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October  2020, 13(5): 1029-1046. doi: 10.3934/krm.2020036

Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

1. 

School of Mathematics and Statistics, Wuhan University & , Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Lvqiao Liu

Received  January 2020 Revised  May 2020 Published  August 2020

Fund Project: The first author is supported by NSF grant Nos. 11871054, 11771342; Fok Ying Tung Education Foundation (151001) and the Natural Science Foundation of Hubei Province (2019CFA007)

In this work we consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation. For any given solution belonging to weighted Sobolev space, we will show it enjoys at positive time the Gelfand-Shilov smoothing effect for the velocity variable and Gevrey regularizing properties for the spatial variable. This improves the result of Lerner-Morimoto-Pravda-Starov-Xu [J. Funct. Anal. 269 (2015) 459-535] on one-dimensional Boltzmann equation to the physical three-dimensional case. Our proof relies on the elementary $ L^2 $ weighted estimate.

Citation: Wei-Xi Li, Lvqiao Liu. Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off. Kinetic and Related Models, 2020, 13 (5) : 1029-1046. doi: 10.3934/krm.2020036
References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.

[2]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.  doi: 10.1007/s00205-010-0290-1.

[3]

R. AlexandreY. MorimotoS. UkaiC.-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.

[4]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661.  doi: 10.1007/s00205-011-0432-0.

[5]

R. AlexandreY. MorimotoS. UkaiC.-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.

[6]

R. Alexandre and M. Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., 15 (2005), 907-920.  doi: 10.1142/S0218202505000613.

[7]

R. Alexandre and M. Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. Ⅱ. Non cutoff case and non Maxwellian molecules, Discrete Contin. Dyn. Syst., 24 (2009), 1-11.  doi: 10.3934/dcds.2009.24.1.

[8]

R. AlexandreF. Hérau and W.-X. Li, Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff, J. Math. Pures Appl. (9), 126 (2019), 1-71.  doi: 10.1016/j.matpur.2019.04.013.

[9]

J.-M. BarbarouxD. HundertmarkT. Ried and S. Vugalter, Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules, Archive for Rational Mechanics and Analysis, 225 (2017), 601-661.  doi: 10.1007/s00205-017-1101-8.

[10]

H. Cao, H.-G. Li, C.-J. Xu and J. Xu, The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space, arXiv E-prints, arXiv: 1902.06699. doi: 10.1016/j.jde.2019.12.025.

[11]

H. Chen, X. Hu, W.-X. Li and J. Zhan, Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off, arXiv E-prints, arXiv: 1805.12543.

[12]

H. ChenW.-X. Li and C.-J. Xu, Propagation of Gevrey regularity for solutions of Landau equations, Kinet. Relat. Models, 1 (2008), 355-368.  doi: 10.3934/krm.2008.1.355.

[13]

H. ChenW.-X. Li and and C.-J. Xu, Analytic smoothness effect of solutions for spatially homogeneous Landau equation, J. Differential Equations, 248 (2010), 77-94.  doi: 10.1016/j.jde.2009.08.006.

[14]

L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys., 168 (1995), 417-440.  doi: 10.1007/BF02101556.

[15]

L. Desvillettes, Regularization properties of the $2$-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Transport Theory Statist. Phys., 26 (1997), 341-357.  doi: 10.1080/00411459708020291.

[16]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.

[17]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations, 29 (2004), 133-155.  doi: 10.1081/PDE-120028847.

[18]

L. GlangetasH.-G. Li and C.-J. Xu, Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation, Kinet. Relat. Models, 9 (2016), 299-371.  doi: 10.3934/krm.2016.9.299.

[19]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8.

[20]

Z. HuoY. MorimotoS. 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.

[21]

N. Lerner, Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators, Birkhäuser Verlag, Basel, 2010. doi: 10.1007/978-3-7643-8510-1.

[22]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, J. Differential Equations, 256 (2014), 797-831.  doi: 10.1016/j.jde.2013.10.001.

[23]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation, J. Funct. Anal., 269 (2015), 459-535.  doi: 10.1016/j.jfa.2015.04.017.

[24]

H.-G. Li and C.-J. Xu, The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect, J. Differential Equations, 263 (2017), 5120-5150.  doi: 10.1016/j.jde.2017.06.010.

