Skip to main content
SpringerLink
Log in

Hypocoercivity for perturbation theory and perturbation of hypocoercivity for confined Boltzmann-type collisional equations

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

The main concept presented in this review is the hypocoercivity arising in several kinetic equations that are related to the Boltzmann equation. Let us emphasize that it only deals with collisional kernels which feature hard potential and angular cutoff, and when the space variable lives in a confined domain—either bounded or periodic. The review aims at:

  • Giving an overview of the use of hypocoercive properties for Boltzmann-type linear equations and their applications to nonlinear perturbative settings;

  • Describing the most recent techniques developed to recover the lack of coercivity of linear kinetic operators;

  • Showing the robustness of such methods with respect to coefficients and their applications to hydrodynamical limit;

  • Extending the previous points to what we shall see as a perturbation theory for hypocoercivity.

After giving a general presentation of different collisional models in kinetic theory—classical mono-species Boltzmann equation, Boltzmann system for multi-species mixtures and Boltzmann equation with a force—in Sect. 1, we motivate the four points above in Sect. 2 which reviews Cauchy theories and situates hypocoercivity works in the litterature. The linearized operators which are at the heart of the present review are investigated in Sect. 3: both the results which have been obtained and information they taught us in recent models. Then Sect. 4 explains hypocoercivity in natural spaces of linearization while Sect. 5 deals with other functional spaces. Lastly, Sect. 6 shows the robustness of hypocoercivity under different types of perturbation out of equilibrium.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Addala, L., Dolbeault, J., Li, X., Tayeb, M.L.: \(l^2\)-hypocoercivity and large time asymptotics of the linearized Vlasov–Poisson–Fokker–Planck system. J. Stat. Phys. 184(1) (2021). https://doi.org/10.1007/s10955-021-02784-4

  2. Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152(4), 327–355 (2000). https://doi.org/10.1007/s002050000083

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexandre, R., Hérau, F., Li, W.X.: Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff. J. Math. Pures Appl. 9(126), 1–71 (2019). https://doi.org/10.1016/j.matpur.2019.04.013

    Article  MathSciNet  MATH  Google Scholar 

  4. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.J., Yang, T.: Global existence and full regularity of the Boltzmann equation without angular cutoff. Commun. Math. Phys. 304(2), 513–581 (2011). https://doi.org/10.1007/s00220-011-1242-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55(1), 30–70 (2002). https://doi.org/10.1002/cpa.10012

    Article  MathSciNet  MATH  Google Scholar 

  6. Arkeryd, L.: Existence theorems for certain kinetic equations and large data. Arch. Rational Mech. Anal. 103(2), 139–149 (1988). https://doi.org/10.1007/BF00251505

    Article  MathSciNet  MATH  Google Scholar 

  7. Arkeryd, L.: Stability in \(L^1\) for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103(2), 151–167 (1988). https://doi.org/10.1007/BF00251506

    Article  MathSciNet  MATH  Google Scholar 

  8. Arkeryd, L.: On the strong \(L^1\) trend to equilibrium for the Boltzmann equation. Stud. Appl. Math. 87(3), 283–288 (1992). https://doi.org/10.1002/sapm1992873283

    Article  MathSciNet  MATH  Google Scholar 

  9. Arkeryd, L.: Some examples of NSA methods in kinetic theory. In: Nonequilibrium problems in many-particle systems (Montecatini, 1992), Lecture Notes in Math., vol. 1551, pp. 14–57. Springer, Berlin (1993). https://doi.org/10.1007/BFb0090928

  10. Arkeryd, L.: A quantum Boltzmann equation for Haldane statistics and hard forces; the space-homogeneous initial value problem. Commun. Math. Phys. 298(2), 573–583 (2010). https://doi.org/10.1007/s00220-010-1046-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Arkeryd, L., Cercignani, C.: A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rational Mech. Anal. 125(3), 271–287 (1993). https://doi.org/10.1007/BF00383222

    Article  MathSciNet  MATH  Google Scholar 

  12. Arkeryd, L., Cercignani, C., Illner, R.: Measure solutions of the steady Boltzmann equation in a slab. Commun. Math. Phys. 142(2), 285–296 (1991). http://projecteuclid.org/euclid.cmp/1104248586

  13. Arkeryd, L., Heintz, A.: On the solvability and asymptotics of the Boltzmann equation in irregular domains. Commun. Partial Differ. Equ. 22(11–12), 2129–2152 (1997). https://doi.org/10.1080/03605309708821334

    Article  MathSciNet  MATH  Google Scholar 

  14. Arkeryd, L., Maslova, N.: On diffuse reflection at the boundary for the Boltzmann equation and related equations. J. Stat. Phys. 77(5–6), 1051–1077 (1994). https://doi.org/10.1007/BF02183152

    Article  MathSciNet  MATH  Google Scholar 

  15. Arkeryd, L., Nouri, A.: A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory. Indiana Univ. Math. J. 44(3), 815–839 (1995). https://doi.org/10.1512/iumj.1995.44.2010

    Article  MathSciNet  MATH  Google Scholar 

  16. Arkeryd, L., Nouri, A.: Boltzmann asymptotics with diffuse reflection boundary conditions. Monatsh. Math. 123(4), 285–298 (1997). https://doi.org/10.1007/BF01326764

    Article  MathSciNet  MATH  Google Scholar 

  17. Arkeryd, L., Nouri, A.: Well-posedness of the Cauchy problem for a space-dependent anyon Boltzmann equation. SIAM J. Math. Anal. 47(6), 4720–4742 (2015). https://doi.org/10.1137/15M1012335

    Article  MathSciNet  MATH  Google Scholar 

  18. Arkeryd, L., Nouri, A.: On the Cauchy problem with large data for a space-dependent Boltzmann–Nordheim boson equation. Commun. Math. Sci. 15(5), 1247–1264 (2017). https://doi.org/10.4310/CMS.2017.v15.n5.a4

    Article  MathSciNet  MATH  Google Scholar 

  19. Arsénio, D.: From Boltzmann’s equation to the incompressible Navier–Stokes–Fourier system with long-range interactions. Arch. Ration. Mech. Anal. 206(2), 367–488 (2012). https://doi.org/10.1007/s00205-012-0557-9

    Article  MathSciNet  MATH  Google Scholar 

  20. Arsénio, D., Saint-Raymond, L.: From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. Vol. 1. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2019). https://doi.org/10.4171/193

  21. Asano, K.: Local solutions to the initial and initial-boundary value problem for the Boltzmann equation with an external force. I. J. Math. Kyoto Univ. 24(2), 225–238 (1984). https://doi.org/10.1215/kjm/1250521326

    Article  MathSciNet  MATH  Google Scholar 

  22. Asano, K., Shizuta, Y.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Jpn Acad. Ser. A Math. Sci. 53(1), 3–5 (1977). http://projecteuclid.org/euclid.pja/1195518147

  23. Asano, K., Ukai, S.: On the initial-boundary value problem of the linearized Boltzmann equation in an exterior domain. Proc. Jpn. Acad. Ser. A Math. Sci. 56(1), 12–17 (1980). http://projecteuclid.org/euclid.pja/1195517029

  24. Asano, K., Ukai, S.: Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I. Existence. Arch. Rational Mech. Anal. 84(3), 249–291 (1983). https://doi.org/10.1007/BF00281521

    Article  MathSciNet  MATH  Google Scholar 

  25. Asano, K., Ukai, S.: Steady solutions of the Boltzmann equation for a gas flow past an obstacle. II. Stability. Publ. Res. Inst. Math. Sci. 22(6), 1035–1062 (1986). https://doi.org/10.2977/prims/1195177061

