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 & 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.  doi: 10.1016/S0294-1449(03)00030-1.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/S0252-9602(09)60063-1.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.08.006.  Google Scholar

[5]

C. Cercignani, The Boltzmann equation and its applications,, Applied Mathematical Sciences, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[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.  doi: 10.1142/S0218202592000119.  Google Scholar

[7]

L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing,, Transport Theory Statist. Phys., 21 (1992), 259.  doi: 10.1080/00411459208203923.  Google Scholar

[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.  doi: 10.1080/03605300008821512.  Google Scholar

[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, (1968).   Google Scholar

[10]

T. Gramchev, S. Pilipović and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces,, in New Developments in Pseudo-Differential Operators, 189 (2009), 15.  doi: 10.1007/978-3-7643-8969-7_2.  Google Scholar

[11]

L. D. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung,, Phys. Z. Sowjet., 10 (1936), 163.   Google Scholar

[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.  doi: 10.3934/krm.2013.6.625.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.10.001.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.01.028.  Google Scholar

[15]

F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces,, Pseudo-Differential Operators, 4 (2010).  doi: 10.1007/978-3-7643-8512-5.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2012.06.025.  Google Scholar

[17]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules,, Math. Models Methods Appl. Sci., 8 (1998), 957.  doi: 10.1142/S0218202598000433.  Google Scholar

[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.  doi: 10.1007/s002050050106.  Google Scholar

show all references

References:
[1]

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

[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.   Google Scholar

[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.  doi: 10.1016/S0252-9602(09)60063-1.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.08.006.  Google Scholar

[5]

C. Cercignani, The Boltzmann equation and its applications,, Applied Mathematical Sciences, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[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.  doi: 10.1142/S0218202592000119.  Google Scholar

[7]

L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing,, Transport Theory Statist. Phys., 21 (1992), 259.  doi: 10.1080/00411459208203923.  Google Scholar

[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.  doi: 10.1080/03605300008821512.  Google Scholar

[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, (1968).   Google Scholar

[10]

T. Gramchev, S. Pilipović and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces,, in New Developments in Pseudo-Differential Operators, 189 (2009), 15.  doi: 10.1007/978-3-7643-8969-7_2.  Google Scholar

[11]

L. D. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung,, Phys. Z. Sowjet., 10 (1936), 163.   Google Scholar

[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.  doi: 10.3934/krm.2013.6.625.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.10.001.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.01.028.  Google Scholar

[15]

F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces,, Pseudo-Differential Operators, 4 (2010).  doi: 10.1007/978-3-7643-8512-5.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2012.06.025.  Google Scholar

[17]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules,, Math. Models Methods Appl. Sci., 8 (1998), 957.  doi: 10.1142/S0218202598000433.  Google Scholar

[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.  doi: 10.1007/s002050050106.  Google Scholar

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