# American Institute of Mathematical Sciences

December  2013, 6(4): 715-727. doi: 10.3934/krm.2013.6.715

## A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation

 1 Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501 2 Université de Cergy-Pontoise, CNRS UMR 8088, Mathématiques, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise 3 Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray

Received  February 2013 Revised  June 2013 Published  November 2013

We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class $S_{1/2}^{1/2}(\mathbb{R}^d)$, implying the ultra-analyticity and the production of exponential moments of the fluctuation, for any positive time.
Citation: Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation. Kinetic and Related Models, 2013, 6 (4) : 715-727. doi: 10.3934/krm.2013.6.715
##### References:
 [1] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95. doi: 10.1016/S0294-1449(03)00030-1. [2] A. A. Arsen'ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478. [3] H. Chen, W.-X. Li and C.-J. Xu, Gevrey regularity for solution of the spatially homogeneous Landau equation, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 673-686. doi: 10.1016/S0252-9602(09)60063-1. [4] H. Chen, W.-X. Li 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. [5] C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [6] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., 2 (1992), 167-182. doi: 10.1142/S0218202592000119. [7] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923. [8] 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. [9] I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 2. Spaces of fundamental and generalized functions. Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. [10] T. Gramchev, S. Pilipović and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces, in New Developments in Pseudo-Differential Operators, Oper. Theory Adv. Appl. Birkhäuser, Basel, 189 (2009), 15-31. doi: 10.1007/978-3-7643-8969-7_2. [11] L. D. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjet., 10 (1936) 154. Translation: The transport equation in the case of Coulomb interactions, D. ter Haar (Ed.), Collected papers of L. D. Landau, Pergamon Press, Oxford (1981), 163-170. [12] 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. [13] N. Lerner, Y. Morimoto, K. 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. [14] 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. [15] F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces, Pseudo-Differential Operators, Theory and Applications, 4, Birkhäuser Verlag, Basel, 2010. doi: 10.1007/978-3-7643-8512-5. [16] J. Toft, A. Khrennikov, B. Nilsson and S. Nordebo, Decompositions of Gelfand-Shilov kernels into kernels of similar class, J. Math. Anal. Appl., 396 (2012), 315-322. doi: 10.1016/j.jmaa.2012.06.025. [17] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983. doi: 10.1142/S0218202598000433. [18] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.

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##### References:
 [1] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95. doi: 10.1016/S0294-1449(03)00030-1. [2] A. A. Arsen'ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478. [3] H. Chen, W.-X. Li and C.-J. Xu, Gevrey regularity for solution of the spatially homogeneous Landau equation, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 673-686. doi: 10.1016/S0252-9602(09)60063-1. [4] H. Chen, W.-X. Li 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. [5] C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [6] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., 2 (1992), 167-182. doi: 10.1142/S0218202592000119. [7] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923. [8] 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. [9] I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 2. Spaces of fundamental and generalized functions. Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. [10] T. Gramchev, S. Pilipović and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces, in New Developments in Pseudo-Differential Operators, Oper. Theory Adv. Appl. Birkhäuser, Basel, 189 (2009), 15-31. doi: 10.1007/978-3-7643-8969-7_2. [11] L. D. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjet., 10 (1936) 154. Translation: The transport equation in the case of Coulomb interactions, D. ter Haar (Ed.), Collected papers of L. D. Landau, Pergamon Press, Oxford (1981), 163-170. [12] 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. [13] N. Lerner, Y. Morimoto, K. 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. [14] 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. [15] F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces, Pseudo-Differential Operators, Theory and Applications, 4, Birkhäuser Verlag, Basel, 2010. doi: 10.1007/978-3-7643-8512-5. [16] J. Toft, A. Khrennikov, B. Nilsson and S. Nordebo, Decompositions of Gelfand-Shilov kernels into kernels of similar class, J. Math. Anal. Appl., 396 (2012), 315-322. doi: 10.1016/j.jmaa.2012.06.025. [17] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983. doi: 10.1142/S0218202598000433. [18] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.
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