[25]

P.-L. Lions, Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 37-41.  doi: 10.1016/S0764-4442(97)82709-7.

[26]

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.

[27]

Y. Morimoto and T. Yang, Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 429-442.  doi: 10.1016/j.anihpc.2013.12.004.

[28]

Y. MorimotoS. UkaiC.-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.

[29]

Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differential Equations, 247 (2009), 596-617.  doi: 10.1016/j.jde.2009.01.028.

[30]

C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous boltzmann equation without cut-off, Rev. Mat. Iberoam., 15 (1999), 335-352.  doi: 10.4171/RMI/259.

show all references

References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.

[2]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.  doi: 10.1007/s00205-010-0290-1.

[3]

R. AlexandreY. MorimotoS. UkaiC.-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.

[4]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661.  doi: 10.1007/s00205-011-0432-0.

[5]

R. AlexandreY. MorimotoS. UkaiC.-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.

[6]

R. Alexandre and M. Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., 15 (2005), 907-920.  doi: 10.1142/S0218202505000613.

[7]

R. Alexandre and M. Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. Ⅱ. Non cutoff case and non Maxwellian molecules, Discrete Contin. Dyn. Syst., 24 (2009), 1-11.  doi: 10.3934/dcds.2009.24.1.

[8]

R. AlexandreF. Hérau and W.-X. Li, Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff, J. Math. Pures Appl. (9), 126 (2019), 1-71.  doi: 10.1016/j.matpur.2019.04.013.

[9]

J.-M. BarbarouxD. HundertmarkT. Ried and S. Vugalter, Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules, Archive for Rational Mechanics and Analysis, 225 (2017), 601-661.  doi: 10.1007/s00205-017-1101-8.

[10]

H. Cao, H.-G. Li, C.-J. Xu and J. Xu, The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space, arXiv E-prints, arXiv: 1902.06699. doi: 10.1016/j.jde.2019.12.025.

[11]

H. Chen, X. Hu, W.-X. Li and J. Zhan, Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off, arXiv E-prints, arXiv: 1805.12543.

[12]

H. ChenW.-X. Li and C.-J. Xu, Propagation of Gevrey regularity for solutions of Landau equations, Kinet. Relat. Models, 1 (2008), 355-368.  doi: 10.3934/krm.2008.1.355.

[13]

H. ChenW.-X. Li and and C.-J. Xu, Analytic smoothness effect of solutions for spatially homogeneous Landau equation, J. Differential Equations, 248 (2010), 77-94.  doi: 10.1016/j.jde.2009.08.006.

[14]

L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys., 168 (1995), 417-440.  doi: 10.1007/BF02101556.

[15]

L. Desvillettes, Regularization properties of the $2$-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Transport Theory Statist. Phys., 26 (1997), 341-357.  doi: 10.1080/00411459708020291.

[16]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.

[17]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations, 29 (2004), 133-155.  doi: 10.1081/PDE-120028847.

[18]

L. GlangetasH.-G. Li and C.-J. Xu, Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation, Kinet. Relat. Models, 9 (2016), 299-371.  doi: 10.3934/krm.2016.9.299.

[19]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8.

[20]

Z. HuoY. MorimotoS. 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.

[21]

N. Lerner, Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators, Birkhäuser Verlag, Basel, 2010. doi: 10.1007/978-3-7643-8510-1.

[22]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, J. Differential Equations, 256 (2014), 797-831.  doi: 10.1016/j.jde.2013.10.001.

[23]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation, J. Funct. Anal., 269 (2015), 459-535.  doi: 10.1016/j.jfa.2015.04.017.

[24]

H.-G. Li and C.-J. Xu, The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect, J. Differential Equations, 263 (2017), 5120-5150.  doi: 10.1016/j.jde.2017.06.010.

[25]

P.-L. Lions, Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 37-41.  doi: 10.1016/S0764-4442(97)82709-7.

[26]

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.

[27]

Y. Morimoto and T. Yang, Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 429-442.  doi: 10.1016/j.anihpc.2013.12.004.

[28]

Y. MorimotoS. UkaiC.-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.

[29]

Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differential Equations, 247 (2009), 596-617.  doi: 10.1016/j.jde.2009.01.028.

[30]

C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous boltzmann equation without cut-off, Rev. Mat. Iberoam., 15 (1999), 335-352.  doi: 10.4171/RMI/259.

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