    Article  MathSciNet  MATH  Google Scholar 

  26. Baranger, C., Bisi, M., Brull, S., Desvillettes, L.: On the Chapman–Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinet. Relat. Models 11(4), 821–858 (2018). https://doi.org/10.3934/krm.2018033

    Article  MathSciNet  MATH  Google Scholar 

  27. Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoamericana 21(3), 819–841 (2005). https://doi.org/10.4171/RMI/436

    Article  MathSciNet  MATH  Google Scholar 

  28. Bardos, C.: What use for the mathematical theory of the Navier-Stokes equations. In: Mathematical fluid mechanics. Adv. Math. Fluid Mech. pp. 1–25. Birkhäuser, Basel (2001)

  29. Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46(5), 667–753 (1993). https://doi.org/10.1002/cpa.3160460503

    Article  MathSciNet  MATH  Google Scholar 

  30. Bardos, C., Golse, F., Levermore, C.D.: Acoustic and Stokes limits for the Boltzmann equation. C. R. Acad. Sci. Paris Sér. I Math. 327(3), 323–328 (1998). https://doi.org/10.1016/S0764-4442(98)80154-7

    Article  MathSciNet  MATH  Google Scholar 

  31. Bardos, C., Golse, F., Levermore, C.D.: The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153(3), 177–204 (2000). https://doi.org/10.1007/s002050000080

    Article  MathSciNet  MATH  Google Scholar 

  32. Bardos, C., Golse, F., Levermore, D.: Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles. C. R. Acad. Sci. Paris Sér. I Math. 309(11), 727–732 (1989)

    MathSciNet  MATH  Google Scholar 

  33. Bardos, C., Golse, F., Levermore, D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63(1–2), 323–344 (1991). https://doi.org/10.1007/BF01026608

    Article  MathSciNet  MATH  Google Scholar 

  34. Bardos, C., Ukai, S.: The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models Methods Appl. Sci. 1(2), 235–257 (1991). https://doi.org/10.1142/S0218202591000137

    Article  MathSciNet  MATH  Google Scholar 

  35. Bastea, S., Esposito, R., Lebowitz, J.L., Marra, R.: Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations. J. Stat. Phys. 101(5–6), 1087–1136 (2000). https://doi.org/10.1023/A:1026481706240

    Article  MathSciNet  MATH  Google Scholar 

  36. Beals, R., Protopopescu, V.: Abstract time-dependent transport equations. J. Math. Anal. Appl. 121(2), 370–405 (1987). https://doi.org/10.1016/0022-247X(87)90252-6

    Article  MathSciNet  MATH  Google Scholar 

  37. Bianca, C., Dogbe, C.: Recovering Navier-Stokes equations from asymptotic limits of the Boltzmann gas mixture equation. Commun. Theor. Phys. (Beijing) 65(5), 553–562 (2016). https://doi.org/10.1088/0253-6102/65/5/553

    Article  MathSciNet  MATH  Google Scholar 

  38. Biryuk, A., Craig, W., Panferov, V.: Strong solutions of the Boltzmann equation in one spatial dimension. C. R. Math. Acad. Sci. Paris 342(11), 843–848 (2006). https://doi.org/10.1016/j.crma.2006.04.005

    Article  MathSciNet  MATH  Google Scholar 

  39. Bisi, M., Desvillettes, L.: Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases. ESAIM Math. Model. Numer. Anal. 48(4), 1171–1197 (2014). https://doi.org/10.1051/m2an/2013135

    Article  MathSciNet  MATH  Google Scholar 

  40. Bobylëv, A.V.: The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules. Dokl. Akad. Nauk SSSR 225(6), 1041–1044 (1975)

    MathSciNet  Google Scholar 

  41. Bobylëv, A.V.: 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., vol. 7, pp. 111–233. Harwood Academic Publ., Chur (1988)

  42. Bobylev, A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88(5–6), 1183–1214 (1997). https://doi.org/10.1007/BF02732431

    Article  MathSciNet  MATH  Google Scholar 

  43. Bobylev, A.V., Carrillo, J.A., Gamba, I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98(3–4), 743–773 (2000). https://doi.org/10.1023/A:1018627625800

    Article  MathSciNet  MATH  Google Scholar 

  44. Bobylev, A.V., Cercignani, C., Toscani, G.: Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Stat. Phys. 111(1–2), 403–417 (2003). https://doi.org/10.1023/A:1022273528296

    Article  MathSciNet  MATH  Google Scholar 

  45. Bobylev, A.V., Gamba, I.M.: Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails. J. Stat. Phys. 124(2–4), 497–516 (2006). https://doi.org/10.1007/s10955-006-9044-8

    Article  MathSciNet  MATH  Google Scholar 

  46. Bobylev, A.V., Gamba, I.M., Panferov, V.A.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5–6), 1651–1682 (2004). https://doi.org/10.1023/B:JOSS.0000041751.11664.ea

    Article  MathSciNet  MATH  Google Scholar 

  47. Bodineau, T., Gallagher, I., Saint-Raymond, L., Simonella, S.: One-sided convergence in the Boltzmann-Grad limit. Ann. Fac. Sci. Toulouse Math. (6) 27(5), 985–1022 (2018). https://doi.org/10.5802/afst.1589

    Article  MathSciNet  MATH  Google Scholar 

  48. Bondesan, A., Boudin, L., Briant, M., Grec, B.: Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium Maxwell distributions. Commun. Pure Appl. Anal. 19(5), 2549–2573 (2020). https://doi.org/10.3934/cpaa.2020112

    Article  MathSciNet  MATH  Google Scholar 

  49. Bondesan, A., Briant, M.: Perturbative cauchy theory for a flux-incompressible maxwell–stefan system (2019). arXiv:1910.03279

  50. Bondesan, A., Briant, M.: Stability of the Maxwell–Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation. Commun. Math. Phys. 382(1), 381–440 (2021). https://doi.org/10.1007/s00220-021-03976-5

    Article  MathSciNet  MATH  Google Scholar 

  51. Bose, C., Grzegorczyk, P., Illner, R.: Asymptotic behavior of one-dimensional discrete-velocity models in a slab. Arch. Rational Mech. Anal. 127(4), 337–360 (1994). https://doi.org/10.1007/BF00375020

    Article  MathSciNet  MATH  Google Scholar 

  52. Bothe, D.: On the Maxwell-Stefan approach to multicomponent diffusion. In: Parabolic problems, Progr. Nonlinear Differential Equations Appl., vol. 80, pp. 81–93. Birkhäuser/Springer Basel AG, Basel (2011). https://doi.org/10.1007/978-3-0348-0075-4_5

  53. Bouchut, F., Desvillettes, L.: A proof of the smoothing properties of the positive part of Boltzmann’s kernel. Rev. Mat. Iberoamericana 14(1), 47–61 (1998). https://doi.org/10.4171/RMI/233

    Article  MathSciNet  MATH  Google Scholar 

  54. Bouchut, F., Desvillettes, L.: Averaging lemmas without time Fourier transform and application to discretized kinetic equations. Proc. R. Soc. Edinburgh Sect. A 129(1), 19–36 (1999). https://doi.org/10.1017/S030821050002744X

    Article  MathSciNet  MATH  Google Scholar 

  55. Boudin, L., Grec, B., Pavan, V.: The Maxwell–Stefan diffusion limit for a kinetic model of mixtures with general cross sections. Nonlinear Anal. 159, 40–61 (2017). https://doi.org/10.1016/j.na.2017.01.010

