Free Access
Math. Model. Nat. Phenom.
Volume 10, Number 3, 2015
Model Reduction
Page(s) 6 - 15
Published online 22 June 2015
  1. T. Arima, A. Mentrelli, T. Ruggeri. Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments. Annals of Physics, 345 (2014), 111–140. [CrossRef] [Google Scholar]
  2. A.V. Boby’lev. Instabilities in the Chapman-Enskog expansion and hyprbolic Burnett equations. J. Statistical Physics 124 (2-4) (2006), 371–399. [CrossRef] [Google Scholar]
  3. A.V. Boby’lev. The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Soviet Phys. Dokl., 27 (1982), 29–31. [Google Scholar]
  4. A.V. Boby’lev. Generalized Burnett hydrodynamics. J. Statistical Physics, 132 (3) (2008), 569–580. [CrossRef] [Google Scholar]
  5. C. Borgnakke, P.S. Larsen. Statistical collision model for Monte-Carlo simulation of polyatomic gas mixtures. J. Computational Physics, 18 (1975), 405–420. [Google Scholar]
  6. J-F. Bourgat, L. Desvillettes, P. Le Tallec, B. Perthalme. Microreversible collisions for polyatomic gases and the Boltzmann theorem. European J. Mech. Fluids, 13 (1994), 237–254. [Google Scholar]
  7. D. Burnett. The distibution of velocities in a slightly non-uniform gas. Proc. London Math. Soc., S2-39 (1) (1935), 385–430. [Google Scholar]
  8. D. Burnett. The distibution of molecular velocities and the mean motion in a non-uniform gas. Proc. London Math. Soc., S2-40 (1) (1936), 382–435. [CrossRef] [Google Scholar]
  9. J.W. Cahn. Free energy of a non-uniform system. II. Thermodynamic basis. J. Chemical Physics, 30 (1959), 1121–1124. [CrossRef] [Google Scholar]
  10. J.W. Cahn, J.E. Hilliard. Free energy of a non-uniform system II. Thermodynamic basis. J. Chemical Physics, 28 (1958), 258–266. [Google Scholar]
  11. C. Cercignani. The Boltzmann equation and its applications. Series Applied Mathematical Sciences, V. 67. Springer, New York, 1988. [Google Scholar]
  12. S. Chapman, T.G. Cowling. The Mathematical Theory of Nonuniform Gases, third edition. Series Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. [Google Scholar]
  13. I-K. Chen, T-P. Liu, S. Takata. Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation. Archive for Rational Mechanics and Analysis, 212 (2014), 575–595. [CrossRef] [Google Scholar]
  14. J.E. Dunn, J. Serrin. On thermomechanics of interstitial working. Archive for Rational Mechanics and Analysis, 88 (2) (1985), 95–133. [Google Scholar]
  15. A.N. Gorban, I.V. Karlin. Hilbert’s 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. Bulletin of the American Mathematical Society (N.S.), 51 (2) (2014), 187–246. [Google Scholar]
  16. A.N. Gorban, I.V. Karlin. Structure and approximations of the Chapman-Enskog expansion. Soviet Physics JETP, 73 (1991), 637–641. [Google Scholar]
  17. A.N. Gorban, I.V. Karlin. Structure and approximations of the Chapman-Enskog for the linearized Grad equations. Transport Theory and Statistical Physics, 21 (1-2) (1992), 101–117. [CrossRef] [Google Scholar]
  18. A.N. Gorban, I.V. Karlin. Short wave limit of hydrodynamics: a soluble model. Phys. Rev. Letters, 77 (1996), 282–285. [CrossRef] [Google Scholar]
  19. H. Grad. On the kinetic theory of rarefied gases. Communications on Pure and Applied Mathematics, 2 (1949), 331-407. [Google Scholar]
  20. H. Grad. Principles of the kinetic theory of gases. In Handbuch der Physik ( heraus-gegeben von S.Flugge), Bd. 12, Thermodynamik der Gase. Springer, Berlin, 205–294, 1958. [Google Scholar]
  21. D. Hilbert. Mathematical problems. Bulletin of the American Mathematical Society, 8 (10) (1902) 437–479. [CrossRef] [MathSciNet] [Google Scholar]
  22. D. Hilbert. Begrundung der kinetische Gastheorie. Math. Ann., 72 (1912), 562–577. [CrossRef] [MathSciNet] [Google Scholar]
  23. I.V. Karlin, A.N. Gorban. Hydrodynamics from Grad’s equations:what can we learn from exact solutions? Ann. Phys., 11 (10-11) (2002), 783–833. [CrossRef] [Google Scholar]
  24. I.V. Karlin, A.N. Gorban, G. Dukek, T.F. Nonnenmacher. Dynamic correction to oment approximations. Phys. Rev. E, 57 (2) (1998), 1668–1672. [CrossRef] [Google Scholar]
  25. Y-J. Kim, M-G. Lee, M. Slemrod. Thermal creep of a rarefied gas on the basis of non-linear Korteweg-theory. Archive for Rational Mechanics and Analysis, 215 (2) (2015), 353–379. [CrossRef] [Google Scholar]
  26. D.J. Korteweg. Sur la forme que prennent les equations du mouvements des fluides si l’on tient compte des forces capillaires causes par des variations de densite. Arch. Neerl. Sci. Exactes. Ser II 6 (1901), 1–24. [Google Scholar]
