Boltzmann diffusive limit beyond the Navier-Stokes approximation

Communications on Pure and Applied Mathematics

Volumn 59, Issue 5, 2006, Pages 626-687

  Guo, Yan ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33645967200     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20121     Document Type: Article

References (38)

1. Bardos, C.; Golse, F.; Levermore, D. C. Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statist. Phys. 63 (1991), no. 1-2, 323-344.
2. Bardos, C.; Golse, F.; Levermore, D. C. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46 (1993), no. 5, 667-753.
3. Bardos, C.; Golse, F.; Levermore, D. C. The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153 (2000), no. 3, 177-204.
4. Bardos, C.; Ukai, S. The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models Methods Appl. Sci. 1 (1991), no. 2, 235-257.
5. Bobylëv, A. V. On the Chapman-Enskog and Grad methods for solving the Boltzmann equation. (Russian) Dokl. Akad. Nauk SSSR 262 (1982), no. 1, 71-75.
6. Caflisch, R. E. The Boltzmann equation with a soft potential. I, II. Commun. Math. Phys. 74 (1980), no. 2, 71-109.
7. Caflisch, R. E. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33 (1980), no. 5, 651-666.
8. Cercignani, C. The Boltzmann equation and its applications. Applied Mathematical Sciences, 67. Springer, New York, 1988.
9. Cercignani, C.; Illner, R.; Pulvirenti, M. The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer, New York, 1994.
10. Chapman, S.; Cowling, T. G. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. 3rd ed. Cambridge University Press, London, 1970.
11. De Masi, A.; Esposito, R.; Lebowitz, J. L. Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math. 42 (1989), no. 8, 1189-1214.
12. DiPerna, R. J.; Lions, P.-L. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2) 130 (1989), no. 2, 321-366.
13. Enskog, D. The kinetic theory of phenomena in fairly rare gas. Dissertation, Uppsala, 1917.
14. Golse, F.; Levermore, D. C. Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs. Comm. Pure Appl. Math. 55 (2002), no. 3, 336-393.
15. Golse, F.; Saint-Raymond, L. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155 (2004), no. 1, 81-161.
16. Golse, F.; Saint-Raymond, L. Lecture, Fields Institute for Research in Mathematical Sciences, Toronto, 2004.
17. Grad, H. Principles of the kinetic theory of gases. Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12. Thermodynamik der Gase, 205-294. Springer, Berlin-Göttingen-Heidelberg, 1958.
18. Grad, H. Asymptotic theory of the Boltzmann equation. II. Rarefied gas dynamics (Proceedings of the 3rd International Symposium, Palais de l'UNESCO, Paris, 1962), Vol. I, 26-59. Academic Press, New York, 1963.
19. Guo, Y. The Landau equation in a periodic box. Comm. Math. Phys. 231 (2002), no. 3, 391-434.
20. Guo, Y. The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55 (2002), no. 9, 1104-1135.
21. Guo, Y. Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169 (2003), no. 4, 305-353.
22. Guo, Y. The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153 (2003), no. 3, 593-630.
23. Guo, Y. The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53 (2004), no. 4, 1081-1094.
24. Hilbert, D. Begründung der kinetischen Gasstheorie. Math. Ann. 22 (1912), 561-577.
25. Lachowicz, M. On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci. 9 (1987), no. 3, 342-366.
26. Levermore, D.; Masmoudi, N. From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Preprint.
27. Lions, P.-L.; Masmoudi, N. From the Boltzmann equations to the equations of incompressible fluid mechanics. I. Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173-193.
28. Lions, P.-L.; Masmoudi, N. From the Boltzmann equations to the equations of incompressible fluid mechanics. II. Arch. Ration. Mech. Anal. 158 (2001), no. 3, 195-211.
29. Masmoudi, N. Hydrodynamic limits of the Boltzmann equation: recent developments. Bal. Soc. Esp. Mat. Apl. 26 (2003) 57-78.
30. Masmoudi, N.; Saint-Raymond, L. From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Comm. Pure Appl. Math. 56 (2003), no. 9, 1263-1293.
31. Saint-Raymond, L. From the BGK model to the Navier-Stokes equations. Ann. Sci. École Norm. Sup. 4 36 (2003), no. 2, 271-317.
32. Slemrod, M. Constitutive relations for monatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion. Arch. Ration. Mech. Anal. 150 (1999), no. 1, 1-22.
33. Strain, R. M.; Guo, Y. Stability of the relativistic Maxwellian in a collisional plasma. Comm. Math. Phys. 251 (2004), no. 2, 263-320.
34. Strain, R. M.; Guo, Y. Exponential decay for soft potentials. Preprint, 2005.
35. Strain, R. M.; Guo, Y. Almost exponential decay near Maxwellians. Comm. Partial Differential Equations, in press.
36. Ukai, S. Solutions of the Boltzmann equation. Patterns and waves, 37-96. Studies in Mathematics and Its Applications, 18. North-Holland. Amsterdam, 1986.
37. Ukai, S.; Asano, K. The Euler limit and initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J. 12 (1983), no. 3, part 1, 311-332.
38. Villani, C. A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, 71-305. North-Holland, Amsterdam, 2002.