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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 and 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), 203-230. doi: 10.1007/BF01197880.

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.

[3]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012.

[4]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95.

[5]

R. Alonso, J. Cañizo, I. Gamba and C. Mouhot, Exponential moments for the spatially homogeneous Boltzmann equation,, Work in progress., (). 

[6]

L. Arkeryd, R. Esposito and M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data, Comm. Math. Phys., 111 (1987), 393-407. doi: 10.1007/BF01238905.

[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), 435-446.

[8]

D. Bakry and M. Emery, Diffusions hypercontractives, Sém. Proba. XIX, Lecture Notes in Math., 1123 (1985), 177-206. doi: 10.1007/BFb0075847.

[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), 819-841.

[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), 280-310.

[11]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, In "Mathematical Physics Reviews," Vol. 7 of "Soviet Sci. Rev. Sect. C Math. Phys. Rev.," Harwood Academic Publ., Chur, 1988, pp. 111-233.

[12]

A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Stat. Phys., 88 (1997), 1183-1214. doi: 10.1007/BF02732431.

[13]

A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Stat. Phys., 94 (1999), 603-618. doi: 10.1023/A:1004537522686.

[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), 1651-1682. doi: 10.1023/B:JOSS.0000041751.11664.ea.

[15]

L. Boltzmann, Weitere studien uber das wärme gleichgenicht unfer gasmoläkuler, Sitzungsberichte der Akademie der Wissenschaften, 66 (1872), 265-370. Translation: Further studies on the thermal equilibrium of gas molecules, in Kinetic Theory 2, 88-174, Ed. S.G. Brush, Pergamon, Oxford (1966).

[16]

R. E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95.

[17]

T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz," Almqvist & Wiksell, 1957.

[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), 575-608. doi: 10.1007/BF01049721.

[19]

E. A. Carlen and M. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Stat. Phys., 74 (1994), 743-782. doi: 10.1007/BF02188578.

[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), 370-397. doi: 10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0.

[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), 362-387. doi: 10.1016/j.jfa.2004.06.011.

[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), 85-122. doi: 10.3934/krm.2010.3.85.

[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), 1-54. doi: 10.1007/BF02392695.

[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), 681-736. doi: 10.1007/s10955-009-9741-1.

[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), 373-378. doi: 10.1080/00411459708020293.

[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), 521-546. doi: 10.1007/s002200050511.

[27]

E. A. Carlen and X. Lu, Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums, J. Stat. Phys., 112 (2003), 59-134. doi: 10.1023/A:1023623503092.

[28]

C. Cercignani, "Theory and Application of the Boltzmann Equation," Elsevier, New York, 1975.

[29]

C. Cercignani, $H$-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech. (Arch. Mech. Stos.), 34 (1982), 231-241 (1983).

[30]

I. Csiszar, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung., 2 (1967), 299-318.

[31]

P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Rational Mech. Anal., 138 (1997), 137-167. doi: 10.1007/s002050050038.

[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), 167-182. doi: 10.1142/S0218202592000119.

[33]

L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations, Comm. Math. Phys., 123 (1989), 687-702. doi: 10.1007/BF01218592.

[34]

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.

[35]

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal., 123 (1993), 387-404. doi: 10.1007/BF00375586.

[36]

L. Desvillettes and C. Mouhot, About Cercignani's conjecture,, Work in progress., (). 

[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), 179-259.

[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), 261-298.

[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), 1-42. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[40]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.

[41]

P. Diaconis and L. Saloff-Coste, Bounds for Kac's master equation, Comm. Math. Phys., 209 (2000), 729-755. doi: 10.1007/s002200050036.

[42]

I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282. doi: 10.1007/s00205-009-0250-9.

[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), 583-586.

[44]

H. Grad, Asymptotic theory of the Boltzmann equation. II, In "Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I," Academic Press, New York, 1963, pp. 26-59.

[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), 5744-5749. doi: 10.1073/pnas.1001185107.

[46]

L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form, Duke Math. J., 42 (1975), 383-396. doi: 10.1215/S0012-7094-75-04237-4.

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M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, preprint arXiv:1006.5523 (2010).

[48]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.

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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), 703-746. doi: 10.1007/s10955-006-9097-8.

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C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.

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C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672. doi: 10.1007/s00220-005-1455-x.

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C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications, Comm. Math. Sci. suppl., 1 (2007), 73-86.

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C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

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C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures,, To appear in {\it J. Math. Pures. Appl.}, (). 

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G. Toscani, Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation, Quart. Appl. Math., 57 (1999), 521-541.

[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), 667-706. doi: 10.1007/s002200050631.

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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), 1279-1309. doi: 10.1023/A:1018623930325.

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C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.

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C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off, Rev. Mat. Iberoamericana, 15 (1999), 335-352.