    Article  MathSciNet  MATH  Google Scholar 

  56. Boudin, L., Grec, B., Salvarani, F.: A mathematical and numerical analysis of the Maxwell–Stefan diffusion equations. Discrete Contin. Dyn. Syst. Ser. B 17(5), 1427–1440 (2012). https://doi.org/10.3934/dcdsb.2012.17.1427

    Article  MathSciNet  MATH  Google Scholar 

  57. Boudin, L., Grec, B., Salvarani, F.: The Maxwell–Stefan diffusion limit for a kinetic model of mixtures. Acta Appl. Math. 136, 79–90 (2015). https://doi.org/10.1007/s10440-014-9886-z

    Article  MathSciNet  MATH  Google Scholar 

  58. Bouin, E., Dolbeault, J., Mischler, S., Mouhot, C., Schmeiser, C.: Hypocoercivity without confinement. Pure and Applied Analysis (2019). https://hal.archives-ouvertes.fr/hal-01575501

  59. Bourgat, J.F., Desvillettes, L., Le Tallec, P., Perthame, B.: Microreversible collisions for polyatomic gases and Boltzmann’s theorem. Eur. J. Mech. B Fluids 13(2), 237–254 (1994)

    MathSciNet  MATH  Google Scholar 

  60. Briant, M.: From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: a quantitative error estimate. J. Differ. Equ. 259(11), 6072–6141 (2015). https://doi.org/10.1016/j.jde.2015.07.022

    Article  MathSciNet  MATH  Google Scholar 

  61. Briant, M.: Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. Kinet. Relat. Models 8(2), 281–308 (2015). https://doi.org/10.3934/krm.2015.8.281

    Article  MathSciNet  MATH  Google Scholar 

  62. Briant, M.: Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains. Arch. Ration. Mech. Anal. 218(2), 985–1041 (2015). https://doi.org/10.1007/s00205-015-0874-x

    Article  MathSciNet  MATH  Google Scholar 

  63. Briant, M.: Stability of global equilibrium for the multi-species Boltzmann equation in \(L^\infty \) settings. Discrete Contin. Dyn. Syst. 36(12), 6669–6688 (2016). https://doi.org/10.3934/dcds.2016090

    Article  MathSciNet  MATH  Google Scholar 

  64. Briant, M.: Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinet. Relat. Models 10(2), 329–371 (2017). https://doi.org/10.3934/krm.2017014

    Article  MathSciNet  MATH  Google Scholar 

  65. Briant, M., Daus, E.S.: The Boltzmann equation for a multi-species mixture close to global equilibrium. Arch. Ration. Mech. Anal. 222(3), 1367–1443 (2016). https://doi.org/10.1007/s00205-016-1023-x

    Article  MathSciNet  MATH  Google Scholar 

  66. Briant, M., Debussche, A., Vovelle, J.: The boltzmann equation with an external force on the torus: Incompressible navier–stokes-fourier hydrodynamical limit (2019). Preprint arXiv:1906.02960

  67. Briant, M., Einav, A.: On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments. J. Stat. Phys. 163(5), 1108–1156 (2016). https://doi.org/10.1007/s10955-016-1517-9

    Article  MathSciNet  MATH  Google Scholar 

  68. Briant, M., Grec, B.: Rigorous derivation of the fick system from the multi-species boltzmann equation in the diffusive scaling (2019). Preprint arXiv:2003.07891

  69. Briant, M., Guo, Y.: Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions. J. Differ. Equ. 261(12), 7000–7079 (2016). https://doi.org/10.1016/j.jde.2016.09.014

    Article  MathSciNet  MATH  Google Scholar 

  70. Briant, M., Merino-Aceituno, S., Mouhot, C.: From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight. Anal. Appl. (Singap.) 17(1), 85–116 (2019). https://doi.org/10.1142/S021953051850015X

    Article  MathSciNet  MATH  Google Scholar 

  71. Brull, S.: Problem of evaporation-condensation for a two component gas in the slab. Kinet. Relat. Models 1(2), 185–221 (2008). https://doi.org/10.3934/krm.2008.1.185

    Article  MathSciNet  MATH  Google Scholar 

  72. Brull, S.: The stationary Boltzmann equation for a two-component gas in the slab with different molecular masses. Adv. Differ. Equ. 15(11–12), 1103–1124 (2010)

    MathSciNet  MATH  Google Scholar 

  73. Cañizo, J.A., Cao, C., Evans, J., Yoldaş, H.: Hypocoercivity of linear kinetic equations via Harris’s theorem. Kinet. Relat. Models 13(1), 97–128 (2020). https://doi.org/10.3934/krm.2020004

    Article  MathSciNet  MATH  Google Scholar 

  74. Caflisch, R.E.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math. 33(5), 651–666 (1980). https://doi.org/10.1002/cpa.3160330506

    Article  MathSciNet  MATH  Google Scholar 

  75. Cao, Y., Kim, C., Lee, D.: Global strong solutions of the Vlasov–Poisson–Boltzmann system in bounded domains. Arch. Ration. Mech. Anal. 233(3), 1027–1130 (2019). https://doi.org/10.1007/s00205-019-01374-9

    Article  MathSciNet  MATH  Google Scholar 

  76. Carleman, T.: Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60(1), 91–146 (1933). https://doi.org/10.1007/BF02398270

    Article  MathSciNet  MATH  Google Scholar 

  77. Carleman, T.: Problèmes mathématiques dans la théorie cinétique des gaz. Publ. Sci. Inst. Mittag-Leffler. 2. Almqvist & Wiksells Boktryckeri Ab, Uppsala (1957)

  78. Carlen, E.A., Carvalho, M.C.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys. 67(3–4), 575–608 (1992). https://doi.org/10.1007/BF01049721

    Article  MathSciNet  MATH  Google Scholar 

  79. Carlen, E.A., Carvalho, M.C.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74(3–4), 743–782 (1994). https://doi.org/10.1007/BF02188578

    Article  MathSciNet  MATH  Google Scholar 

  80. Carrapatoso, K.: Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials. Bull. Sci. Math. 139(7), 777–805 (2015). https://doi.org/10.1016/j.bulsci.2014.12.002

    Article  MathSciNet  MATH  Google Scholar 

  81. Carrapatoso, K., Tristani, I., Wu, K.C.: Cauchy problem and exponential stability for the inhomogeneous Landau equation. Arch. Ration. Mech. Anal. 221(1), 363–418 (2016). https://doi.org/10.1007/s00205-015-0963-x

    Article  MathSciNet  MATH  Google Scholar 

  82. Cercignani, C.: The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67. Springer, New York (1988)

    Book  MATH  Google Scholar 

  83. Cercignani, C.: On the initial-boundary value problem for the Boltzmann equation. Arch. Rational Mech. Anal. 116(4), 307–315 (1992). https://doi.org/10.1007/BF00375670

    Article  MathSciNet  MATH  Google Scholar 

  84. Cercignani, C.: Errata: Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett. 8(5), 95–99 (1995). https://doi.org/10.1016/0893-9659(95)00074-Z

  85. Cercignani, C.: Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett. 8(2), 53–59 (1995). https://doi.org/10.1016/0893-9659(95)00011-E

    Article  MathSciNet  MATH  Google Scholar 

  86. Cercignani, C.: Initial-boundary value problems for the Boltzmann equation. In: Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), 25, 425–436 (1996). https://doi.org/10.1080/00411459608220711

