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Global classical solutions of the Boltzmann equation without angular cut-off


Authors: Philip T. Gressman and Robert M. Strain
Journal: J. Amer. Math. Soc. 24 (2011), 771-847
MSC (2010): Primary 35Q20, 35R11, 76P05, 82C40, 35H20, 35B65, 26A33
DOI: https://doi.org/10.1090/S0894-0347-2011-00697-8
Published electronically: March 18, 2011
MathSciNet review: 2784329
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Abstract: This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $ r^{-(p-1)}$ with $ p>2$, for initial perturbations of the Maxwellian equilibrium states, as announced in an earlier paper by the authors. We more generally cover collision kernels with parameters $ s\in (0,1)$ and $ \gamma$ satisfying $ \gamma > -n$ in arbitrary dimensions $ \mathbb{T}^n \times \mathbb{R}^n$ with $ n\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $ H$-theorem. When $ \gamma \ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $ \gamma <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $ \gamma \ge -2s$, as conjectured by Mouhot and Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.


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  • [1] Radjesvarane Alexandre, Une définition des solutions renormalisées pour l'équation de Boltzmann sans troncature angulaire, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 11, 987-991. MR 1696193
  • [2] Radjesvarane Alexandre, Around $ 3D$ Boltzmann non linear operator without angular cutoff, a new formulation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 3, 575-590. MR 1763526 (2002a:82052)
  • [3] Radjesvarane Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys. 104 (2001), no. 1-2, 327-358. MR 1851391 (2002i:82077)
  • [4] Radjesvarane Alexandre and Mouhamad El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci. 15 (2005), no. 6, 907-920. MR 2149928 (2006c:82027)
  • [5] Radjesvarane Alexandre, Integral estimates for a linear singular operator linked with the Boltzmann operator. I. Small singularities $ 0<\nu<1$, Indiana Univ. Math. J. 55 (2006), no. 6, 1975-2021. MR 2284553 (2008j:35023)
  • [6] Radjesvarane Alexandre, A Review of Boltzmann Equation with Singular Kernels, Kinetic and Related Models 2 (December 2009), no. 4, 551-646. MR 2556715
  • [7] R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000), no. 4, 327-355. MR 1765272 (2001c:82061)
  • [8] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math. 55 (2002), no. 1, 30-70. MR 1857879 (2002f:82026)
  • [9] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, Uncertainty principle and kinetic equations, J. Funct. Anal. 255 (2008), no. 8, 2013-2066. MR 2462585 (2010b:35330)
  • [10] Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 39-123, 10.1007/s00205-010-0290-1. MR 2679369
  • [11] Radjesvarane Alexandre, Y. Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, preprint (Dec. 27, 2009), available at arXiv:0912.1426v2.
  • [12] Radjesvarane Alexandre, Y. Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, The Boltzmann equation without angular cutoff in the whole space: I. An essential coercivity estimate, preprint (May 28, 2010), available at arXiv:1005.0447v2.
  • [13] Radjesvarane Alexandre, Y. Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Boltzmann equation without angular cutoff in the whole space: II. Global existence for soft potential, preprint (July 2, 2010), available at arXiv:1007.0304v1.
  • [14] Ricardo J. Alonso and Irene M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys. 137 (2009), no. 5-6, 1147-1165, 10.1007/s10955-009-9873-3. MR 2570766
  • [15] Ricardo J. Alonso, Emanuel Carneiro, and Irene M. Gamba, Convolution inequalities for the Boltzmann collision operator, Comm. Math. Phys. 298 (2010), 293-322, available at arXiv:0902.0507v2. MR 2669437
  • [16] Leif Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), no. 1, 11-21. MR 630119 (83k:76057)
  • [17] Leif Arkeryd, Asymptotic behaviour of the Boltzmann equation with infinite range forces, Comm. Math. Phys. 86 (1982), no. 4, 475-484. MR 679196 (85d:76020)
  • [18] Laurent Bernis and Laurent Desvillettes, Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation, Discrete Contin. Dyn. Syst. 24 (2009), no. 1, 13-33. MR 2476678 (2009k:82102)
  • [19] A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical Physics Reviews, Vol. 7 Soviet Sci. Rev. Sect. C Math. Phys. Rev., vol. 7, Harwood Academic Publ., Chur, 1988, pp. 111-233. MR 1128328 (92m:82112)
  • [20] Ludwig Boltzmann, Lectures on gas theory, Translated by Stephen G. Brush, University of California Press, Berkeley, 1964. Reprint of the 1896-1898 Edition. MR 0158708 (28:1931)
  • [21] Laurent Boudin and Laurent Desvillettes, On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math. 131 (2000), no. 2, 91-108. MR 1798557 (2001m:35014)
  • [22] Torsten Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1933), no. 1, 91-146 (French). MR 1555365
  • [23] Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028 (95i:82082)
