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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 and 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), 551-646. doi: 10.3934/krm.2009.2.551. [2] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083. [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), 39-123. doi: 10.1007/s00205-010-0290-1. [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, 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), 915-1010. doi: 10.1016/j.jfa.2011.10.007. [6] 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. [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), 465-478. [8] A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in "Mathematical Physics Reviews," Vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, (1988), 111-233. [9] C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique I," Hermann, Éditeurs des Sciences et des Arts, 1992. [10] C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique II," Hermann, Éditeurs des Sciences et des Arts, 1992. [11] C. Cercignani, "Mathematical Methods in Kinetic Theory," Plenum Press, New York, 1969. [12] C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [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), 167-182. doi: 10.1142/S0218202592000119. [14] 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. [15] L. Desvillettes, About the regularization properties of the non-cut-off Kac equation, Comm. Math. Phys., 168 (1995), 417-440. doi: 10.1007/BF02101556. [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), 47-68. [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), 771-847. doi: 10.1090/S0894-0347-2011-00697-8. [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), 2349-2384. doi: 10.1016/j.aim.2011.05.005. [19] L. D. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjet. 10 (1936), 154; translation in "The Transport Equation in the Case of Coulomb Interactions" (ed. D. ter Haar), Collected papers of L. D. Landau, Pergamon Press, Oxford, (1981), 163-170. [20] N. N. Lebedev, "Special Functions and their Applications," Revised edition, translated from the Russian and edited by Richard A. Silverman, Dover Publications, Inc., New York, 1972. [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), 234-284. doi: 10.1080/03605302.2011.625462. [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. [23] Y. Morimoto and C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto Univ., 47 (2007), 129-152. [24] Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Diff. Equations, 247 (2009), 596-617. doi: 10.1016/j.jde.2009.01.028. [25] C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Part. Diff. Equat., 31 (2006), 1321-1348. doi: 10.1080/03605300600635004. [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), 515-535. doi: 10.1016/j.matpur.2007.03.003. [27] Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math., 27 (1974), 407-428. doi: 10.1002/cpa.3160270402. [28] Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math., 27 (1974), 559-581. doi: 10.1002/cpa.3160270402. [29] G. Szegö, "Orthogonal Polynomials," Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. [30] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106. [31] C. Villani, "Contribution à l'Étude Mathématique des Collisions en Théorie Cinétique," Habilitation dissertation, Université Paris-Dauphine, France, 2000. [32] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics," Vol. I, North-Holland, Amsterdam, (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0. [33] C. S. Wang Chang and G. E. Uhlenbeck, "On the Propagation of Sound in Monoatomic Gases," Univ. of Michigan Press, Ann Arbor, Michigan; Reprinted in 1970 in "Studies in Statistical Mechanics," Vol. V (eds. J. L. Lebowitz and E. Montroll), North-Holland, 1970. [34] B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Part. Diff. Equat., 19 (1994), 2057-2074. doi: 10.1080/03605309408821082.

show all references

##### References:
 [1] R. Alexandre, A review of Boltzmann equation with singular kernels, Kinet. Relat. Models, 2 (2009), 551-646. doi: 10.3934/krm.2009.2.551. [2] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083. [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), 39-123. doi: 10.1007/s00205-010-0290-1. [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, 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), 915-1010. doi: 10.1016/j.jfa.2011.10.007. [6] 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. [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), 465-478. [8] A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in "Mathematical Physics Reviews," Vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, (1988), 111-233. [9] C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique I," Hermann, Éditeurs des Sciences et des Arts, 1992. [10] C. Cohen-Tannoudji, B. Diu and F. Laloë, "Mécanique Quantique II," Hermann, Éditeurs des Sciences et des Arts, 1992. [11] C. Cercignani, "Mathematical Methods in Kinetic Theory," Plenum Press, New York, 1969. [12] C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [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), 167-182. doi: 10.1142/S0218202592000119. [14] 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. [15] L. Desvillettes, About the regularization properties of the non-cut-off Kac equation, Comm. Math. Phys., 168 (1995), 417-440. doi: 10.1007/BF02101556. [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), 47-68. [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), 771-847. doi: 10.1090/S0894-0347-2011-00697-8. [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), 2349-2384. doi: 10.1016/j.aim.2011.05.005. [19] L. D. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjet. 10 (1936), 154; translation in "The Transport Equation in the Case of Coulomb Interactions" (ed. D. ter Haar), Collected papers of L. D. Landau, Pergamon Press, Oxford, (1981), 163-170. [20] N. N. Lebedev, "Special Functions and their Applications," Revised edition, translated from the Russian and edited by Richard A. Silverman, Dover Publications, Inc., New York, 1972. [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), 234-284. doi: 10.1080/03605302.2011.625462. [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. [23] Y. Morimoto and C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto Univ., 47 (2007), 129-152. [24] Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Diff. Equations, 247 (2009), 596-617. doi: 10.1016/j.jde.2009.01.028. [25] C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Part. Diff. Equat., 31 (2006), 1321-1348. doi: 10.1080/03605300600635004. [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), 515-535. doi: 10.1016/j.matpur.2007.03.003. [27] Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math., 27 (1974), 407-428. doi: 10.1002/cpa.3160270402. [28] Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math., 27 (1974), 559-581. doi: 10.1002/cpa.3160270402. [29] G. Szegö, "Orthogonal Polynomials," Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. [30] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106. [31] C. Villani, "Contribution à l'Étude Mathématique des Collisions en Théorie Cinétique," Habilitation dissertation, Université Paris-Dauphine, France, 2000. [32] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics," Vol. I, North-Holland, Amsterdam, (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0. [33] C. S. Wang Chang and G. E. Uhlenbeck, "On the Propagation of Sound in Monoatomic Gases," Univ. of Michigan Press, Ann Arbor, Michigan; Reprinted in 1970 in "Studies in Statistical Mechanics," Vol. V (eds. J. L. Lebowitz and E. Montroll), North-Holland, 1970. [34] B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Part. Diff. Equat., 19 (1994), 2057-2074. doi: 10.1080/03605309408821082.
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