  87. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106. Springer, New York (1994)

    Book  MATH  Google Scholar 

  88. Chai, X.: The Boltzmann equation near Maxwellian in the whole space. Commun. Pure Appl. Anal. 10(2), 435–458 (2011). https://doi.org/10.3934/cpaa.2011.10.435

    Article  MathSciNet  MATH  Google Scholar 

  89. Chen, X., Jüngel, A.: Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system. Commun. Math. Phys. 340(2), 471–497 (2015). https://doi.org/10.1007/s00220-015-2472-z

    Article  MathSciNet  MATH  Google Scholar 

  90. Daus, E.S., Jüngel, A., Mouhot, C., Zamponi, N.: Hypocoercivity for a linearized multispecies Boltzmann system. SIAM J. Math. Anal. 48(1), 538–568 (2016). https://doi.org/10.1137/15M1017934

    Article  MathSciNet  MATH  Google Scholar 

  91. Daus, E.S., Jüngel, A., Tang, B.Q.: Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type. Arch. Ration. Mech. Anal. 235(2), 1059–1104 (2020). https://doi.org/10.1007/s00205-019-01439-9

    Article  MathSciNet  MATH  Google Scholar 

  92. De Masi, A., Esposito, R., Lebowitz, J.L.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Commun. Pure Appl. Math. 42(8), 1189–1214 (1989). https://doi.org/10.1002/cpa.3160420810

    Article  MathSciNet  MATH  Google Scholar 

  93. Denlinger, R.: The propagation of chaos for a rarefied gas of hard spheres in the whole space. Arch. Ration. Mech. Anal. 229(2), 885–952 (2018). https://doi.org/10.1007/s00205-018-1229-1

    Article  MathSciNet  MATH  Google Scholar 

  94. Desvillettes, L.: Une minoration du terme de dissipation d’entropie pour le modèle de Kac de la cinétique des gaz. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 307(19), 1955–1960 (1988)

    MathSciNet  MATH  Google Scholar 

  95. Desvillettes, L.: Entropy dissipation rate and convergence in kinetic equations. Comm. Math. Phys. 123(4), 687–702 (1989). http://projecteuclid.org/euclid.cmp/1104178990

  96. Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Rational Mech. Anal. 110(1), 73–91 (1990). https://doi.org/10.1007/BF00375163

    Article  MathSciNet  MATH  Google Scholar 

  97. Desvillettes, L., Monaco, R., Salvarani, F.: A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B. Fluids 24(2), 219–236 (2005). https://doi.org/10.1016/j.euromechflu.2004.07.004

    Article  MathSciNet  MATH  Google Scholar 

  98. Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54(1), 1–42 (2001). 10.1002/1097-0312(200101)54:1\(<\)1::AID-CPA1\(>\)3.0.CO;2-Q

  99. Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005). https://doi.org/10.1007/s00222-004-0389-9

    Article  MathSciNet  MATH  Google Scholar 

  100. DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. (2) 130(2), 321–366 (1989). https://doi.org/10.2307/1971423

    Article  MathSciNet  MATH  Google Scholar 

  101. DiPerna, R.J., Lions, P.L.: Global solutions of Boltzmann’s equation and the entropy inequality. Arch. Rational Mech. Anal. 114(1), 47–55 (1991). https://doi.org/10.1007/BF00375684

    Article  MathSciNet  MATH  Google Scholar 

  102. DiPerna, R.J., Lions, P.L., Meyer, Y.: \(L^p\) regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(3–4), 271–287 (1991). https://doi.org/10.1016/S0294-1449(16)30264-5

    Article  MathSciNet  MATH  Google Scholar 

  103. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Math. Acad. Sci. Paris 347(9–10), 511–516 (2009). https://doi.org/10.1016/j.crma.2009.02.025

    Article  MathSciNet  MATH  Google Scholar 

  104. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass. Trans. Am. Math. Soc. 367(6), 3807–3828 (2015). https://doi.org/10.1090/S0002-9947-2015-06012-7

    Article  MathSciNet  MATH  Google Scholar 

  105. Drange, H.B.: On the Boltzmann equation with external forces. SIAM J. Appl. Math. 34(3), 577–592 (1978). https://doi.org/10.1137/0134045

    Article  MathSciNet  MATH  Google Scholar 

  106. Duan, R.: Stability of the Boltzmann equation with potential forces on torus. Phys. D 238(17), 1808–1820 (2009). https://doi.org/10.1016/j.physd.2009.06.007

    Article  MathSciNet  MATH  Google Scholar 

  107. Duan, R., Huang, F., Wang, Y., Yang, T.: Global well-posedness of the Boltzmann equation with large amplitude initial data. Arch. Ration. Mech. Anal. 225(1), 375–424 (2017). https://doi.org/10.1007/s00205-017-1107-2

    Article  MathSciNet  MATH  Google Scholar 

  108. Duan, R., Liu, S., Yang, T., Zhao, H.: Stability of the nonrelativistic Vlasov–Maxwell–Boltzmann system for angular non-cutoff potentials. Kinet. Relat. Models 6(1), 159–204 (2013). https://doi.org/10.3934/krm.2013.6.159

    Article  MathSciNet  MATH  Google Scholar 

  109. Duan, R., Strain, R.M.: Optimal time decay of the Vlasov-Poisson-Boltzmann system in \(\mathbb{R}^3\). Arch. Ration. Mech. Anal. 199(1), 291–328 (2011). https://doi.org/10.1007/s00205-010-0318-6

    Article  MathSciNet  MATH  Google Scholar 

  110. Duan, R., Ukai, S., Yang, T., Zhao, H.: Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications. Commun. Math. Phys. 277(1), 189–236 (2008). https://doi.org/10.1007/s00220-007-0366-4

    Article  MathSciNet  MATH  Google Scholar 

  111. Duan, R., Yang, T., Zhao, H.: The Vlasov-Poisson-Boltzmann system in the whole space: the hard potential case. J. Differ. Equ. 252(12), 6356–6386 (2012). https://doi.org/10.1016/j.jde.2012.03.012

    Article  MathSciNet  MATH  Google Scholar 

  112. Duan, R., Yang, T., Zhao, H.: The Vlasov-Poisson-Boltzmann system for soft potentials. Math. Models Methods Appl. Sci. 23(6), 979–1028 (2013). https://doi.org/10.1142/S0218202513500012

    Article  MathSciNet  MATH  Google Scholar 

  113. Duan, R., Yang, T., Zhu, C.: Global existence to Boltzmann equation with external force in infinite vacuum. J. Math. Phys. 46(5), 053307, 13 (2005). https://doi.org/10.1063/1.1899985

  114. Dudyński, M.: Spectral properties of the linearized Boltzmann operator in \(L^p\) for \(1\le p\le \infty \). J. Stat. Phys. 153(6), 1084–1106 (2013). https://doi.org/10.1007/s10955-013-0873-y

    Article  MathSciNet  MATH  Google Scholar 

  115. Ellis, R.S., Pinsky, M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 9(54), 125–156 (1975)

    MathSciNet  MATH  Google Scholar 

  116. Escobedo, M., Velázquez, J.J.L.: On the blow up and condensation of supercritical solutions of the Nordheim equation for bosons. Commun. Math. Phys. 330(1), 331–365 (2014). https://doi.org/10.1007/s00220-014-2034-9

