We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fluid Dynamic Limits of the Kinetic Theory of Gases | SpringerLink
Skip to main content
SpringerLink
Log in

Fluid Dynamic Limits of the Kinetic Theory of Gases

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

  • First Online:
From Particle Systems to Partial Differential Equations

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

These three lectures introduce the reader to recent progress on the hydrodynamic limits of the kinetic theory of gases. Lecture 1 outlines the main mathematical results in this direction, and explain in particular how the Euler or Navier-Stokes equations for compressible as well as incompressible fluids, can be derived from the Boltzmann equation. It also presents the notion of renormalized solution of the Boltzmann equation, due to P.-L. Lions and R. DiPerna, together with the mathematical methods used in the proofs of the fluid dynamic limits. Lecture 2 gives a detailed account of the derivation by L. Saint-Raymond of the incompressible Euler equations from the BGK model with constant collision frequency (Saint-Raymond, Bull Sci Math 126:493–506, 2002). Finally, Lecture 3 sketches the main steps in the proof of the incompressible Navier-Stokes limit of the Boltzmann equation, connecting the DiPerna-Lions theory of renormalized solutions of the Boltzmann equation with Leray’s theory of weak solutions of the Navier-Stokes system, following (Golse and Saint-Raymond, J Math Pures Appl 91:508–552, 2009). As is the case of all mathematical results in continuum mechanics, the fluid dynamic limits of the Boltzmann equation involve some basic properties of isotropic tensor fields that are recalled in Appendices 1 and 2.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    If X is a topological space and E is a Banach space, the notation C(X, w − E) designates the set of continuous maps from X to E equipped with its weak topology.

  2. 2.

    This is not completely true, however, since the velocity averaging method is at the heart of the kinetic formulation of hyperbolic conservation laws. Unfortunately, while this approach is rather successful in the case of scalar conservation laws, it seems so far limited to some very special kind of hyperbolic systems: see P.-L. Lions-B. Perthame-E. Tadmor [74], P.-E. Jabin-B. Perthame [61], B. Perthame [83].

  3. 3.

    Let X be a topological space and E a topological vector space. The notation C b (X, E) designates the set of bounded continuous maps from X to E.

  4. 4.

    The Legendre dual f ∗ of a function f: I → R, where I is an interval of R, is defined for all p ∈ R by the formula

    $$\displaystyle{{f}^{{\ast}}(p) =\sup _{ x\in I}(\mathit{px} - f(x))\,.}$$
  5. 5.

    Consider the endomorphism of \({({\mathbf{R}}^{N})}^{\otimes 2}\) defined by

    $$\displaystyle{u \otimes v\mapsto {(u \otimes v)}^{\sigma } = v \otimes u\,.}$$

    An element T of \({({\mathbf{R}}^{N})}^{\otimes 2}\) is said to be symmetric if and only if T σ = T.

  6. 6.

    Let G be a subgroup of O N (R) and V be a linear subspace of R N. Assume that G leaves V invariant, i.e. gV ⊂ V for each g ∈ G, and that G acts transitively on the spheres of V centered at 0, i.e. if for each v 1, v 2 ∈ V such that | v 1 |  =  | v 2 | , there exists g ∈ G satisfying gv 1 = v 2. Then, the only vector v ∈ V such that gv = v for each g ∈ G is v = 0. Indeed, if v ≠ 0, one has | v |  =  | − v | and therefore there exists g ∈ G such that gv = −v ≠ v.

  7. 7.

    Let G be a subgroup of O N (R) and let V be a linear subspace of R N. Assume that G acts transitively on spheres of V centered at 0. Then the only linear subspaces W of V such that gW ⊂ W for each g ∈ G are {0} and V. Indeed, if W is a linear subspace of V different from either {0} or V, let w ∈ W and z ∈ V ∖ W satisfy | w |  =  | z | ≠ 0. By the transitivity assumption above, there exists g ∈ G such that gw = z and therefore gW is not included in W.

References

  1. Agoshkov, V.I.: Spaces of functions with differential difference characteristics and smoothness of the solutions of the transport equations. Dokl. Akad. Nauk SSSR 276, 1289–1293 (1984)

    MathSciNet  Google Scholar 

  2. Arkeryd, L., Nouri, A.: The stationary Boltzmann equation in R n with given indata. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 1, 359–385 (2002)

    Google Scholar 

  3. Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361 (1966)

    Article  Google Scholar 

  4. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer, New York (1998)

    MATH  Google Scholar 

  5. Bardos, C., Golse, F.: Différents aspects de la notion d’entropie au niveau de l’équation de Boltzmann et de Navier-Stokes. C. R. Math. Acad. Sci. Paris 299, 291–294 (1984)