  • [24] Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620 (96g:82046)
  • [25] Yemin Chen, Laurent Desvillettes, and Lingbing He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 21-55. MR 2506070 (2010m:35062)
  • [26] Yemin Chen and Lingbing He, Smoothing effect for Boltzmann equation with full-range interactions; Part I and Part II, arXiv preprint (July 22, 2010), available at arXiv:1007.3892.
  • [27] Laurent Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys. 168 (1995), no. 2, 417-440. MR 1324404 (96d:82052)
  • [28] Laurent Desvillettes, Regularization for the non-cutoff $ 2$D radially symmetric Boltzmann equation with a velocity dependent cross section, Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), 1996, pp. 383-394. MR 1407542 (97j:82126)
  • [29] Laurent Desvillettes, Regularization properties of the $ 2$-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Transport Theory Statist. Phys. 26 (1997), no. 3, 341-357. MR 1475459 (98h:82058)
  • [30] Laurent Desvillettes, About the use of the Fourier transform for the Boltzmann equation, Riv. Mat. Univ. Parma (7) 2* (2003), 1-99. In: Summer School on ``Methods and Models of Kinetic Theory'' (M&MKT 2002). MR 2052786 (2005a:35040)
  • [31] L. Desvillettes and F. Golse, On a model Boltzmann equation without angular cutoff, Differential Integral Equations 13 (2000), no. 4-6, 567-594. MR 1750040 (2001g:76051)
  • [32] Laurent Desvillettes and Clément Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal. 193 (2009), no. 2, 227-253. MR 2525118 (2010h:35288)
  • [33] Laurent Desvillettes and Bernt Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations 29 (2004), no. 1-2, 133-155. MR 2038147 (2004k:82086)
  • [34] Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations 25 (2000), no. 1-2, 179-259. MR 1737547 (2001c:82065)
  • [35] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2005), no. 2, 245-316. MR 2116276 (2005j:82070)
  • [36] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321-366. MR 1014927 (90k:82045)
  • [37] Renjun Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $ L\sp 2\sb {\xi}(H\sp N\sb x)$, J. Differential Equations 244 (2008), no. 12, 3204-3234. MR 2420519 (2009g:82051)
  • [38] Renjun Duan, Meng-Rong Li, and Tong Yang, Propagation of singularities in the solutions to the Boltzmann equation near equilibrium, Math. Models Methods Appl. Sci. 18 (2008), no. 7, 1093-1114. MR 2435186 (2009g:76132)
  • [39] Renjun Duan and Robert M. Strain, Optimal Time Decay of the Vlasov-Poisson-Boltzmann System in $ {\mathbb{R}}^3$, Arch. Rational Mech. Anal. 199 (2011), no. 1, 291-328, 10.1007/s00205-010-0318-6, available at arXiv:0912.1742.
  • [40] Renjun Duan and Robert M. Strain, Optimal Large-Time Behavior of the Vlasov-Maxwell-Boltzmann System in the Whole Space, preprint (2010), available at arXiv:1006.3605v1.
  • [41] Nicolas Fournier and Hélène Guérin, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys. 131 (2008), no. 4, 749-781. MR 2398952 (2009c:35027)
  • [42] Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129-206. MR 707957 (85f:35001)
  • [43] 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. MR 2533928
  • [44] Robert T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR 1379589 (97i:82070)
  • [45] T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions, J. Statist. Phys. 89 (1997), no. 3-4, 751-776. MR 1484062 (98k:82152)
  • [46] Harold Grad, Asymptotic theory of the Boltzmann equation. II, Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I Academic Press, New York, 1963, pp. 26-59. MR 0156656 (27:6577)
  • [47] Philip T. Gressman and Robert M. Strain, Global Strong Solutions of the Boltzmann Equation without Angular Cut-off (Dec. 4, 2009), 55, available at arXiv:0912.0888v1.
  • [48] Philip T. Gressman and Robert M. Strain, Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions, Proc. Nat. Acad. Sci. U. S. A. 107 (March 30, 2010), no. 13, 5744-5749, available at doi: 10.1073/pnas.1001185107. MR 2629879
  • [49] Philip T. Gressman and Robert M. Strain, Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft-Potentials (Feb. 15, 2010), 51, available at arXiv:1002.3639v1.
  • [50] Philip T. Gressman and Robert M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, submitted (July 8, 2010), 29pp., available at arXiv:1007.1276v1.
  • [51] Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391-434. MR 1946444 (2004c:82121)
  • [52] Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal. 169 (2003), no. 4, 305-353. MR 2013332 (2004i:82054)
  • [53] Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math. 153 (2003), no. 3, 593-630. MR 2000470 (2004m:82123)
  • [54] Yan Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53 (2004), no. 4, 1081-1094. MR 2095473 (2005g:35028)
  • [55] Reinhard Illner and Marvin Shinbrot, The Boltzmann equation: global existence for a rare gas in an infinite vacuum, Comm. Math. Phys. 95 (1984), no. 2, 217-226. MR 760333 (86a:82019)
