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October  2020, 13(5): 951-978. doi: 10.3934/krm.2020033

Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

1. 

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

2. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, P. R. China

3. 

Université de Rouen-Normandie, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France

* Corresponding author: morimoto.yoshinori.74r@st.kyoto-u.ac.jp

Received  December 2019 Revised  March 2020 Published  August 2020

Fund Project: The research of the first author is supported by JSPS Kakenhi Grant No.17K05318. The research of the second author is supported partially by "The Fundamental Research Funds for Central Universities"

We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if $ H^r_x(L^2_v), r >3/2 $ norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position $ x $ and velocity $ v $ variables for any time $ t>0 $. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight of exponential type associated with the kinetic transport operator.

Citation: Yoshinori Morimoto, Chao-Jiang Xu. Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules. Kinetic & Related Models, 2020, 13 (5) : 951-978. doi: 10.3934/krm.2020033
References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interaction, 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, 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

[4]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.  doi: 10.1007/s00220-011-1242-9.  Google Scholar

[5]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 21 (2004), 61-95.  doi: 10.1016/S0294-1449(03)00030-1.  Google Scholar

[6]

H. BarbarouxD. HundermarkT. Ried and S. Vugalter, Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equation without cutoff for Maxwellian molecules, Arch. Rational Mech. Anal., 225 (2017), 601-661.  doi: 10.1007/s00205-017-1101-8.  Google Scholar

[7]

A. V. Bobylev, The expansion of the Boltzmann collision integral in a Landau series, Dokl. Akad. Nauk SSSR, 225 (1975), 535-538.   Google Scholar

[8]

A. V. BobylevI. M. Gamba and I. F. Potapenko, On some properties of the Landau kinetic equation, J. Stat. Phys., 161 (2015), 1327-1338.  doi: 10.1007/s10955-015-1311-0.  Google Scholar

[9]

A. V. BobylevM. Pulvirenti and C. Saffrio, From practical system to the Landau equation: A consistency result, Commun. Math. Phys., 319 (2013), 683-702.  doi: 10.1007/s00220-012-1633-6.  Google Scholar

[10]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theory, Editions scientifiques et médicales Elsevier, Paris, 2000.  Google Scholar

[11]

S. CameronL. Silvestre and S. Snelson, Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials, Ann. Inst. H. Poincaré Anal. Non linéaire, 35 (2018), 625-642.  doi: 10.1016/j.anihpc.2017.07.001.  Google Scholar

[12]

H.-M. CaoH.-G. LiC.-J. Xu and J. Xu, Well-posedness of Cauchy problem for Landau equation in critical Besov space, Kinetic and Related Models, 12 (2019), 829-884.  doi: 10.3934/krm.2019032.  Google Scholar

[13]

K. Carrapatoso and S. Mischler, Landau equation for very soft and Coulomb potentials near Maxwellians, Ann. Partial Differential Equations, 3 (2017), 1-65.  doi: 10.1007/s40818-017-0021-0.  Google Scholar

[14]

K. CarrapatosoI. Tristani and K.-C. Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., 221 (2016), 363-418.  doi: 10.1007/s00205-015-0963-x.  Google Scholar

[15]

L. Desvillettes, On asymptotics of the Boltzmann equation when collisions became grazing, Transp. Theory Stat. Phys., 21 (1992), 259-276.  doi: 10.1080/00411459208203923.  Google Scholar

[16]

L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., 269 (2015), 1359-1403.  doi: 10.1016/j.jfa.2015.05.009.  Google Scholar

[17]

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

[18]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Ⅱ. H-theorem and applications, Commun. Partial Differ. Equ., 25 (2000), 261-298.  doi: 10.1080/03605300008821513.  Google Scholar

[19]

R. Duan, S. Liu, S. Sakamoto and R. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations, Comm. Pure Appl. Math., (2020). doi: 10.1002/cpa.21920.  Google Scholar

[20]

F. GolseC. ImbertC. Mouhot and A. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19 (2019), 253-295.   Google Scholar

[21]

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

[22]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phy., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.  Google Scholar

[23]

C. Henderson and S. Snelson, $C^\infty$ smoothing for weak solutions of the inhomogeneous Landau equations, Arch. Ration. Mech. Anal., 236 (2020), 113-143.  doi: 10.1007/s00205-019-01465-7.  Google Scholar

[24]

L. D. Landau, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10 (1936), 154-164.   Google Scholar

[25]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators, Kinet. Relat. Models, 6 (2013), 625-648.  doi: 10.3934/krm.2013.6.625.  Google Scholar

[26]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov 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

[27]

P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London, Ser. A, 346 (1994), 191-204.  doi: 10.1098/rsta.1994.0018.  Google Scholar

[28]

Y. MorimotoK. Pravda-Starov and C.-J. Xu, A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equations, Kinet. trlat. Models, 6 (2013), 715-727.  doi: 10.3934/krm.2013.6.715.  Google Scholar

[29]

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

[30]

Y. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst. A, 24 (2009), 187-212.  doi: 10.3934/dcds.2009.24.187.  Google Scholar

[31]

Y. MorimotoS. Wang and T. Yang, Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, J. Stat. Phys., 165 (2016), 866-906.  doi: 10.1007/s10955-016-1655-0.  Google Scholar

[32]

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

[33]

R. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.  doi: 10.1007/s00205-007-0067-3.  Google Scholar

[34]

