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

Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation

Léo Glangetas 1, and  Mohamed Najeme 1,

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.
Keywords: Kac's equation, Fourier analysis., Boltzmann equation, smoothing effect of Cauchy problem, non-cutoff cross-sections, Gevrey regularizing, analytical regularizing.
Mathematics Subject Classification: 35A20, 35B65, 35H20, 76P05, 82C4.
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), 327-355. 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), 2013-2066. 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), 915-1010. 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), 113-134. 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), 599-661. 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), 417-440.  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), 1731-1747. 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), 567-594.  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), 133-155. 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, Springer-Verlag, Berlin, 1994.  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), 453-489. 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), 647-666. 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), 191-204. 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), 139-159. 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), 187-212. 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), 141-156. 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), 273-307. 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), 1172-1190. 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), 327-355. 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), 2013-2066. 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), 915-1010. 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), 113-134. 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), 599-661. 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), 417-440.  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), 1731-1747. 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), 567-594.  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), 133-155. 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, Springer-Verlag, Berlin, 1994.  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), 453-489. 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), 647-666. 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), 191-204. 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), 139-159. 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), 187-212. 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), 141-156. 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), 273-307. 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), 1172-1190. doi: 10.1016/j.jde.2012.04.023.  Google Scholar

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