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September  2012, 5(3): 551-561. doi: 10.3934/krm.2012.5.551

A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules

1. 

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan

Received  January 2012 Revised  March 2012 Published  August 2012

The purpose of this paper is to extend the result concerning the existence and the uniqueness of infinite energy solutions, given by Cannone-Karch, of the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecules without Grad's angular cutoff assumption in the mild singularity case, to the strong singularity case. This extension follows from a simple observation of the symmetry on the unit sphere for the Bobylev formula which is the Fourier transform of the Boltzmann collision term.
Citation: Yoshinori Morimoto. A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinetic & Related Models, 2012, 5 (3) : 551-561. doi: 10.3934/krm.2012.5.551
References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327.   Google Scholar

[2]

R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff and Maxwellian molecules,, Math. Models Methods Appl. Sci., 15 (2005), 907.   Google Scholar

[3]

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.   Google Scholar

[4]

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

[5]

Z. H. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff,, Kinetic and Related Models, 1 (2008), 453.   Google Scholar

[6]

N. Jacob, "Pseudo-Differential Operators and Markov Process. Vol. 1. Fourier Analysis and Semigroups,", Imperial College Press, (2001).   Google Scholar

[7]

Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,, Discrete and Continuous Dynamical Systems, 24 (2009), 187.   Google Scholar

[8]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Wahrsch. Verw. Geb., 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[9]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas,, J. Statist. Phys., 94 (1999), 619.   Google Scholar

[10]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273.   Google Scholar

[11]

C. Villani, "A Review of Mathematical Topics in Collisional Kinetic Theory,", in, (2002), 71.   Google Scholar

[12]

C. Villani, private communication,, Kyoto, (2008).   Google Scholar

[13]

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

show all references

References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327.   Google Scholar

[2]

R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff and Maxwellian molecules,, Math. Models Methods Appl. Sci., 15 (2005), 907.   Google Scholar

[3]

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.   Google Scholar

[4]

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

[5]

Z. H. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff,, Kinetic and Related Models, 1 (2008), 453.   Google Scholar

[6]

N. Jacob, "Pseudo-Differential Operators and Markov Process. Vol. 1. Fourier Analysis and Semigroups,", Imperial College Press, (2001).   Google Scholar

[7]

Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,, Discrete and Continuous Dynamical Systems, 24 (2009), 187.   Google Scholar

[8]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Wahrsch. Verw. Geb., 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[9]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas,, J. Statist. Phys., 94 (1999), 619.   Google Scholar

[10]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273.   Google Scholar

[11]

C. Villani, "A Review of Mathematical Topics in Collisional Kinetic Theory,", in, (2002), 71.   Google Scholar

[12]

C. Villani, private communication,, Kyoto, (2008).   Google Scholar

[13]

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

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