  27. L.D. Landau, E.M. Lifschitz. Statistical Physics, Addison-Wesley, Reading, MA, 1969. [Google Scholar]
  28. P.D. Lax, C.D. Levermore. The small dispersion limit of the Korteweg-deVries equation I. Communications in Pure and Applied Mathematics, 36 (1983), 253–290. [CrossRef] [Google Scholar]
  29. J.C. Maxwell. On the stresses in rarefied gases arising from inequalities in temperature. Phil. Trans. Roy. Soc. (London), 170 (1879), 231–256. [CrossRef] [Google Scholar]
  30. J.C. Maxwell. On the dynamic theory of gases. Phil. Trans. Roy. Soc. (London), 157 (1879) 49–88. [Google Scholar]
  31. T. Nishida. Fluid dynamic limit of the nonlinear Boltzmann equation at the level of the compressible Euler equations. Comm. Math. Physics, 61 (1978), 119–148. [CrossRef] [MathSciNet] [Google Scholar]
  32. T. Ohwada, Y. Sone, K. Aoki. Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard sphere molecules. Physics of Fluids A-Fluid Dynamics, 1(9) (1989), 1588–1599. [CrossRef] [Google Scholar]
  33. T. Ohwada, Y. Sone, K. Aoki. Numerical analysis of the Poiseuille and thermal transpiration flows between 2 parallel plates on the basis of the Boltzmann equation for hard sphere molecules. Physics of Fluids A-Fluid Dynamics, 1 (12) (1989), 2042–2049. [CrossRef] [Google Scholar]
  34. M. Pavic, T. Ruggeri, S. Simic. Maximum entropy principle for rarefied polyatomic gases. Physica A, 392 (2013), 1302–1317. [CrossRef] [Google Scholar]
  35. P. Rosenau. Extension of the Landau-Ginzburg free energy functions to high gradient domains. Physical Review A, 3 (1989), 6614–6617. [CrossRef] [Google Scholar]
  36. P. Rosenau. Extending hydrodynamics via regularization of the Chapman-Enskog expansion. Physical Review A, 90 (1989), 7193–7196. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  37. L. Saint-Raymond. A mathematical PDE perspective of the Chapman-Enskog expansion. Bulletin of the American Mathematical Society (N.S.), 51 (2) (2014), 247–276. [CrossRef] [Google Scholar]
  38. L. Saint-Raymond. Hydrodynamic limits of the Boltzmann equation. Lecture Notes in Mathematics, V. 1971, Springer, Berlin 2009. [Google Scholar]
  39. L. Saint-Raymond. From Boltzmann’s kinetic theory to Euler’s equations. Physica D, 237 (2008), 2028–2036.; [CrossRef] [Google Scholar]
  40. M.E. Schonbek. Convergence of solutions to nonlinear dispersive equations. Communications in Partial Differential Equations, 7 (1982), 959–1000. [Google Scholar]
  41. M. Slemrod. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Archive for Rational Mechanics and Analysis, 81 (4) (1983), 301–315. [CrossRef] [Google Scholar]
  42. M. Slemrod. Dynamic phase transitions in a van der Waals fluid. J. Differential Equations, 52 (1) (1984), 1–23. [CrossRef] [MathSciNet] [Google Scholar]
  43. M. Slemrod. Chapman-Enskog=⇒viscosity-capillarity. Quarterly of Applied Mathematics, 70 (3)(2012), 613–624. [CrossRef] [MathSciNet] [Google Scholar]
  44. M. Slemrod. Admissibility of weak solutions for the compressible Euler equations, n ≥ 2. Philosophical Transactions Roy. Soc. A, 371 (3) (2013), 1–11. [CrossRef] [Google Scholar]
  45. M. Slemrod. From Boltzmann to Euler: Hilbert’s 6th problem revisited. Computers and Mathematics with Applications, 65 (10) (2013), 1497–1501. [Google Scholar]
  46. H. Struchtrup. Macroscopic transport equations for rarefied gas flows. Series Interaction of Mathematics and Mechanics. Springer, Berlin, 2005. [Google Scholar]
  47. H. Struchtrup. Macroscopic transport models for rarefied gas flows: a brief review. IMA J. Appl.Math., 76 (5) (2011), 672–697. [CrossRef] [MathSciNet] [Google Scholar]
  48. P. Taheri, A.S. Rana, M. Torrilhon, H. Struchtrup. Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics. Continuum Mechanics and Thermodynamics, 21 (6) (2009), 423–443. [CrossRef] [MathSciNet] [Google Scholar]
  49. S. Takata, H. Funagane. Singular behavior of a rarefied gas on a plane boundary. J. Fluid. Mechanics, 717 (2013), 30–47. [CrossRef] [Google Scholar]
  50. S. Ukai, K. Asano. The Euler limit and initial layers of the nonlinear Boltzmann equation. Hokkaido Math. J., 12 (1983), 311–332. [MathSciNet] [Google Scholar]
  51. J.D. van der Waals. Theorie thermodynamique de la capillarite, dans l’hypothese d’une variation continue de densite. Archives Neerlandaises des Sciences exactes et naturelles, 28 (1895), 121–209. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.