June  2013, 6(2): 407-427. doi: 10.3934/krm.2013.6.407

Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation

1. 

Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l'Université, BP.12, 76801 Saint Etienne du Rouvray, France, France

Received  June 2012 Revised  September 2012 Published  February 2013

In this paper, we consider a class of spatially homogeneous Boltzmann equation without angular cutoff. We prove that any radial symmetric weak solution of the Cauchy problem become analytic for positive time.
Citation: Léo Glangetas, Mohamed Najeme. Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 407-427. doi: 10.3934/krm.2013.6.407
References:
[1]

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

[2]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Uncertainty principle and kinetic equations,, J. Funct. Anal., 255 (2008), 2013.  doi: 10.1016/j.jfa.2008.07.004.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential,, J. Funct. Anal., 262 (2012), 915.  doi: 10.1016/j.jfa.2011.10.007.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential,, Analysis and Applications, 9 (2011), 113.  doi: 10.1142/S0219530511001777.  Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions,, Arch. Rat. Mech. Anal., 202 (2011), 599.  doi: 10.1007/s00205-011-0432-0.  Google Scholar

[6]

L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation,, Comm. Math. Phys., 168 (1995), 417.   Google Scholar

[7]

L. Desvillettes, G. Furioli and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules,, Trans. Amer. Math. Soc., 361 (2009), 1731.  doi: 10.1090/S0002-9947-08-04574-1.  Google Scholar

[8]

L. Desvillettes and F. Golse, On a model Boltzmann equation without angular cutoff,, Diff. Int. Eq., 13 (2000), 567.   Google Scholar

[9]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff,, Comm. Part. Diff. Equations, 29 (2004), 133.  doi: 10.1081/PDE-120028847.  Google Scholar

[10]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators,", Corrected reprint of the 1985 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275 (1985).   Google Scholar

[11]

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.  doi: 10.3934/krm.2008.1.453.  Google Scholar

[12]

N. Lekrine and C.-J. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation,, Kinetic and Related Models, 2 (2009), 647.  doi: 10.3934/krm.2009.2.647.  Google Scholar

[13]

P.-L. Lions, On Boltzmann and Landau equations,, Philos. Trans. Roy. Soc. London A, 346 (1994), 191.  doi: 10.1098/rsta.1994.0018.  Google Scholar

[14]

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

[15]

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

[16]

S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,, Japan J. Appl. Math., 1 (1984), 141.  doi: 10.1007/BF03167864.  Google Scholar

[17]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273.  doi: 10.1007/s002050050106.  Google Scholar

[18]

T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff,, J. Differential Equations, 253 (2012), 1172.  doi: 10.1016/j.jde.2012.04.023.  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.  doi: 10.1007/s002050000083.  Google Scholar

[2]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Uncertainty principle and kinetic equations,, J. Funct. Anal., 255 (2008), 2013.  doi: 10.1016/j.jfa.2008.07.004.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential,, J. Funct. Anal., 262 (2012), 915.  doi: 10.1016/j.jfa.2011.10.007.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential,, Analysis and Applications, 9 (2011), 113.  doi: 10.1142/S0219530511001777.  Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions,, Arch. Rat. Mech. Anal., 202 (2011), 599.  doi: 10.1007/s00205-011-0432-0.  Google Scholar

[6]

L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation,, Comm. Math. Phys., 168 (1995), 417.   Google Scholar

[7]

L. Desvillettes, G. Furioli and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules,, Trans. Amer. Math. Soc., 361 (2009), 1731.  doi: 10.1090/S0002-9947-08-04574-1.  Google Scholar

[8]

L. Desvillettes and F. Golse, On a model Boltzmann equation without angular cutoff,, Diff. Int. Eq., 13 (2000), 567.   Google Scholar

[9]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff,, Comm. Part. Diff. Equations, 29 (2004), 133.  doi: 10.1081/PDE-120028847.  Google Scholar

[10]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators,", Corrected reprint of the 1985 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275 (1985).   Google Scholar

[11]

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.  doi: 10.3934/krm.2008.1.453.  Google Scholar

[12]

N. Lekrine and C.-J. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation,, Kinetic and Related Models, 2 (2009), 647.  doi: 10.3934/krm.2009.2.647.  Google Scholar

[13]

P.-L. Lions, On Boltzmann and Landau equations,, Philos. Trans. Roy. Soc. London A, 346 (1994), 191.  doi: 10.1098/rsta.1994.0018.  Google Scholar

[14]

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

[15]

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

[16]

S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,, Japan J. Appl. Math., 1 (1984), 141.  doi: 10.1007/BF03167864.  Google Scholar

[17]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273.  doi: 10.1007/s002050050106.  Google Scholar

[18]

T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff,, J. Differential Equations, 253 (2012), 1172.  doi: 10.1016/j.jde.2012.04.023.  Google Scholar

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