    Article  MathSciNet  MATH  Google Scholar 

  117. Escobedo, M., Velázquez, J.J.L.: Finite time blow-up and condensation for the bosonic Nordheim equation. Invent. Math. 200(3), 761–847 (2015). https://doi.org/10.1007/s00222-014-0539-7

    Article  MathSciNet  MATH  Google Scholar 

  118. Esposito, R., Guo, Y., Kim, C., Marra, R.: Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun. Math. Phys. 323(1), 177–239 (2013). https://doi.org/10.1007/s00220-013-1766-2

    Article  MathSciNet  MATH  Google Scholar 

  119. Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4(1), Art. 1, 119 (2018). https://doi.org/10.1007/s40818-017-0037-5

  120. Esposito, R., Guo, Y., Marra, R.: Phase transition in a Vlasov-Boltzmann binary mixture. Commun. Math. Phys. 296(1), 1–33 (2010). https://doi.org/10.1007/s00220-010-1009-8

    Article  MathSciNet  MATH  Google Scholar 

  121. Esposito, R., Guo, Y., Marra, R.: Hydrodynamic limit of a kinetic gas flow past an obstacle. Commun. Math. Phys. 364(2), 765–823 (2018). https://doi.org/10.1007/s00220-018-3173-1

    Article  MathSciNet  MATH  Google Scholar 

  122. Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)

  123. Fournier, N.: Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff. J. Stat. Phys. 125(4), 927–946 (2006). https://doi.org/10.1007/s10955-006-9208-6

    Article  MathSciNet  MATH  Google Scholar 

  124. Fournier, N., Guérin, H.: On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131(4), 749–781 (2008). https://doi.org/10.1007/s10955-008-9511-5

    Article  MathSciNet  MATH  Google Scholar 

  125. Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: hard spheres and short-range potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), PAGES = xii+137. Zürich (2013)

  126. Gallagher, I., Tristani, I.: On the convergence of smooth solutions from Boltzmann to Navier-Stokes. Ann. H. Lebesgue 3, 561–614 (2020). https://doi.org/10.5802/ahl.40

    Article  MathSciNet  MATH  Google Scholar 

  127. Gamba, I.M., Panferov, V., Villani, C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246(3), 503–541 (2004). https://doi.org/10.1007/s00220-004-1051-5

    Article  MathSciNet  MATH  Google Scholar 

  128. Gamba, I.M., Pavić-Čolić, M.: On existence and uniqueness to homogeneous Boltzmann flows of monatomic gas mixtures. Arch. Ration. Mech. Anal. 235(1), 723–781 (2020). https://doi.org/10.1007/s00205-019-01428-y

    Article  MathSciNet  MATH  Google Scholar 

  129. Gerasimenko, V.I., Gapyak, I.V.: Hard sphere dynamics and the Enskog equation. Kinet. Relat. Models 5(3), 459–484 (2012). https://doi.org/10.3934/krm.2012.5.459

    Article  MathSciNet  MATH  Google Scholar 

  130. Giovangigli, V.: Multicomponent flow modeling. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston, MA (1999). https://doi.org/10.1007/978-1-4612-1580-6

  131. Golse, F.: From kinetic to macroscopic models (1998). Lecture notes

  132. Golse, F.: Fluid dynamic limits of the kinetic theory of gases. In: From particle systems to partial differential equations, Springer Proc. Math. Stat., vol. 75, pp. 3–91. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54271-8_1

  133. Golse, F., Levermore, C.D.: Stokes–Fourier and acoustic limits for the Boltzmann equation: convergence proofs. Commun. Pure Appl. Math. 55(3), 336–393 (2002). https://doi.org/10.1002/cpa.3011

    Article  MathSciNet  MATH  Google Scholar 

  134. Golse, F., Saint-Raymond, L.: Velocity averaging in \(L^1\) for the transport equation. C. R. Math. Acad. Sci. Paris 334(7), 557–562 (2002). https://doi.org/10.1016/S1631-073X(02)02302-6

    Article  MathSciNet  MATH  Google Scholar 

  135. Golse, F., Saint-Raymond, L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004). https://doi.org/10.1007/s00222-003-0316-5

    Article  MathSciNet  MATH  Google Scholar 

  136. Golse, F., Saint-Raymond, L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. (9) 91(5), 508–552 (2009). https://doi.org/10.1016/j.matpur.2009.01.013

    Article  MathSciNet  MATH  Google Scholar 

  137. Goudon, T.: Generalized invariant sets for the Boltzmann equation. Math. Models Methods Appl. Sci. 7(4), 457–476 (1997). https://doi.org/10.1142/S0218202597000256

    Article  MathSciNet  MATH  Google Scholar 

  138. Goudon, T.: On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions. J. Stat. Phys. 89(3–4), 751–776 (1997). https://doi.org/10.1007/BF02765543

    Article  MathSciNet  MATH  Google Scholar 

  139. Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, pp. 205–294. Springer, Berlin (1958)

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

  141. Grad, H.: Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations. In: Proc. Sympos. Appl. Math., Vol. XVII, pp. 154–183. Amer. Math. Soc., Providence, R.I. (1965)

  142. Greenberg, W., van der Mee, C., Protopopescu, V.: Boundary value problems in abstract kinetic theory, Operator Theory: Advances and Applications, vol. 23. Birkhäuser Verlag, Basel (1987). https://doi.org/10.1007/978-3-0348-5478-8

  143. Gressman, P.T., Strain, R.M.: Global classical solutions of the Boltzmann equation without angular cut-off. J. Am. Math. Soc. 24(3), 771–847 (2011). https://doi.org/10.1090/S0894-0347-2011-00697-8

    Article  MathSciNet  MATH  Google Scholar 

  144. Gualdani, M.P., Mischler, S., Mouhot, C.: Factorization of non-symmetric operators and exponential \(H\)-theorem. Mém. Soc. Math. Fr. (N.S.) 153, 137 (2017)

    MathSciNet  MATH  Google Scholar 

  145. Guiraud, J.P.: An \(H\) theorem for a gas of rigid spheres in a bounded domain. In: Théories cinétiques classiques et relativistes (Colloq. Internat. Centre Nat. Recherche Sci., No. 236, Paris, 1974), pp. 29–58. CNRS (1975)

  146. Guo, Y.: The Vlasov–Poisson–Boltzmann system near vacuum. Commun. Math. Phys. 218(2), 293–313 (2001). https://doi.org/10.1007/s002200100391

    Article  MathSciNet  MATH  Google Scholar 

  147. Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434 (2002). https://doi.org/10.1007/s00220-002-0729-9

    Article  MathSciNet  MATH  Google Scholar 

  148. Guo, Y.: The Vlasov–Poisson–Boltzmann system near Maxwellians. Commun. Pure Appl. Math. 55(9), 1104–1135 (2002). https://doi.org/10.1002/cpa.10040

    Article  MathSciNet  MATH  Google Scholar 

  149. Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169(4), 305–353 (2003). https://doi.org/10.1007/s00205-003-0262-9

    Article  MathSciNet  MATH  Google Scholar 

  150. Guo, Y.: The Vlasov–Maxwell–Boltzmann system near Maxwellians. Invent. Math. 153(3), 593–630 (2003). https://doi.org/10.1007/s00222-003-0301-z

    Article  MathSciNet  MATH  Google Scholar 

  151. Guo, Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4), 1081–1094 (2004). https://doi.org/10.1512/iumj.2004.53.2574

    Article  MathSciNet  MATH  Google Scholar 

  152. Guo, Y.: Boltzmann diffusive limit beyond the Navier–Stokes approximation. Commun. Pure Appl. Math. 59(5), 626–687 (2006). https://doi.org/10.1002/cpa.20121

