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March  2011, 4(1): 277-294. doi: 10.3934/krm.2011.4.277

## Celebrating Cercignani's conjecture for the Boltzmann equation

 1 ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex 2 University of Cambridge, DPMMS, Wilberforce road, CB3 0WA, United Kingdom 3 Institut Henri Poincaré & Université Claude Bernard Lyon 1, 11 rue Pierre et Marie Curie 75230 Paris Cedex 05, France

Received  September 2010 Revised  October 2010 Published  January 2011

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.
Citation: Laurent Desvillettes, Clément Mouhot, Cédric Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. Kinetic & Related Models, 2011, 4 (1) : 277-294. doi: 10.3934/krm.2011.4.277
##### References:
 [1] M. Aizenman and T. Bak, Convergence to equilibrium in a system of reacting polymers,, Comm. Math. Phys., 65 (1979), 203. doi: 10.1007/BF01197880. Google Scholar [2] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327. doi: 10.1007/s002050000083. Google Scholar [3] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions,, Comm. Pure Appl. Math., 55 (2002), 30. doi: 10.1002/cpa.10012. Google Scholar [4] R. Alexandre and C. Villani, On the Landau approximation in plasma physics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61. Google Scholar [5] R. Alonso, J. Cañizo, I. Gamba and C. Mouhot, Exponential moments for the spatially homogeneous Boltzmann equation,, Work in progress., (). Google Scholar [6] L. Arkeryd, R. Esposito and M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data,, Comm. Math. Phys., 111 (1987), 393. doi: 10.1007/BF01238905. 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,, Mat. Sb., 181 (1990), 435. Google Scholar [8] D. Bakry and M. Emery, Diffusions hypercontractives,, Sém. Proba. XIX, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar [9] C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials,, Rev. Mat. Iberoamericana, 21 (2005), 819. Google Scholar [10] A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwell gas,, Teoret. Mat. Fiz., 60 (1984), 280. Google Scholar [11] A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, In, 7 (1988), 111. Google Scholar [12] A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems,, J. Stat. Phys., 88 (1997), 1183. doi: 10.1007/BF02732431. Google Scholar [13] A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation,, J. Stat. Phys., 94 (1999), 603. doi: 10.1023/A:1004537522686. Google Scholar [14] A. V. Bobylev, I. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions,, J. Statist. Phys., 116 (2004), 1651. doi: 10.1023/B:JOSS.0000041751.11664.ea. Google Scholar [15] L. Boltzmann, Weitere studien uber das wärme gleichgenicht unfer gasmoläkuler,, Sitzungsberichte der Akademie der Wissenschaften, 66 (1872), 265. Google Scholar [16] R. E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous,, Comm. Math. Phys., 74 (1980), 71. Google Scholar [17] T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz,", Almqvist & Wiksell, (1957). Google Scholar [18] E. A. Carlen and M. Carvalho, Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation,, J. Stat. Phys., 67 (1992), 575. doi: 10.1007/BF01049721. Google Scholar [19] E. A. Carlen and M. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels,, J. Stat. Phys., 74 (1994), 743. doi: 10.1007/BF02188578. Google Scholar [20] E. A. Carlen, M. C. Carvalho and E. Gabetta, Central limit theorem for Maxwellian molecules and truncation of the Wild expansion,, Comm. Pure Appl. Math., 53 (2000), 370. doi: 10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0. Google Scholar [21] E. A. Carlen, M. C. Carvalho and E. Gabetta, On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation,, J. Funct. Anal., 220 (2005), 362. doi: 10.1016/j.jfa.2004.06.011. Google Scholar [22] E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model,, Kinet. Relat. Models, 3 (2010), 85. doi: 10.3934/krm.2010.3.85. Google Scholar [23] E. A. Carlen, M. C. Carvalho and M. Loss, Determination of the spectral gap for Kac's master equation and related stochastic evolution,, Acta Math., 191 (2003), 1. doi: 10.1007/BF02392695. Google Scholar [24] E. A. Carlen, M. C. Carvalho and X. Lu, On strong convergence to equilibrium for the Boltzmann equation with soft potentials,, J. Stat. Phys., 135 (2009), 681. doi: 10.1007/s10955-009-9741-1. Google Scholar [25] E. A. Carlen, M. C. Carvalho and B. Wennberg, Entropic convergence for solutions of the Boltzmann equation with general physical initial data,, Transport Theory Statist. Phys., 26 (1997), 373. doi: 10.1080/00411459708020293. Google Scholar [26] E. A. Carlen, E. Gabetta and G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas,, Comm. Math. Phys., 199 (1999), 521. doi: 10.1007/s002200050511. Google Scholar [27] E. A. Carlen and X. Lu, Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums,, J. Stat. Phys., 112 (2003), 59. doi: 10.1023/A:1023623503092. Google Scholar [28] C. Cercignani, "Theory and Application of the Boltzmann Equation,", Elsevier, (1975). Google Scholar [29] C. Cercignani, $H$-theorem and trend to equilibrium in the kinetic theory of gases,, Arch. Mech. (Arch. Mech. Stos.), 34 (1982), 231. Google Scholar [30] I. Csiszar, Information-type measures of difference of probability distributions and indirect observations,, Stud. Sci. Math. Hung., 2 (1967), 299. Google Scholar [31] P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation,, Arch. Rational Mech. Anal., 138 (1997), 137. doi: 10.1007/s002050050038. Google Scholar [32] 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 [33] L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations,, Comm. Math. Phys., 123 (1989), 687. doi: 10.1007/BF01218592. Google Scholar [34] 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 [35] L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations,, Arch. Rational Mech. Anal., 123 (1993), 387. doi: 10.1007/BF00375586. Google Scholar [36] L. Desvillettes and C. Mouhot, About Cercignani's conjecture,, Work in progress., (). Google Scholar [37] 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. Google Scholar [38] L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications,, Comm. Partial Differential Equations, 25 (2000), 261. Google Scholar [39] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation,, Comm. Pure Appl. Math., 54 (2001), 1. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. Google Scholar [40] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation,, Invent. Math., 159 (2005), 245. doi: 10.1007/s00222-004-0389-9. Google Scholar [41] P. Diaconis and L. Saloff-Coste, Bounds for Kac's master equation,, Comm. Math. Phys., 209 (2000), 729. doi: 10.1007/s002200050036. Google Scholar [42] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation,, Arch. Ration. Mech. Anal., 194 (2009), 253. doi: 10.1007/s00205-009-0250-9. Google Scholar [43] F. Golse and F. Poupaud, Un résultat de compacité pour l'équation de Boltzmann avec potentiel mou. Application au problème de demi-espace,, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 583. Google Scholar [44] H. Grad, Asymptotic theory of the Boltzmann equation. II,, In, (1962), 26. Google Scholar [45] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions,, Proc. Natl. Acad. Sci. USA, 107 (2010), 5744. doi: 10.1073/pnas.1001185107. Google Scholar [46] L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form,, Duke Math. J., 42 (1975), 383. doi: 10.1215/S0012-7094-75-04237-4. Google Scholar [47] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, preprint, \arXiv{1006.5523} (2010)., (2010). Google Scholar [48] Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391. doi: 10.1007/s00220-002-0729-9. Google Scholar [49] F. Hérau and K. Pravda-Starov, Anisotropic hypoelliptic estimates for Landau-type operators, preprint, \arXiv{1003.3265} (2010)., (2010). Google Scholar [50] D. Hilbert, "Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen,", Math. Ann., 72 (1912). Google Scholar [51] E. Janvresse, Spectral gap for Kac's model of Boltzmann equation,, Ann. Probab., 29 (2001), 288. doi: 10.1214/aop/1008956330. Google Scholar [52] M. Kac, Foundations of kinetic theory,, In, (1956), 1954. Google Scholar [53] M. Klaus, Boltzmann collision operator without cut-off,, Helv. Phys. Acta, 50 (1977), 893. Google Scholar [54] S. Kullback, A lower bound for discrimination information in terms of variation,, IEEE Trans. Inf. The., 4 (1967), 126. doi: 10.1109/TIT.1967.1053968. Google Scholar [55] L. Landau, Die kinetische gleichung für den fall coulombscher wechselwirkung,, Phys. Z. Sowjet., 154 (1936), 163. Google Scholar [56] P.-L. Lions, Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 37. Google Scholar [57] D. K. Maslen, The eigenvalues of Kac's master equation,, Math. Z., 243 (2003), 291. doi: 10.1007/s00209-002-0466-y. Google Scholar [58] J. C. Maxwell, On the dynamical theory of gases,, Philos. Trans. Roy. Soc. London Ser. A, 157 (1867), 49. doi: 10.1098/rstl.1867.0004. Google Scholar [59] S. Mischler and C. Mouhot, Quantitative uniform in time chaos propagation for Boltzmann collision processes, preprint, \arXiv{1001.2994} (2010)., (2010). Google Scholar [60] S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior,, J. Stat. Phys., 124 (2006), 703. doi: 10.1007/s10955-006-9097-8. Google Scholar [61] C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Partial Differential Equations, 31 (2006), 1321. Google Scholar [62] C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials,, Comm. Math. Phys., 261 (2006), 629. doi: 10.1007/s00220-005-1455-x. Google Scholar [63] C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications,, Comm. Math. Sci. suppl., 1 (2007), 73. Google Scholar [64] C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus,, Nonlinearity, 19 (2006), 969. doi: 10.1088/0951-7715/19/4/011. Google Scholar [65] C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures,, To appear in {\it J. Math. Pures. Appl.