    MathSciNet  Google Scholar 

  6. Bardos, C., Ukai, S.: The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models Methods Appl. Sci. 1, 235–257 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284, 617–649 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bardos, C., Caflisch, R.E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Commun. Pure Appl. Math. 39, 323–352 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of the Boltzmann equation I. J. Stat. Phys. 63, 323–344 (1991)

    Article  MathSciNet  Google Scholar 

  11. 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, 667–753 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bardos, C., Golse, F., Levermore, C.D.: The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153, 177–204 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bardos, C., Golse, F., Sone, Y.: Half-space problems for the Boltzmann equation: a survey. J. Stat. Phys. 124, 275–300 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bardos, C., Levermore, C.D., Ukai, S., Yang, T.: Kinetic equations: fluid dynamical limits and viscous heating. Bull. Inst. Math. Acad. Sin. (New Ser.) 3, 1–49 (2008)

    Google Scholar 

  15. Bardos, C., Golse, F., Paillard, L.: The incompressible Euler limit of the Boltzmann equation with accommodation boundary condition. Commun. Math. Sci. 10, 159–190 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    MATH  Google Scholar 

  17. Bobylev, A.V.: Quasistationary hydrodynamics for the Boltzmann equation. J. Stat. Phys. 80, 1063–1085 (2006)

    Article  MathSciNet  Google Scholar 

  18. Bobylev, A.V.: Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations. J. Stat. Phys. 124, 371–399 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Bobylev, A.V.: Generalized Burnett hydrodynamics. J. Stat. Phys. 132, 569–580 (2008)

    MATH  MathSciNet  Google Scholar 

  20. Bouchut, F., Golse, F., Pulvirenti, M.: Stationary quasivariational inequalities with gradient constraint and nonhomogeneous boundary conditions. In: Desvillettes, L., Perthame, B. (eds.) Kinetic Equations and Asymptotic Theory (Editions scientifiques et médicales). Elsevier, Paris (2000)

    Google Scholar 

  21. Bourgain, J., Golse, F., Wennberg, B.: On the distribution of free path lengths for the periodic Lorentz gas. Commun. Math. Phys. 190, 491–508 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52, 411–452 (1999)

    Article  MathSciNet  Google Scholar 

  23. Brenier, Y., DeLellis, C., Székélyhidi, L.: Weak-strong uniqueness for measure-valued solutions. Commun. Math. Phys. 305, 351–361 (2011)

    Article  MATH  Google Scholar 

  24. Caflisch, R.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math. 33, 651–666 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  25. Caglioti, E., Golse, F.: The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions. C. R. Math. Acad. Sci. Paris 346, 477–482 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Caglioti, E., Golse, F.: On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas. J. Stat. Phys. 141, 264–317 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Cercignani, C.: Bifurcation problems in fluid mechanics. Meccanica J. Ital. Assoc. Theor. Appl. Mech. 5, 7–16 (1970)

    MATH  MathSciNet  Google Scholar 

  28. Cercignani, C.: Theory and Applications of the Boltzmann Equation. Scottish Academic Press, Edinburgh/London (1975)

    Google Scholar 

  29. Chen, G.Q., Levermore, C.D., Liu, T.P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Constantin, P., Foias, C.: Navier-Stokes Equations. University of Chicago Press, Chicago (1988)

    MATH  Google Scholar 

  31. Coron, F., Golse, F., Sulem, C.: A classification of well-posed kinetic layer problems. Commun. Pure Appl. Math. 41, 409–435 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  32. DeLellis, C., Szekelyhidi, L.: The Euler equations as a differential inclusion. Ann. Math. (2) 170, 1417–1436 (2009)

    Google Scholar 

  33. DeMasi, A., Esposito, R., Lebowitz, J.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Commun. Pure Appl. Math. 42, 1189–1214 (1990)

    Article  MathSciNet  Google Scholar 

  34. DiPerna, R., Lions, P.-L.: On the Cauchy problem for the Boltzmann equation: global existence and weak stability results. Ann. Math. 130, 321–366 (1990)

    Article  MathSciNet  Google Scholar 

  35. DiPerna, R.J., Majda, A.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667–689 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  36. DiPerna, R., Lions, P.-L., Meyer, Y.: L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 271–288 (1991)

    MATH  MathSciNet  Google Scholar 

  37. Gallavotti, G.: Divergences and approach to equilibrium in the Lorentz and the wind-tree-models. Phys. Rev. 185, 308–322 (1969)

    Article  Google Scholar 

  38. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  39. Golse, F.: The Boltzmann equation and its hydrodynamic limits. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 2, pp. 161–301. Elsevier B.V., Amsterdam/Boston/Paris (2005)