    Article  MathSciNet  MATH  Google Scholar 

  153. Guo, Y.: Erratum: “Boltzmann diffusive limit beyond the Navier–Stokes approximation” [Comm. Pure Appl. Math. 59 (2006), no. 5, 626–687; mr2172804]. Commun. Pure Appl. Math. 60(2), 291–293 (2007). https://doi.org/10.1002/cpa.20171

  154. Guo, Y.: Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 713–809 (2010). https://doi.org/10.1007/s00205-009-0285-y

    Article  MathSciNet  MATH  Google Scholar 

  155. Guo, Y., Jang, J., Jiang, N.: Acoustic limit for the Boltzmann equation in optimal scaling. Commun. Pure Appl. Math. 63(3), 337–361 (2010). https://doi.org/10.1002/cpa.20308

    Article  MathSciNet  MATH  Google Scholar 

  156. Guo, Y., Kim, C., Tonon, D., Trescases, A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Ration. Mech. Anal. 220(3), 1045–1093 (2016). https://doi.org/10.1007/s00205-015-0948-9

    Article  MathSciNet  MATH  Google Scholar 

  157. Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains. Invent. Math. 207(1), 115–290 (2017). https://doi.org/10.1007/s00222-016-0670-8

    Article  MathSciNet  MATH  Google Scholar 

  158. Hamdache, K.: Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions. Arch. Rational Mech. Anal. 119(4), 309–353 (1992). https://doi.org/10.1007/BF01837113

    Article  MathSciNet  MATH  Google Scholar 

  159. Henderson, C., Snelson, S., Tarfulea, A.: Self-generating lower bounds and continuation for the Boltzmann equation. Calc. Var. Partial Differ. Equ. 59(6), Paper No. 191, 13 (2020). https://doi.org/10.1007/s00526-020-01856-9

  160. Hérau, F.: Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46(3–4), 349–359 (2006)

    MathSciNet  MATH  Google Scholar 

  161. Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171(2), 151–218 (2004). https://doi.org/10.1007/s00205-003-0276-3

    Article  MathSciNet  MATH  Google Scholar 

  162. Hérau, F., Tonon, D., Tristani, I.: Short time diffusion properties of inhomogeneous kinetic equations with fractional collision kernel (2018). https://hal.archives-ouvertes.fr/hal-01596009. Working paper or preprint

  163. Hérau, F., Tonon, D., Tristani, I.: Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off. Commun. Math. Phys. 377(1), 697–771 (2020). https://doi.org/10.1007/s00220-020-03682-8

    Article  MathSciNet  MATH  Google Scholar 

  164. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967). https://doi.org/10.1007/BF02392081

    Article  MathSciNet  MATH  Google Scholar 

  165. Hutridurga, H., Salvarani, F.: On the Maxwell–Stefan diffusion limit for a mixture of monatomic gases. Math. Methods Appl. Sci. 40(3), 803–813 (2017). https://doi.org/10.1002/mma.4013

    Article  MathSciNet  MATH  Google Scholar 

  166. Illner, R., Pulvirenti, M.: A derivation of the BBGKY-hierarchy for hard sphere particle systems. Transp. Theory Stat. Phys. 16(7), 997–1012 (1987). https://doi.org/10.1080/00411458708204603

    Article  MathSciNet  MATH  Google Scholar 

  167. Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: “Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum” [Comm. Math. Phys. 105 (1986), no. 2, 189–203; MR0849204 (88d:82061)] and “Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum” [ibid. 113, (1987), no. 1, 79–85; MR0918406 (89b:82052)] by Pulvirenti. Comm. Math. Phys. 121(1), 143–146 (1989). http://projecteuclid.org/euclid.cmp/1104178007

  168. Illner, R., Shinbrot, M.: The Boltzmann equation: global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95(2), 217–226 (1984). http://projecteuclid.org/euclid.cmp/1103941523

  169. Imbert, C., Mouhot, C., Silvestre, L.: Gaussian lower bounds for the Boltzmann equation without cutoff. SIAM J. Math. Anal. 52(3), 2930–2944 (2020). https://doi.org/10.1137/19M1252375

    Article  MathSciNet  MATH  Google Scholar 

  170. Jüngel, A., Stelzer, I.V.: Existence analysis of Maxwell–Stefan systems for multicomponent mixtures. SIAM J. Math. Anal. 45(4), 2421–2440 (2013). https://doi.org/10.1137/120898164

    Article  MathSciNet  MATH  Google Scholar 

  171. Kaniel, S., Shinbrot, M.: The Boltzmann equation. I. Uniqueness and local existence. Commun. Math. Phys. 58(1), 65–84 (1978). http://projecteuclid.org/euclid.cmp/1103901367

  172. Kim, C.: Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308(3), 641–701 (2011). https://doi.org/10.1007/s00220-011-1355-1

    Article  MathSciNet  MATH  Google Scholar 

  173. Kim, C.: Boltzmann equation with a large potential in a periodic box. Commun. Partial Differ. Equ. 39(8), 1393–1423 (2014). https://doi.org/10.1080/03605302.2014.903278

    Article  MathSciNet  MATH  Google Scholar 

  174. Kim, C., Lee, D.: The Boltzmann equation with specular boundary condition in convex domains. Commun. Pure Appl. Math. 71(3), 411–504 (2018). https://doi.org/10.1002/cpa.21705

    Article  MathSciNet  MATH  Google Scholar 

  175. Kim, C., Lee, D.: Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains. Arch. Ration. Mech. Anal. 230(1), 49–123 (2018). https://doi.org/10.1007/s00205-018-1241-5

    Article  MathSciNet  MATH  Google Scholar 

  176. Kim, C., Yun, S.B.: The Boltzmann equation near a rotational local Maxwellian. SIAM J. Math. Anal. 44(4), 2560–2598 (2012). https://doi.org/10.1137/11084981X

    Article  MathSciNet  MATH  Google Scholar 

  177. Klaus, M.: The linear Boltzmann operator-spectral properties and short-wavelength limit. Helv. Phys. Acta 48, 99–129 (1975)

    MathSciNet  Google Scholar 

  178. Krishna, R., Wesselingh, J.A.: The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 52, 861–911 (1997)

    Article  Google Scholar 

  179. Lachowicz, M.: On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci. 9(3), 342–366 (1987). https://doi.org/10.1002/mma.1670090127

    Article  MathSciNet  MATH  Google Scholar 

  180. Lanford III, O.E.: Time evolution of large classical systems. In: Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1–111. Lecture Notes in Phys., Vol. 38. Springer (1975)

  181. Levermore, C.D., Masmoudi, N.: From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 196(3), 753–809 (2010). https://doi.org/10.1007/s00205-009-0254-5

    Article  MathSciNet  MATH  Google Scholar 

  182. Li, F., Yu, H.: Global existence of classical solutions to the Boltzmann equation with external force for hard potentials. Int. Math. Res. Not. IMRN pp. Art. ID rnn112, 22 (2008). https://doi.org/10.1093/imrn/rnn112

  183. Lions, P.L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. I, II. J. Math. Kyoto Univ. 34(2), 391–427, 429–461 (1994). https://doi.org/10.1215/kjm/1250519017

  184. Lions, P.L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. III. J. Math. Kyoto Univ. 34(3), 539–584 (1994). https://doi.org/10.1215/kjm/1250518932

    Article  MathSciNet  MATH  Google Scholar 

  185. Lions, P.L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 158(3), 173–193, 195–211 (2001). https://doi.org/10.1007/s002050100143

