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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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