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 & 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.  doi: 10.1007/s002050000083.  Google Scholar

[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.  doi: 10.1215/21562261-1625154.  Google Scholar

[3]

P. Biler, G. Karch and W. A. Woyczynski, Asymptotics for multifractal conservation laws,, Studia Math., 135 (1999), 231.   Google Scholar

[4]

R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes,, Trans. Amer. Math. Soc., 95 (1960), 263.  doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[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.   Google Scholar

[6]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, Mathematical physics reviews, 7 (1988), 111.   Google Scholar

[7]

A. V. Bobylev and C. Cercignani, Exact eternal solutions of the Boltzmann equation,, J. Statist. Phys., 106 (2002), 1019.  doi: 10.1023/A:1014085719973.  Google Scholar

[8]

A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications,, J. Statist. Phys., 106 (2002), 1039.  doi: 10.1023/A:1014037804043.  Google Scholar

[9]

M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation,, Comm. Pure Appl. Math., 63 (2010), 747.  doi: 10.1002/cpa.20298.  Google Scholar

[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.  doi: 10.3934/krm.2008.1.453.  Google Scholar

[11]

N. Jacob, Pseudo-differential Operators and Markov Processes. Vol. I,, Fourier analysis and semigroups. Imperial College Press, (2001).  doi: 10.1142/9781860949746.  Google Scholar

[12]

Y. Morimoto, A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules,, Kinet. Relat. Models, 5 (2012), 551.  doi: 10.3934/krm.2012.5.551.  Google Scholar

[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.  doi: 10.1007/s11868-010-0008-z.  Google Scholar

[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.  doi: 10.3934/dcds.2009.24.187.  Google Scholar

[15]

Y. Morimoto and T. Yang, Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum,, , ().   Google Scholar

[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.  doi: 10.1023/A:1004589506756.  Google Scholar

[17]

C. Villani, A review of mathematical topics in collisional kinetic theory,, In Handbook of Mathematical Fluid Dynamics, I (2002), 71.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

show all references

References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Ration. Mech. Anal., 152 (2000), 327.  doi: 10.1007/s002050000083.  Google Scholar

[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.  doi: 10.1215/21562261-1625154.  Google Scholar

[3]

P. Biler, G. Karch and W. A. Woyczynski, Asymptotics for multifractal conservation laws,, Studia Math., 135 (1999), 231.   Google Scholar

[4]

R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes,, Trans. Amer. Math. Soc., 95 (1960), 263.  doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[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.   Google Scholar

[6]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, Mathematical physics reviews, 7 (1988), 111.   Google Scholar

[7]

A. V. Bobylev and C. Cercignani, Exact eternal solutions of the Boltzmann equation,, J. Statist. Phys., 106 (2002), 1019.  doi: 10.1023/A:1014085719973.  Google Scholar

[8]

A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications,, J. Statist. Phys., 106 (2002), 1039.  doi: 10.1023/A:1014037804043.  Google Scholar

[9]

M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation,, Comm. Pure Appl. Math., 63 (2010), 747.  doi: 10.1002/cpa.20298.  Google Scholar

[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.  doi: 10.3934/krm.2008.1.453.  Google Scholar

[11]

N. Jacob, Pseudo-differential Operators and Markov Processes. Vol. I,, Fourier analysis and semigroups. Imperial College Press, (2001).  doi: 10.1142/9781860949746.  Google Scholar

[12]

Y. Morimoto, A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules,, Kinet. Relat. Models, 5 (2012), 551.  doi: 10.3934/krm.2012.5.551.  Google Scholar

[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.  doi: 10.1007/s11868-010-0008-z.  Google Scholar

[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.  doi: 10.3934/dcds.2009.24.187.  Google Scholar

[15]

Y. Morimoto and T. Yang, Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum,, , ().   Google Scholar

[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.  doi: 10.1023/A:1004589506756.  Google Scholar

[17]

C. Villani, A review of mathematical topics in collisional kinetic theory,, In Handbook of Mathematical Fluid Dynamics, I (2002), 71.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

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