    Google Scholar 

  40. Golse, F.: On the periodic Lorentz gas in the Boltzmann-Grad scaling. Ann. Fac. Sci. Toulouse 17, 735–749 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Golse, F.: Analysis of the boundary layer equation in the kinetic theory of gases. Bull. Inst. Math. Acad. Sin. (New Series) 3, 211–242 (2008)

    Google Scholar 

  42. Golse, F.: From the kinetic theory of gases to continuum mechanics. In: Harold Grad Lecture, 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove. AIP Conference Proceedings, vol. 1333, pp. 15–27 (2011)

    Article  Google Scholar 

  43. Golse, F.: From the Boltzmann equation to hydrodynamic equations in thin layers. Boll. Unione Mat. Ital. (9) 4, 163–186 (2011)

    Google Scholar 

  44. Golse, F.: From the Boltzmann equation to the Euler equations in the presence of boundaries. Comput. Math. Appl. 65, 815–830 (2013)

    Article  MathSciNet  Google Scholar 

  45. Golse, F., Levermore, C.D.: The Stokes-Fourier and acoustic limits for the Boltzmann equation. Commun. Pure Appl. Math. 55, 336–393 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  46. Golse, F., Levermore, C.D.: Hydrodynamic limits of kinetic models. In: Topics in Kinetic Theory. Fields Institute Communications, vol. 48, pp. 1–75. American Mathematical Society, Providence (2005)

    Google Scholar 

  47. Golse, F., Saint-Raymond, L.: Velocity averaging in L 1 for the transport equation. C. R. Acad. Sci. 334, 557–562 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  48. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  49. Golse, F., Saint-Raymond, L.: Hydrodynamic limits for the Boltzmann equation. Riv. Mat. Univ. Parma (7) 4* *, 1–144 (2005)

    Google Scholar 

  50. Golse, F., Saint-Raymond, L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91, 508–552 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  51. Golse, F., Perthame, B., Sentis, R.: Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale de l’opérateur de transport. C. R. Acad. Sci. 301, 341–344 (1985)

    MATH  MathSciNet  Google Scholar 

  52. Golse, F., Lions, P.-L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76, 110–125 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  53. Golse, F., Perthame, B., Sulem, C.: On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Ration. Mech. Anal. 103, 81–96 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  54. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  55. Grad, H.: Principles of the kinetic theory of gases. In: Flügge, S. (ed.) Handbook der Physik Band XII, pp. 205–294. Springer, Berlin (1958)

    Google Scholar 

  56. Grad, H.: Asymptotic theory of the Boltzmann equation II. In: Laurmann, J.A. (ed.) Rarefied Gas Dynamics, vol. I, pp. 26–59. Academic, New York (1963)

    Google Scholar 

  57. Guiraud, J.-P.: Problème aux limites intérieur pour l’équation de Boltzmann linéaire. J. Mécanique 9, 443–490 (1970)

    MATH  MathSciNet  Google Scholar 

  58. Guiraud, J.-P.: Problème aux limites intérieur pour l’équation de Boltzmann en régime stationnaire, faiblement non linéaire. J. Mécanique 11, 183–231 (1972)

    MATH  MathSciNet  Google Scholar 

  59. Hilbert, D.: Begründung der kinetischen Gastheorie. Math. Ann. 72, 562–577 (1912)

    Article  MATH  MathSciNet  Google Scholar 

  60. Hilbert, D.: Mathematical problems. In: International Congress of Mathematicians, Paris 1900. (Translated and reprinted in Bull. Am. Math. Soc. 37, 407–436 (2000))

    Google Scholar 

  61. Jabin, P.-E., Perthame, B.: Regularity in kinetic formulations via averaging lemmas, a tribute to J. L. Lions. ESAIM Control Optim. Calc. Var. 8, 761–774 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  62. Jin, S., Slemrod, M.: Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103, 1009–1033 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  63. Kogan, M.N., Galkin, V.S., Fridlender, O.G.: Stresses produced in gases by temperature and concentration inhomogeneities. New types of free convection. Sov. Phys. Usp. 19, 420–428 (1976)

    Article  Google Scholar 

  64. Lachowicz, M.: On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci. 9, 342–366 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  65. Lanford, O.E.: The evolution of large classical system. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol. 38, pp. 1–111. Springer, Heidelberg (1975)

    Chapter  Google Scholar 

  66. Larsen, E.W., Keller, J.B.: Asymptotics solutions of neutron transport problems for small mean free paths. J. Math. Phys. 15, 75–81 (1974)

    Article  MathSciNet  Google Scholar 

  67. Leray, J.: Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  68. Levermore, C.D., Masmoudi, N.: From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 196, 753–809 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  69. Lifhsitz, E., Pitayevski, L.: Physical Kinetics. Pergamon Press, Oxford/New York (1981)

