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September  2013, 6(3): 625-648. doi: 10.3934/krm.2013.6.625

Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators

1. 

Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris VI), 4 Place Jussieu, 75252 Paris cedex 05, France

2. 

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

3. 

Université de Cergy-Pontoise, CNRS UMR 8088, Mathématiques, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

4. 

Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray

Received  November 2012 Revised  February 2013 Published  May 2013

We prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
Citation: Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinetic & Related Models, 2013, 6 (3) : 625-648. doi: 10.3934/krm.2013.6.625
References:
[1]

R. Alexandre, A review of Boltzmann equation with singular kernels,, Kinet. Relat. Models, 2 (2009), 551. doi: 10.3934/krm.2009.2.551. Google Scholar

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Ration. Mech. Anal., 152 (2000), 327. doi: 10.1007/s002050000083. Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation,, Arch. Ration. Mech. Anal., 198 (2010), 39. doi: 10.1007/s00205-010-0290-1. Google Scholar

[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. doi: 10.1007/s00220-011-1242-9. Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-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. doi: 10.1016/j.jfa.2011.10.007. Google Scholar

[6]

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

[7]

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

[8]

A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, in, 7 (1988), 111. Google Scholar

[9]

C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique I,", Hermann, (1992). Google Scholar

[10]

C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique II,", Hermann, (1992). Google Scholar

[11]

C. Cercignani, "Mathematical Methods in Kinetic Theory,", Plenum Press, (1969). Google Scholar

[12]

C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (1988). doi: 10.1007/978-1-4612-1039-9. Google Scholar

[13]

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

[14]

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

[15]

L. Desvillettes, About the regularization properties of the non-cut-off Kac equation,, Comm. Math. Phys., 168 (1995), 417. doi: 10.1007/BF02101556. Google Scholar

[16]

E. Dolera, On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules,, Boll. Unione Mat. Ital. (9), 4 (2011), 47. Google Scholar

[17]

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. doi: 10.1090/S0894-0347-2011-00697-8. Google Scholar

[18]

P.-T. Gressman and R.-M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production,, Adv. in Math., 227 (2011), 2349. doi: 10.1016/j.aim.2011.05.005. Google Scholar

[19]

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

[20]

N. N. Lebedev, "Special Functions and their Applications,", Revised edition, (1972). Google Scholar

[21]

N. Lerner, Y. Morimoto and K. Pravda-Starov, Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff,, Comm. Part. Diff. Equat., 37 (2012), 234. doi: 10.1080/03605302.2011.625462. Google Scholar

[22]

N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Spectral and phase space analysis of the linearized non-cutoff Kac collision operator,, to appear in J. Math. Pures Appl., (2013). doi: 10.1016/j.matpur.2013.03.005. Google Scholar

[23]

Y. Morimoto and C.-J. Xu, Hypoellipticity for a class of kinetic equations,, J. Math. Kyoto Univ., 47 (2007), 129. Google Scholar

[24]

Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations,, J. Diff. Equations, 247 (2009), 596. doi: 10.1016/j.jde.2009.01.028. Google Scholar

[25]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Part. Diff. Equat., 31 (2006), 1321. doi: 10.1080/03605300600635004. Google Scholar

[26]

C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff,, J. Math. Pures Appl. (9), 87 (2007), 515. doi: 10.1016/j.matpur.2007.03.003. Google Scholar

[27]

Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I,, Comm. Pure Appl. Math., 27 (1974), 407. doi: 10.1002/cpa.3160270402. Google Scholar

[28]

Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II,, Comm. Pure Appl. Math., 27 (1974), 559. doi: 10.1002/cpa.3160270402. Google Scholar

[29]

G. Szegö, "Orthogonal Polynomials,", Fourth edition, (1975). Google Scholar

[30]

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

[31]

C. Villani, "Contribution à l'Étude Mathématique des Collisions en Théorie Cinétique,", Habilitation dissertation, (2000). Google Scholar

[32]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

[33]

C. S. Wang Chang and G. E. Uhlenbeck, "On the Propagation of Sound in Monoatomic Gases,", Univ. of Michigan Press, (1970). Google Scholar