  • [56] Shmuel Kaniel and Marvin Shinbrot, The Boltzmann equation. I. Uniqueness and local existence, Comm. Math. Phys. 58 (1978), no. 1, 65-84. MR 0475532 (57:15133)
  • [57] Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math. 7 (1990), no. 2, 301-320. MR 1057534 (91i:82020)
  • [58] Juhi Jang, Vlasov-Maxwell-Boltzmann diffusive limit, Arch. Ration. Mech. Anal. 184 (2009), no. 2, 531-584. MR 2563638
  • [59] S. Klainerman and I. Rodnianski, A geometric approach to the Littlewood-Paley theory, Geom. Funct. Anal. 16 (2006), no. 1, 126-163. MR 2221254 (2007e:58046)
  • [60] P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 (1994), no. 1679, 191-204. MR 1278244 (95d:82050)
  • [61] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, III, J. Math. Kyoto Univ. 34 (1994), no. 2 & 3, 391-427, 429-461, 539-584. MR 1284432 (96f:35133); MR 1295942 (96f:35134)
  • [62] Pierre-Louis Lions, Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 1, 37-41. MR 1649477 (99h:82062)
  • [63] Tai-Ping Liu, Tong Yang, and Shih-Hsien Yu, Energy method for Boltzmann equation, Phys. D 188 (2004), no. 3-4, 178-192. MR 2043729 (2005a:82091)
  • [64] J. Clerk Maxwell, On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society of London 157 (1867), 49-88.
  • [65] Dietrich Morgenstern, General existence and uniqueness proof for spatially homogeneous solutions of the Maxwell-Boltzmann equation in the case of Maxwellian molecules, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 719-721. MR 0063956 (16:205c)
  • [66] Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst. 24 (2009), no. 1, 187-212. MR 2476686 (2010h:35035)
  • [67] Clément Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1321-1348. MR 2254617 (2007h:35020)
  • [68] Clément Mouhot and Robert M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl. (9) 87 (2007), no. 5, 515-535, available at arXiv:math.AP/0607495. MR 2322149 (2008g:82116)
  • [69] Camil Muscalu, Jill Pipher, Terence Tao, and Christoph Thiele, Multi-parameter paraproducts, Revista Mathemática Iberoamericana 22 (2006), no. 3, 963-976. MR 2320408 (2008b:42037)
  • [70] Young Ping Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, II, Comm. Pure Appl. Math. 27 (1974), 407-428; ibid. 27 (1974), 559-581. MR 0636407 (58:30519)
  • [71] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [72] Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J., 1970. MR 0252961 (40:6176)
  • [73] Robert M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys. 268 (2006), no. 2, 543-567. MR 2259206 (2008b:82085)
  • [74] Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, preprint (2010), available at arXiv:1011.5561v2.
  • [75] Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417-429. MR 2209761 (2006m:82042)
  • [76] Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287-339. MR 2366140 (2008m:82008)
  • [77] Seiji Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad. 50 (1974), 179-184. MR 0363332 (50:15770)
  • [78] Seiji Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math. 1 (1984), no. 1, 141-156. MR 839310 (87j:45025)
  • [79] Seiji Ukai, Solutions of the Boltzmann equation, Patterns and waves Stud. Math. Appl., vol. 18, North-Holland, Amsterdam, 1986, pp. 37-96. MR 882376 (88g:35187)
  • [80] Cédric Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal. 143 (1998), no. 3, 273-307. MR 1650006 (99j:82065)
  • [81] Cédric Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off, Rev. Mat. Iberoamericana 15 (1999), no. 2, 335-352. MR 1715411 (2000f:82090)
  • [82] Cédric Villani, A review of mathematical topics in collisional kinetic theory North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, Vol. I, 2002, pp. 71-305. MR 1942465 (2003k:82087)
  • [83] Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. (2009), no. 202, iv+141, available at arXiv:math/0609050v1.
  • [84] C. S. Wang Chang, G. E. Uhlenbeck, and J. de Boer, On the Propagation of Sound in Monatomic Gases Univ. of Michigan Press, Ann Arbor, Michigan, 1952, pp. 1-56, available at http://deepblue.lib.umich.edu/.

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Additional Information

Philip T. Gressman
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: gressman@math.upenn.edu

Robert M. Strain
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: strain@math.upenn.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00697-8
Keywords: Kinetic theory, Boltzmann equation, long-range interaction, non-cut-off, soft potentials, hard potentials, fractional derivatives, anisotropy, harmonic analysis.
Received by editor(s): February 15, 2010
Received by editor(s) in revised form: July 22, 2010, and January 21, 2011
Published electronically: March 18, 2011
Additional Notes: The first author was partially supported by the NSF grant DMS-0850791 and an Alfred P. Sloan Foundation Research Fellowship.
The second author was partially supported by the NSF grant DMS-0901463 and an Alfred P. Sloan Foundation Research Fellowship.
Article copyright: © Copyright 2011 American Mathematical Society
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