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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 and Related Models, 2020, 13 (5) : 951-978. doi: 10.3934/krm.2020033
##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interaction, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-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. 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. Ration. Mech. Anal., 202 (2011), 599-661.  doi: 10.1007/s00205-011-0432-0. [4] R. Alexandre, Y. Morimoto, S. Ukai, C.-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. [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. [6] H. Barbaroux, D. Hundermark, T. 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. [7] A. V. Bobylev, The expansion of the Boltzmann collision integral in a Landau series, Dokl. Akad. Nauk SSSR, 225 (1975), 535-538. [8] A. V. Bobylev, I. 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. [9] A. V. Bobylev, M. 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. [10] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theory, Editions scientifiques et médicales Elsevier, Paris, 2000. [11] S. Cameron, L. 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. [12] H.-M. Cao, H.-G. Li, C.-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. [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. [14] K. Carrapatoso, I. 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. [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. [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. [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. [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. [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. [20] F. Golse, C. Imbert, C. 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. [21] L. Glangetas, H.-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. [22] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phy., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9. [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. [24] L. D. Landau, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10 (1936), 154-164. [25] N. Lerner, Y. Morimoto, K. 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. [26] N. Lerner, Y. Morimoto, K. 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. [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. [28] Y. Morimoto, K. 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. [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. [30] 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. A, 24 (2009), 187-212.  doi: 10.3934/dcds.2009.24.187. [31] Y. Morimoto, S. 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. [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. [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. [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. [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. [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.

show all references

##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interaction, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-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. 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. Ration. Mech. Anal., 202 (2011), 599-661.  doi: 10.1007/s00205-011-0432-0. [4] R. Alexandre, Y. Morimoto, S. Ukai, C.-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. [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. [6] H. Barbaroux, D. Hundermark, T. 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. [7] A. V. Bobylev, The expansion of the Boltzmann collision integral in a Landau series, Dokl. Akad. Nauk SSSR, 225 (1975), 535-538. [8] A. V. Bobylev, I. 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. [9] A. V. Bobylev, M. 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. [10] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theory, Editions scientifiques et médicales Elsevier, Paris, 2000. [11] S. Cameron, L. 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. [12] H.-M. Cao, H.-G. Li, C.-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. [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. [14] K. Carrapatoso, I. 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. [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. [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. [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. [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. [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. [20] F. Golse, C. Imbert, C. 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. [21] L. Glangetas, H.-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. [22] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phy., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9. [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. [24] L. D. Landau, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10 (1936), 154-164. [25] N. Lerner, Y. Morimoto, K. 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. [26] N. Lerner, Y. Morimoto, K. 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. [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. [28] Y. Morimoto, K. 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. [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. [30] 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. A, 24 (2009), 187-212.  doi: 10.3934/dcds.2009.24.187. [31] Y. Morimoto, S. 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. [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. [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. [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. [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. [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.
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