[34]

B. Wennberg, Regularity in the Boltzmann equation and the Radon transform,, Comm. Part. Diff. Equat., 19 (1994), 2057. doi: 10.1080/03605309408821082. Google Scholar

show all references

References:
[1]

R. Alexandre, A review of Boltzmann equation with singular kernels,, Kinet. Relat. Models, 2 (2009), 551. doi: 10.3934/krm.2009.2.551. Google Scholar

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Ration. Mech. Anal., 152 (2000), 327. doi: 10.1007/s002050000083. Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation,, Arch. Ration. Mech. Anal., 198 (2010), 39. doi: 10.1007/s00205-010-0290-1. Google Scholar

[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. doi: 10.1007/s00220-011-1242-9. Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-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. doi: 10.1016/j.jfa.2011.10.007. Google Scholar

[6]

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

[7]

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

[8]

A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, in, 7 (1988), 111. Google Scholar

[9]

C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique I,", Hermann, (1992). Google Scholar

[10]

C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique II,", Hermann, (1992). Google Scholar

[11]

C. Cercignani, "Mathematical Methods in Kinetic Theory,", Plenum Press, (1969). Google Scholar

[12]

C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (1988). doi: 10.1007/978-1-4612-1039-9. Google Scholar

[13]

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

[14]

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

[15]

L. Desvillettes, About the regularization properties of the non-cut-off Kac equation,, Comm. Math. Phys., 168 (1995), 417. doi: 10.1007/BF02101556. Google Scholar

[16]

E. Dolera, On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules,, Boll. Unione Mat. Ital. (9), 4 (2011), 47. Google Scholar

[17]

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. doi: 10.1090/S0894-0347-2011-00697-8. Google Scholar

[18]

P.-T. Gressman and R.-M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production,, Adv. in Math., 227 (2011), 2349. doi: 10.1016/j.aim.2011.05.005. Google Scholar

[19]

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

[20]

N. N. Lebedev, "Special Functions and their Applications,", Revised edition, (1972). Google Scholar

[21]

N. Lerner, Y. Morimoto and K. Pravda-Starov, Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff,, Comm. Part. Diff. Equat., 37 (2012), 234. doi: 10.1080/03605302.2011.625462. Google Scholar

[22]

N. Lerner, Y. Morimoto, K. Pravda-Starov and C.-J. Xu, Spectral and phase space analysis of the linearized non-cutoff Kac collision operator,, to appear in J. Math. Pures Appl., (2013). doi: 10.1016/j.matpur.2013.03.005. Google Scholar

[23]

Y. Morimoto and C.-J. Xu, Hypoellipticity for a class of kinetic equations,, J. Math. Kyoto Univ., 47 (2007), 129. Google Scholar

[24]

Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations,, J. Diff. Equations, 247 (2009), 596. doi: 10.1016/j.jde.2009.01.028. Google Scholar

[25]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Part. Diff. Equat., 31 (2006), 1321. doi: 10.1080/03605300600635004. Google Scholar

[26]

C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff,, J. Math. Pures Appl. (9), 87 (2007), 515. doi: 10.1016/j.matpur.2007.03.003. Google Scholar

[27]

Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I,, Comm. Pure Appl. Math., 27 (1974), 407. doi: 10.1002/cpa.3160270402. Google Scholar

[28]

Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II,, Comm. Pure Appl. Math., 27 (1974), 559. doi: 10.1002/cpa.3160270402. Google Scholar

[29]

G. Szegö, "Orthogonal Polynomials,", Fourth edition, (1975). Google Scholar

[30]

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

[31]

C. Villani, "Contribution à l'Étude Mathématique des Collisions en Théorie Cinétique,", Habilitation dissertation, (2000). Google Scholar

[32]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

[33]

C. S. Wang Chang and G. E. Uhlenbeck, "On the Propagation of Sound in Monoatomic Gases,", Univ. of Michigan Press, (1970). Google Scholar

[34]

B. Wennberg, Regularity in the Boltzmann equation and the Radon transform,, Comm. Part. Diff. Equat., 19 (1994), 2057. doi: 10.1080/03605309408821082. Google Scholar

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