  186. Liu, S., Yang, X.: The initial boundary value problem for the Boltzmann equation with soft potential. Arch. Ration. Mech. Anal. 223(1), 463–541 (2017). https://doi.org/10.1007/s00205-016-1038-3

    Article  MathSciNet  MATH  Google Scholar 

  187. Liu, T.P., Yang, T., Yu, S.H.: Energy method for Boltzmann equation. Phys. D 188(3–4), 178–192 (2004). https://doi.org/10.1016/j.physd.2003.07.011

    Article  MathSciNet  MATH  Google Scholar 

  188. Liu, T.P., Yu, S.H.: Boltzmann equation: micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004). https://doi.org/10.1007/s00220-003-1030-2

    Article  MathSciNet  MATH  Google Scholar 

  189. Liu, T.P., Yu, S.H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Commun. Pure Appl. Math. 57(12), 1543–1608 (2004). https://doi.org/10.1002/cpa.20011

    Article  MathSciNet  MATH  Google Scholar 

  190. Liu, T.P., Yu, S.H.: Green’s function of Boltzmann equation, 3-D waves. Bull. Inst. Math. Acad. Sin. (N.S.) 1(1), 1–78 (2006)

    MathSciNet  MATH  Google Scholar 

  191. Lods, B.: Semigroup generation properties of streaming operators with noncontractive boundary conditions. Math. Comput. Model. 42(13), 1441–1462 (2005). https://doi.org/10.1016/j.mcm.2004.12.007

    Article  MathSciNet  MATH  Google Scholar 

  192. Lu, X.: A direct method for the regularity of the gain term in the Boltzmann equation. J. Math. Anal. Appl. 228(2), 409–435 (1998). https://doi.org/10.1006/jmaa.1998.6141

    Article  MathSciNet  MATH  Google Scholar 

  193. Lu, X.: A modified Boltzmann equation for Bose–Einstein particles: isotropic solutions and long-time behavior. J. Stat. Phys. 98(5–6), 1335–1394 (2000). https://doi.org/10.1023/A:1018628031233

    Article  MathSciNet  MATH  Google Scholar 

  194. Lu, X.: On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particles. J. Stat. Phys. 116(5–6), 1597–1649 (2004). https://doi.org/10.1023/B:JOSS.0000041750.11320.9c

    Article  MathSciNet  MATH  Google Scholar 

  195. Lu, X.: The Boltzmann equation for Bose–Einstein particles: regularity and condensation. J. Stat. Phys. 156(3), 493–545 (2014). https://doi.org/10.1007/s10955-014-1026-7

    Article  MathSciNet  MATH  Google Scholar 

  196. Masmoudi, N., Saint-Raymond, L.: From the Boltzmann equation to the Stokes–Fourier system in a bounded domain. Commun. Pure Appl. Math. 56(9), 1263–1293 (2003). https://doi.org/10.1002/cpa.10095

    Article  MathSciNet  MATH  Google Scholar 

  197. Mischler, S.: On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210(2), 447–466 (2000). https://doi.org/10.1007/s002200050787

    Article  MathSciNet  MATH  Google Scholar 

  198. Mischler, S.: On the trace problem for solutions of the Vlasov equation. Commun. Partial Differ. Equ. 25(7–8), 1415–1443 (2000). https://doi.org/10.1080/03605300008821554

    Article  MathSciNet  MATH  Google Scholar 

  199. Mischler, S.: Kinetic equations with Maxwell boundary conditions. Ann. Sci. Éc. Norm. Supér. (4) 43(5), 719–760 (2010). https://doi.org/10.24033/asens.2132

    Article  MathSciNet  MATH  Google Scholar 

  200. Mischler, S., Mouhot, C.: Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior. J. Stat. Phys. 124(2–4), 703–746 (2006). https://doi.org/10.1007/s10955-006-9097-8

    Article  MathSciNet  MATH  Google Scholar 

  201. Mischler, S., Mouhot, C.: Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation. Arch. Ration. Mech. Anal. 221(2), 677–723 (2016). https://doi.org/10.1007/s00205-016-0972-4

    Article  MathSciNet  MATH  Google Scholar 

  202. Mischler, S., Mouhot, C., Rodriguez Ricard, M.: Cooling process for inelastic Boltzmann equations for hard spheres. I. The Cauchy problem. J. Stat. Phys. 124(2–4), 655–702 (2006). https://doi.org/10.1007/s10955-006-9096-9

    Article  MathSciNet  MATH  Google Scholar 

  203. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(4), 467–501 (1999). https://doi.org/10.1016/S0294-1449(99)80025-0

    Article  MathSciNet  MATH  Google Scholar 

  204. Mokhtar-Kharroubi, M.: On collisionless transport semigroups with boundary operators of norm one. J. Evol. Equ. 8(2), 327–352 (2008). https://doi.org/10.1007/s00028-007-0360-5

    Article  MathSciNet  MATH  Google Scholar 

  205. Mouhot, C.: Quantitative lower bounds for the full Boltzmann equation. Commun. Partial Differ. Equ. I. Period. Bound. Conditions 30(4–6), 881–917 (2005). https://doi.org/10.1081/PDE-200059299

    Article  MATH  Google Scholar 

  206. Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun. Partial Differ. Equ. 31(7–9), 1321–1348 (2006). https://doi.org/10.1080/03605300600635004

    Article  MathSciNet  MATH  Google Scholar 

  207. Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261(3), 629–672 (2006). https://doi.org/10.1007/s00220-005-1455-x

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  209. Mouhot, C., Strain, R.M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. (9) 87(5), 515–535 (2007). https://doi.org/10.1016/j.matpur.2007.03.003

    Article  MathSciNet  MATH  Google Scholar 

  210. Mouhot, C., Villani, C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal. 173(2), 169–212 (2004). https://doi.org/10.1007/s00205-004-0316-7

    Article  MathSciNet  MATH  Google Scholar 

  211. Nishida, T., Imai, K.: Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. Res. Inst. Math. Sci. 12(1), 229–239 (1976/77). https://doi.org/10.2977/prims/1195190965

  212. Pao, Y.P.: Boltzmann collision operator with inverse-power intermolecular potentials. I. Commun. Pure Appl. Math. 27, 407–428 (1974). https://doi.org/10.1002/cpa.3160270402

    Article  MathSciNet  MATH  Google Scholar 

  213. Pao, Y.P.: Boltzmann collision operator with inverse-power intermolecular potentials. II. Commun. Pure Appl. Math. 27, 559–581 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  214. Perthame, B., Souganidis, P.E.: A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4) 31(4), 591–598 (1998). https://doi.org/10.1016/S0012-9593(98)80108-0

    Article  MathSciNet  MATH  Google Scholar 

  215. Pettersson, R.: On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces. J. Stat. Phys. 59(1–2), 403–440 (1990). https://doi.org/10.1007/BF01015576

    Article  MathSciNet  MATH  Google Scholar 

  216. Poritsky, H.: The billiard ball problem on a table with a convex boundary—an illustrative dynamical problem. Ann. Math. 2(51), 446–470 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  217. Povzner, A.J.: On the Boltzmann equation in the kinetic theory of gases. Mat. Sb. (N.S.) 58(100), 65–86 (1962)

    MathSciNet  MATH  Google Scholar 

  218. Pulvirenti, A., Wennberg, B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183(1), 145–160 (1997). https://doi.org/10.1007/BF02509799