    Google Scholar 

  70. Lions, P.-L.: Conditions at infinity for Boltzmann’s equation. Commun Partial Differ. Equ. 19, 335–367 (1994)

    Article  MATH  Google Scholar 

  71. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. II. J. Math. Kyoto Univ. 34, 429–461 (1994)

    Google Scholar 

  72. Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models. The Clarendon Press/Oxford University Press, New York (1996)

    Google Scholar 

  73. Lions, P.-L., Masmoudi, N.: From Boltzmann equation to the Navier-Stokes and Euler equations I. Arch. Ration. Mech. Anal. 158, 173–193 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  74. Lions, P.-L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7, 169–191 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  75. Liu, T.-P.: Solutions in the large for the equations of nonisentropic gas dynamics. Indiana Univ. Math. J. 26, 147–177 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  76. Marklof, J., Strömbergsson, A.: The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. Math. 174, 225–298 (2011)

    Article  MATH  Google Scholar 

  77. Masmoudi, N., Saint-Raymond, L.: From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Commun. Pure Appl. Math. 56, 1263–1293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  78. Mischler, S.: Kinetic equations with Maxwell boundary conditions. Ann. Sci. Ecole Norm. Sup. 43, 719–760 (2010)

    MATH  MathSciNet  Google Scholar 

  79. Nirenberg, L.: An abstract form of the nonlinear Cauchy-Kowalewski theorem. J. Differ. Geom. 6, 561–576 (1972)

    MATH  MathSciNet  Google Scholar 

  80. Nishida, T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12, 629–633 (1977)

    MATH  MathSciNet  Google Scholar 

  81. Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61, 119–148 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  82. Papanicolaou, G.C.: Asymptotic analysis of transport processes. Bull. Am. Math. Soc. 81, 330–392 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  83. Perthame, B.: Kinetic Formulation of Conservation Laws. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  84. Perthame, B., Pulvirenti, M.: Weighted L ∞ bounds and uniqueness for the Boltzmann BGK model. Arch. Ration. Mech. Anal. 125, 289–295 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  85. Saint-Raymond, L.: Du modèle BGK de l’équation de Boltzmann aux équations d’Euler des fluides incompressibles. Bull. Sci. Math. 126, 493–506 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  86. Saint-Raymond, L.: Discrete time Navier-Stokes limit for the BGK Boltzmann equation. Commun. Partial Differ. Equ. 27, 149–184 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  87. Saint-Raymond, L.: Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Ration. Mech. Anal. 166, 47–80 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  88. Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Lecturer Notes in Mathematics, vol. 1971. Springer, Berlin/Heidelberg (2009)

    Google Scholar 

  89. Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  90. Shnirelman, A.: On the nonuniqueness of weak solutions of the Euler equation. Commun. Pure Appl. Math. 50, 1261–1286 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  91. Sideris, T.: Formation of Singularities in 3D compressible fluids. Commun. Math. Phys. 101, 475–485 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  92. Sone, Y.: Continuum gas dynamics in the light of kinetic theory and new features of rarefied gas flows. In: Shen, C. (ed.) Harold Grad Lecture, Rarefied Gas Dynamics, pp. 3–24. Peking University Press, Beijing (1997)

    Google Scholar 

  93. Sone, Y.: Molecular Gas Dynamics. Theory, Techniques, and Applications. Birkhäuser, Boston (2002)

    Google Scholar 

  94. Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston (2007)

    Google Scholar 

  95. Sone, Y., Aoki, K., Takata, S., Sugimoto, H., Bobylev, A.V.: Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: examination by asymptotic analysis and numerical computation of the Boltzmann equation. Phys. Fluids 8, 628–638 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  96. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear boundary layers of the Boltzmann equation. I. Existence. Commun. Math. Phys. 236, 373–393 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  97. Villani, C.: Limites hydrodynamiques de l’équation de Boltzmann. In: Séminaire Bourbaki, Exp. No. 893, pp. 364–405 (2000–2001)

    Google Scholar 

  98. Wiedemann, E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 727–730 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  99. Yau, H.-T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz (CMLS), 91128, Palaiseau Cedex, France

    François Golse

Authors
  1. François Golse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Golse .

Editor information

Editors and Affiliations

  1. UMR CNRS-UNS 7351, Université de Nice Sophia Antipolis, Laboratoire J.A.Dieudonné, Nice, France

    Cédric Bernardin

  2. Universidade do Minho, Centre of Mathematics, Braga, Portugal

    Patricia Gonçalves

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Golse, F. (2014). Fluid Dynamic Limits of the Kinetic Theory of Gases. In: Bernardin, C., Gonçalves, P. (eds) From Particle Systems to Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54271-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation