2013 Revised  August 2013 Published  November 2013

In this note, our results from [ Comm. Pure Appl. Math. 63 (2010), 747--778] on infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules are discussed, presented in a different context, and improved by using recent observations by Morimoto and Yang. In particular, similarities between the homogeneous Boltzmann equation and the fractional heat equation are emphasized. Moreover, we show that a certain conjecture by Bobylev and Cercignani on regularity of self-similar solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules has a positive answer.
Citation: Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52 (2012), 433-463. doi: 10.1215/21562261-1625154. [3] P. Biler, G. Karch and W. A. Woyczynski, Asymptotics for multifractal conservation laws, Studia Math., 135 (1999), 231-252. [4] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273. doi: 10.1090/S0002-9947-1960-0119247-6. [5] A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044. [6] A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical physics reviews, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, 7 (1988), 111-233. [7] A. V. Bobylev and C. Cercignani, Exact eternal solutions of the Boltzmann equation, J. Statist. Phys., 106 (2002), 1019-1038. doi: 10.1023/A:1014085719973. [8] A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications, J. Statist. Phys., 106 (2002), 1039-1071. doi: 10.1023/A:1014037804043. [9] M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation, Comm. Pure Appl. Math., 63 (2010), 747-778. doi: 10.1002/cpa.20298. [10] Z. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, 1 (2008), 453-489. doi: 10.3934/krm.2008.1.453. [11] N. Jacob, Pseudo-differential Operators and Markov Processes. Vol. I, Fourier analysis and semigroups. Imperial College Press, London, 2001. doi: 10.1142/9781860949746. [12] Y. Morimoto, A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules, Kinet. Relat. Models, 5 (2012), 551-561. doi: 10.3934/krm.2012.5.551. [13] Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseudo-Differ. Oper. Appl., 1 (2010), 139-159. doi: 10.1007/s11868-010-0008-z. [14] Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst., 24 (2009), 187-212. doi: 10.3934/dcds.2009.24.187. [15] Y. Morimoto and T. Yang, Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum, arXiv:1210.0296 . [16] G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756. [17] C. Villani, A review of mathematical topics in collisional kinetic theory, In Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.

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##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52 (2012), 433-463. doi: 10.1215/21562261-1625154. [3] P. Biler, G. Karch and W. A. Woyczynski, Asymptotics for multifractal conservation laws, Studia Math., 135 (1999), 231-252. [4] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273. doi: 10.1090/S0002-9947-1960-0119247-6. [5] A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044. [6] A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical physics reviews, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, 7 (1988), 111-233. [7] A. V. Bobylev and C. Cercignani, Exact eternal solutions of the Boltzmann equation, J. Statist. Phys., 106 (2002), 1019-1038. doi: 10.1023/A:1014085719973. [8] A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications, J. Statist. Phys., 106 (2002), 1039-1071. doi: 10.1023/A:1014037804043. [9] M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation, Comm. Pure Appl. Math., 63 (2010), 747-778. doi: 10.1002/cpa.20298. [10] Z. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, 1 (2008), 453-489. doi: 10.3934/krm.2008.1.453. [11] N. Jacob, Pseudo-differential Operators and Markov Processes. Vol. I, Fourier analysis and semigroups. Imperial College Press, London, 2001. doi: 10.1142/9781860949746. [12] Y. Morimoto, A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules, Kinet. Relat. Models, 5 (2012), 551-561. doi: 10.3934/krm.2012.5.551. [13] Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseudo-Differ. Oper. Appl., 1 (2010), 139-159. doi: 10.1007/s11868-010-0008-z. [14] Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst., 24 (2009), 187-212. doi: 10.3934/dcds.2009.24.187. [15] Y. Morimoto and T. Yang, Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum, arXiv:1210.0296 . [16] G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756. [17] C. Villani, A review of mathematical topics in collisional kinetic theory, In Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.
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