}, (). Google Scholar [66] 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 [67] Y. P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I II,, Comm. Pure Appl. Math., 27 (1974), 407. Google Scholar [68] G. Toscani, Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation,, Quart. Appl. Math., 57 (1999), 521. Google Scholar [69] G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation,, Comm. Math. Phys., 203 (1999), 667. doi: 10.1007/s002200050631. Google Scholar [70] G. Toscani and C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds,, J. Stat. Phys., 98 (2000), 1279. doi: 10.1023/A:1018623930325. Google Scholar [71] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273. doi: 10.1007/s002050050106. Google Scholar [72] C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off,, Rev. Mat. Iberoamericana, 15 (1999), 335. Google Scholar [73] C. Villani, "Contribution à L'étude Mathématique des Collisions en Théorie Cinétique (HDR),", PhD thesis, (2000). Google Scholar [74] C. Villani, A review of mathematical topics in collisional kinetic theory,, In, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar [75] C. Villani, Cercignani's conjecture is sometimes true and always almost true,, Comm. Math. 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show all references

##### References:
 [1] M. Aizenman and T. Bak, Convergence to equilibrium in a system of reacting polymers,, Comm. Math. Phys., 65 (1979), 203. doi: 10.1007/BF01197880. Google Scholar [2] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327. doi: 10.1007/s002050000083. Google Scholar [3] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions,, Comm. Pure Appl. Math., 55 (2002), 30. doi: 10.1002/cpa.10012. Google Scholar [4] R. Alexandre and C. Villani, On the Landau approximation in plasma physics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61. Google Scholar [5] R. Alonso, J. Cañizo, I. Gamba and C. Mouhot, Exponential moments for the spatially homogeneous Boltzmann equation,, Work in progress., (). Google Scholar [6] L. Arkeryd, R. Esposito and M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data,, Comm. Math. Phys., 111 (1987), 393. doi: 10.1007/BF01238905. 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,, Mat. Sb., 181 (1990), 435. Google Scholar [8] D. Bakry and M. Emery, Diffusions hypercontractives,, Sém. Proba. XIX, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar [9] C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials,, Rev. Mat. Iberoamericana, 21 (2005), 819. Google Scholar [10] A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwell gas,, Teoret. Mat. Fiz., 60 (1984), 280. Google Scholar [11] A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules,, In, 7 (1988), 111. Google Scholar [12] A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems,, J. Stat. Phys., 88 (1997), 1183. doi: 10.1007/BF02732431. Google Scholar [13] A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation,, J. Stat. Phys., 94 (1999), 603. doi: 10.1023/A:1004537522686. Google Scholar [14] A. V. Bobylev, I. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions,, J. Statist. Phys., 116 (2004), 1651. doi: 10.1023/B:JOSS.0000041751.11664.ea. Google Scholar [15] L. Boltzmann, Weitere studien uber das wärme gleichgenicht unfer gasmoläkuler,, Sitzungsberichte der Akademie der Wissenschaften, 66 (1872), 265. Google Scholar [16] R. E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous,, Comm. Math. Phys., 74 (1980), 71. Google Scholar [17] T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz,", Almqvist & Wiksell, (1957). Google Scholar [18] E. A. Carlen and M. Carvalho, Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation,, J. Stat. Phys., 67 (1992), 575. doi: 10.1007/BF01049721. Google Scholar [19] E. A. Carlen and M. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels,, J. Stat. Phys., 74 (1994), 743. doi: 10.1007/BF02188578. Google Scholar [20] E. A. Carlen, M. C. Carvalho and E. Gabetta, Central limit theorem for Maxwellian molecules and truncation of the Wild expansion,, Comm. Pure Appl. Math., 53 (2000), 370. doi: 10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0. Google Scholar [21] E. A. Carlen, M. C. Carvalho and E. Gabetta, On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation,, J. Funct. Anal., 220 (2005), 362. doi: 10.1016/j.jfa.2004.06.011. Google Scholar [22] E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model,, Kinet. Relat. Models, 3 (2010), 85. doi: 10.3934/krm.2010.3.85. Google Scholar [23] E. A. Carlen, M. C. Carvalho and M. Loss, Determination of the spectral gap for Kac's master equation and related stochastic evolution,, Acta Math., 191 (2003), 1. doi: 10.1007/BF02392695. Google Scholar [24] E. A. Carlen, M. C. Carvalho and X. Lu, On strong convergence to equilibrium for the Boltzmann equation with soft potentials,, J. Stat. Phys., 135 (2009), 681. doi: 10.1007/s10955-009-9741-1. Google Scholar [25] E. A. Carlen, M. C. Carvalho and B. Wennberg, Entropic convergence for solutions of the Boltzmann equation with general physical initial data,, Transport Theory Statist. Phys., 26 (1997), 373. doi: 10.1080/00411459708020293. Google Scholar [26] E. A. Carlen, E. Gabetta and G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas,, Comm. Math. Phys., 199 (1999), 521. doi: 10.1007/s002200050511. Google Scholar [27] E. A. Carlen and X. Lu, Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums,, J. Stat. Phys., 112 (2003), 59. doi: 10.1023/A:1023623503092. Google Scholar [28] C. Cercignani, "Theory and Application of the Boltzmann Equation,", Elsevier, (1975). Google Scholar [29] C. Cercignani, $H$-theorem and trend to equilibrium in the kinetic theory of gases,, Arch. Mech. (Arch. Mech. Stos.), 34 (1982), 231. Google Scholar [30] I. Csiszar, Information-type measures of difference of probability distributions and indirect observations,, Stud. Sci. Math. Hung., 2 (1967), 299. Google Scholar [31] P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation,, Arch. Rational Mech. Anal., 138 (1997), 137. doi: 10.1007/s002050050038. Google Scholar [32] 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 [33] L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations,, Comm. Math. Phys., 123 (1989), 687. doi: 10.1007/BF01218592. Google Scholar [34] 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 [35] L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations,, Arch. Rational Mech. Anal., 123 (1993), 387. doi: 10.1007/BF00375586. Google Scholar [36] L. Desvillettes and C. Mouhot, About Cercignani's conjecture,, Work in progress., (). Google Scholar [37] 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. Google Scholar [38] L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications,, Comm. Partial Differential Equations, 25 (2000), 261. Google Scholar [39] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation,, Comm. Pure Appl. Math., 54 (2001), 1. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. Google Scholar [40] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation,, Invent. Math., 159 (2005), 245. doi: 10.1007/s00222-004-0389-9. Google Scholar [41] P. Diaconis and L. Saloff-Coste, Bounds for Kac's master equation,, Comm. Math. Phys., 209 (2000), 729. doi: 10.1007/s002200050036. Google Scholar [42] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation,, Arch. Ration. Mech. Anal., 194 (2009), 253. doi: 10.1007/s00205-009-0250-9. Google Scholar [43] F. Golse and F. Poupaud, Un résultat de compacité pour l'équation de Boltzmann avec potentiel mou. Application au problème de demi-espace,, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 583. Google Scholar [44] H. Grad, Asymptotic theory of the Boltzmann equation. II,, In, (1962), 26. Google Scholar [45] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions,, Proc. Natl. Acad. Sci. USA, 107 (2010), 5744. doi: 10.1073/pnas.1001185107. Google Scholar [46] L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form,, Duke Math. J., 42 (1975), 383. doi: 10.1215/S0012-7094-75-04237-4. Google Scholar [47] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, preprint, \arXiv{1006.5523} (2010)., (2010). Google Scholar [48] Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391. doi: 10.1007/s00220-002-0729-9. Google Scholar [49] F. Hérau and K. Pravda-Starov, Anisotropic hypoelliptic estimates for Landau-type operators, preprint, \arXiv{1003.3265} (2010)., (2010). Google Scholar [50] D. Hilbert, "Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen,", Math. Ann., 72 (1912). Google Scholar [51] E. Janvresse, Spectral gap for Kac's model of Boltzmann equation,, Ann. Probab., 29 (2001), 288. doi: 10.1214/aop/1008956330. Google Scholar [52] M. Kac, Foundations of kinetic theory,, In, (1956), 1954. Google Scholar [53] M. Klaus, Boltzmann collision operator without cut-off,, Helv. Phys. Acta, 50 (1977), 893. Google Scholar [54] S. 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