[73]

C. Villani, "Contribution à L'étude Mathématique des Collisions en Théorie Cinétique (HDR)," PhD thesis, Univ. Paris Dauphine, France, 2000.

[74]

C. Villani, A review of mathematical topics in collisional kinetic theory, In "Handbook of Mathematical Fluid Dynamics, Vol. I," North-Holland, Amsterdam, 2002, pp. 71-305. doi: 10.1016/S1874-5792(02)80004-0.

[75]

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B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86 (1997), 1053-1066. doi: 10.1007/BF02183613.

show all references

References:
[1]

M. Aizenman and T. Bak, Convergence to equilibrium in a system of reacting polymers, Comm. Math. Phys., 65 (1979), 203-230. doi: 10.1007/BF01197880.

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.

[3]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012.

[4]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95.

[5]

R. Alonso, J. Cañizo, I. Gamba and C. Mouhot, Exponential moments for the spatially homogeneous Boltzmann equation,, Work in progress., (). 

[6]

L. Arkeryd, R. Esposito and M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data, Comm. Math. Phys., 111 (1987), 393-407. doi: 10.1007/BF01238905.

[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), 435-446.

[8]

D. Bakry and M. Emery, Diffusions hypercontractives, Sém. Proba. XIX, Lecture Notes in Math., 1123 (1985), 177-206. doi: 10.1007/BFb0075847.

[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), 819-841.

[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), 280-310.

[11]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, In "Mathematical Physics Reviews," Vol. 7 of "Soviet Sci. Rev. Sect. C Math. Phys. Rev.," Harwood Academic Publ., Chur, 1988, pp. 111-233.

[12]

A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Stat. Phys., 88 (1997), 1183-1214. doi: 10.1007/BF02732431.

[13]

A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Stat. Phys., 94 (1999), 603-618. doi: 10.1023/A:1004537522686.

[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), 1651-1682. doi: 10.1023/B:JOSS.0000041751.11664.ea.

[15]

L. Boltzmann, Weitere studien uber das wärme gleichgenicht unfer gasmoläkuler, Sitzungsberichte der Akademie der Wissenschaften, 66 (1872), 265-370. Translation: Further studies on the thermal equilibrium of gas molecules, in Kinetic Theory 2, 88-174, Ed. S.G. Brush, Pergamon, Oxford (1966).

[16]

R. E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95.

[17]

T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz," Almqvist & Wiksell, 1957.

[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), 575-608. doi: 10.1007/BF01049721.

[19]

E. A. Carlen and M. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Stat. Phys., 74 (1994), 743-782. doi: 10.1007/BF02188578.

[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), 370-397. doi: 10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0.

[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), 362-387. doi: 10.1016/j.jfa.2004.06.011.

[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), 85-122. doi: 10.3934/krm.2010.3.85.

[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), 1-54. doi: 10.1007/BF02392695.

[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), 681-736. doi: 10.1007/s10955-009-9741-1.

[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), 373-378. doi: 10.1080/00411459708020293.

[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), 521-546. doi: 10.1007/s002200050511.

[27]

E. A. Carlen and X. Lu, Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums, J. Stat. Phys., 112 (2003), 59-134. doi: 10.1023/A:1023623503092.

[28]

C. Cercignani, "Theory and Application of the Boltzmann Equation," Elsevier, New York, 1975.

[29]

C. Cercignani, $H$-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech. (Arch. Mech. Stos.), 34 (1982), 231-241 (1983).

[30]

I. Csiszar, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung., 2 (1967), 299-318.

[31]

P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Rational Mech. Anal., 138 (1997), 137-167. doi: 10.1007/s002050050038.

[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), 167-182. doi: 10.1142/S0218202592000119.

[33]

L. Desvillettes, Entropy dissipation rate and convergence in kinetic equations, Comm. Math. Phys., 123 (1989), 687-702. doi: 10.1007/BF01218592.

[34]

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.

[35]

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal., 123 (1993), 387-404. doi: 10.1007/BF00375586.

[36]

L. Desvillettes and C. Mouhot, About Cercignani's conjecture,, Work in progress., (). 

[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), 179-259.

[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), 261-298.

[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), 1-42. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[40]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.

[41]

P. Diaconis and L. Saloff-Coste, Bounds for Kac's master equation, Comm. Math. Phys., 209 (2000), 729-755. doi: 10.1007/s002200050036.

[42]

I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282. doi: 10.1007/s00205-009-0250-9.

[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), 583-586.

[44]

H. Grad, Asymptotic theory of the Boltzmann equation. II, In "Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I," Academic Press, New York, 1963, pp. 26-59.

[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), 5744-5749. doi: 10.1073/pnas.1001185107.

[46]

L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form, Duke Math. J., 42 (1975), 383-396. doi: 10.1215/S0012-7094-75-04237-4.

[47]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, preprint arXiv:1006.5523 (2010).

[48]

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