T. Yang and H. Yu, Optimal convergence rates of the Landau equation with external forcing in the whole space, Acta Math. Sci., 29B (2009), 1035-1062.  doi: 10.1016/S0252-9602(09)60085-0.  Google Scholar

[35]

C. Villani, On a new class of week solutions to the homogeneous Boltzmann and Landau equations, Arch. Rat. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.  Google Scholar

[36]

C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Handbook of Mathematical Fluid Dynamics, 1, North-Holland, Amsterdam, 2002, 71–305. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

show all references

References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interaction, 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, 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

[4]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.  doi: 10.1007/s00220-011-1242-9.  Google Scholar

[5]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 21 (2004), 61-95.  doi: 10.1016/S0294-1449(03)00030-1.  Google Scholar

[6]

H. BarbarouxD. HundermarkT. Ried and S. Vugalter, Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equation without cutoff for Maxwellian molecules, Arch. Rational Mech. Anal., 225 (2017), 601-661.  doi: 10.1007/s00205-017-1101-8.  Google Scholar

[7]

A. V. Bobylev, The expansion of the Boltzmann collision integral in a Landau series, Dokl. Akad. Nauk SSSR, 225 (1975), 535-538.   Google Scholar

[8]

A. V. BobylevI. M. Gamba and I. F. Potapenko, On some properties of the Landau kinetic equation, J. Stat. Phys., 161 (2015), 1327-1338.  doi: 10.1007/s10955-015-1311-0.  Google Scholar

[9]

A. V. BobylevM. Pulvirenti and C. Saffrio, From practical system to the Landau equation: A consistency result, Commun. Math. Phys., 319 (2013), 683-702.  doi: 10.1007/s00220-012-1633-6.  Google Scholar

[10]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theory, Editions scientifiques et médicales Elsevier, Paris, 2000.  Google Scholar

[11]

S. CameronL. Silvestre and S. Snelson, Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials, Ann. Inst. H. Poincaré Anal. Non linéaire, 35 (2018), 625-642.  doi: 10.1016/j.anihpc.2017.07.001.  Google Scholar

[12]

H.-M. CaoH.-G. LiC.-J. Xu and J. Xu, Well-posedness of Cauchy problem for Landau equation in critical Besov space, Kinetic and Related Models, 12 (2019), 829-884.  doi: 10.3934/krm.2019032.  Google Scholar

[13]

K. Carrapatoso and S. Mischler, Landau equation for very soft and Coulomb potentials near Maxwellians, Ann. Partial Differential Equations, 3 (2017), 1-65.  doi: 10.1007/s40818-017-0021-0.  Google Scholar

[14]

K. CarrapatosoI. Tristani and K.-C. Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., 221 (2016), 363-418.  doi: 10.1007/s00205-015-0963-x.  Google Scholar

[15]

L. Desvillettes, On asymptotics of the Boltzmann equation when collisions became grazing, Transp. Theory Stat. Phys., 21 (1992), 259-276.  doi: 10.1080/00411459208203923.  Google Scholar

[16]

L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., 269 (2015), 1359-1403.  doi: 10.1016/j.jfa.2015.05.009.  Google Scholar

[17]

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

[18]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Ⅱ. H-theorem and applications, Commun. Partial Differ. Equ., 25 (2000), 261-298.  doi: 10.1080/03605300008821513.  Google Scholar

[19]

R. Duan, S. Liu, S. Sakamoto and R. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations, Comm. Pure Appl. Math., (2020). doi: 10.1002/cpa.21920.  Google Scholar

[20]

F. GolseC. ImbertC. Mouhot and A. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19 (2019), 253-295.   Google Scholar

[21]

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

[22]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phy., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.  Google Scholar

[23]

C. Henderson and S. Snelson, $C^\infty$ smoothing for weak solutions of the inhomogeneous Landau equations, Arch. Ration. Mech. Anal., 236 (2020), 113-143.  doi: 10.1007/s00205-019-01465-7.  Google Scholar

[24]

L. D. Landau, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10 (1936), 154-164.   Google Scholar

[25]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators, Kinet. Relat. Models, 6 (2013), 625-648.  doi: 10.3934/krm.2013.6.625.  Google Scholar

[26]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov 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

[27]

P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London, Ser. A, 346 (1994), 191-204.  doi: 10.1098/rsta.1994.0018.  Google Scholar

[28]

Y. MorimotoK. Pravda-Starov and C.-J. Xu, A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equations, Kinet. trlat. Models, 6 (2013), 715-727.  doi: 10.3934/krm.2013.6.715.  Google Scholar

[29]

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

[30]

Y. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst. A, 24 (2009), 187-212.  doi: 10.3934/dcds.2009.24.187.  Google Scholar

[31]

Y. MorimotoS. Wang and T. Yang, Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, J. Stat. Phys., 165 (2016), 866-906.  doi: 10.1007/s10955-016-1655-0.  Google Scholar

[32]

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

[33]

R. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.  doi: 10.1007/s00205-007-0067-3.  Google Scholar

[34]

T. Yang and H. Yu, Optimal convergence rates of the Landau equation with external forcing in the whole space, Acta Math. Sci., 29B (2009), 1035-1062.  doi: 10.1016/S0252-9602(09)60085-0.  Google Scholar

[35]

C. Villani, On a new class of week solutions to the homogeneous Boltzmann and Landau equations, Arch. Rat. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.  Google Scholar

[36]

C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Handbook of Mathematical Fluid Dynamics, 1, North-Holland, Amsterdam, 2002, 71–305. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

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