    Article  MathSciNet  MATH  Google Scholar 

  219. Pulvirenti, M., Saffirio, C., Simonella, S.: On the validity of the Boltzmann equation for short range potentials. Rev. Math. Phys. 26(2), 1450001, 64 (2014). https://doi.org/10.1142/S0129055X14500019

  220. Pulvirenti, M., Simonella, S.: The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error. Invent. Math. 207(3), 1135–1237 (2017). https://doi.org/10.1007/s00222-016-0682-4

    Article  MathSciNet  MATH  Google Scholar 

  221. Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Lecture Notes in Mathematics, vol. 1971. Springer, Berlin (2009)

  222. Speck, J., Strain, R.M.: Hilbert expansion from the Boltzmann equation to relativistic fluids. Commun. Math. Phys. 304(1), 229–280 (2011). https://doi.org/10.1007/s00220-011-1207-z

    Article  MathSciNet  MATH  Google Scholar 

  223. Spohn, H.: Kinetics of the Bose–Einstein condensation. Phys. D 239(10), 627–634 (2010). https://doi.org/10.1016/j.physd.2010.01.018

    Article  MathSciNet  MATH  Google Scholar 

  224. Strain, R.M., Guo, Y.: Almost exponential decay near Maxwellian. Commun. Partial Differ. Equ. 31(1–3), 417–429 (2006). https://doi.org/10.1080/03605300500361545

    Article  MathSciNet  MATH  Google Scholar 

  225. Strain, R.M., Guo, Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008). https://doi.org/10.1007/s00205-007-0067-3

    Article  MathSciNet  MATH  Google Scholar 

  226. Tabachnikov, S.: Billiards. Panor. Synth. 1, vi+142 (1995)

    Google Scholar 

  227. Tabachnikov, S.: Geometry and billiards, Student Mathematical Library, vol. 30. American Mathematical Society, Providence, RI (2005)

  228. Tabata, M.: Decay of solutions to the Cauchy problem for the linearized Boltzmann equation with some external-force potential. Jpn. J. Indust. Appl. Math. 10(2), 237–253 (1993). https://doi.org/10.1007/BF03167574

    Article  MathSciNet  MATH  Google Scholar 

  229. Tabata, M.: Decay of solutions to the Cauchy problem for the linearized Boltzmann equation with an unbounded external-force potential. Transp. Theory Stat. Phys. 23(6), 741–780 (1994). https://doi.org/10.1080/00411459408203926

    Article  MathSciNet  MATH  Google Scholar 

  230. Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94(3–4), 619–637 (1999). https://doi.org/10.1023/A:1004589506756

    Article  MathSciNet  MATH  Google Scholar 

  231. Toscani, G., Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203(3), 667–706 (1999). https://doi.org/10.1007/s002200050631

    Article  MathSciNet  MATH  Google Scholar 

  232. Toscani, G., Villani, C.: On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Stat. Phys. 98(5–6), 1279–1309 (2000). https://doi.org/10.1023/A:1018623930325

    Article  MathSciNet  MATH  Google Scholar 

  233. Tristani, I.: Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off. J. Stat. Phys. 157(3), 474–496 (2014). https://doi.org/10.1007/s10955-014-1066-z

    Article  MathSciNet  MATH  Google Scholar 

  234. Tristani, I.: Fractional Fokker-Planck equation. Commun. Math. Sci. 13(5), 1243–1260 (2015). https://doi.org/10.4310/CMS.2015.v13.n5.a8

    Article  MathSciNet  MATH  Google Scholar 

  235. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974). http://projecteuclid.org/euclid.pja/1195519027

  236. Ukai, S.: Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demi-espace. C. R. Acad. Sci. Paris Sér. A-B 282(6), Ai, A317–A320 (1976)

  237. Ukai, S.: Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Jpn. J. Appl. Math. 1(1), 141–156 (1984). https://doi.org/10.1007/BF03167864

    Article  MathSciNet  MATH  Google Scholar 

  238. Ukai, S.: Solutions of the Boltzmann equation. In: Patterns and waves, Stud. Math. Appl., vol. 18, pp. 37–96. North-Holland, Amsterdam (1986). https://doi.org/10.1016/S0168-2024(08)70128-0

  239. Ukai, S., Yang, T.: Mathematical theory of the Boltzmann equation (2006). Lecture Notes Series, no. 8, Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong

  240. Ukai, S., Yang, T., Zhao, H.: Global solutions to the Boltzmann equation with external forces. Anal. Appl. (Singap.) 3(2), 157–193 (2005). https://doi.org/10.1142/S0219530505000522

    Article  MathSciNet  MATH  Google Scholar 

  241. Vidav, I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970). https://doi.org/10.1016/0022-247X(70)90160-5

    Article  MathSciNet  MATH  Google Scholar 

  242. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143(3), 273–307 (1998). https://doi.org/10.1007/s002050050106

    Article  MathSciNet  MATH  Google Scholar 

  243. Villani, C.: Limites hydrodynamiques de l’équation de Boltzmann (d’après C. Bardos, F. Golse, C. D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond). In: Séminaire Bourbaki, Vol. 2000/2001, 282, pp. Exp. No. 893, ix, 365–405. Société Mathématique de France (2002)

  244. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, Vol. I, pp. 71–305. North-Holland, Amsterdam (2002). https://doi.org/10.1016/S1874-5792(02)80004-0

  245. Villani, C.: Hypocoercive diffusion operators. In: International Congress of Mathematicians. Vol. III, pp. 473–498. Eur. Math. Soc., Zürich (2006)

  246. Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950), iv+141 (2009). https://doi.org/10.1090/S0065-9266-09-00567-5

    Article  MathSciNet  MATH  Google Scholar 

  247. Wennberg, B.: Regularity in the Boltzmann equation and the Radon transform. Commun. Partial Differ. Equ. 19(11–12), 2057–2074 (1994). https://doi.org/10.1080/03605309408821082

    Article  MathSciNet  MATH  Google Scholar 

  248. Xiao, Q., Xiong, L., Zhao, H.: The Vlasov–Poisson–Boltzmann system with angular cutoff for soft potentials. J. Differ. Equ. 255(6), 1196–1232 (2013). https://doi.org/10.1016/j.jde.2013.05.005

    Article  MathSciNet  MATH  Google Scholar 

  249. Xiao, Q., Xiong, L., Zhao, H.: The Vlasov–Poisson–Boltzmann system for the whole range of cutoff soft potentials. J. Funct. Anal. 272(1), 166–226 (2017). https://doi.org/10.1016/j.jfa.2016.09.017

    Article  MathSciNet  MATH  Google Scholar 

  250. Yu, H.: Global classical solutions of the Boltzmann equation near Maxwellians. Acta Math. Sci. Ser. B (Engl. Ed.) 26(3), 491–501 (2006). https://doi.org/10.1016/S0252-9602(06)60074-X

    Article  MathSciNet  MATH  Google Scholar 

  251. Yu, H.: Global classical solutions to the Boltzmann equation with external force. Commun. Pure Appl. Anal. 8(5), 1647–1668 (2009). https://doi.org/10.3934/cpaa.2009.8.1647

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire MAP5, CNRS, Université de Paris, 75006, Paris, France

    Marc Briant

Authors
  1. Marc Briant
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Briant, M. Hypocoercivity for perturbation theory and perturbation of hypocoercivity for confined Boltzmann-type collisional equations. SeMA 80, 27–83 (2023). https://doi.org/10.1007/s40324-021-00281-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-021-00281-y

Search

Navigation