The stationary nonlinear boltzmann equation in a couette setting with multiple, isolated Lq -solutions and hydrodynamic limits

Journal of Statistical Physics

Volumn 118, Issue 5-6, 2005, Pages 849-881

  Arkeryd, L ^a   Nouri, A ^b  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Asymptotic techniques; Couette setting; Hydrodynamic limit; Isolated solutions; Multiple solutions; Stationary nonlinear Boltzmann equation

Indexed keywords

EID: 17444401720     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-004-2708-3     Document Type: Article

References (30)

1. K. Aoki, S. Takata, and S. Taguchi, Vapor flows with evaporation and condensation in the continuum limit: effect of a trace of noncondensable gas, preprint 2002.
2. L. Arkeryd and A. Nouri, The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27:533-556 (1998).
3. L. Arkeryd and A. Nouri, On the stationary Povzner equation in three space variables, J. Math. Kyoto Univ. 39:115-153 (1999).
4. 1 solutions to the stationary Boltzmann equation in a slab, Ann. Fac. Sci. Toulouse Math. 9:375-413 (2000).
5. n with given indata, Ann. Scuola Norm. Sup. Pisa 31:1-28 (2002).
6. L. Arkeryd and A. Nouri, A large data existence result for the stationary Boltzmann equation in a cylindrical geometry, to appear in Arkiv för Matematik, (2005).
7. C. Bardos, E. Caflish, and B. Nicolaenko, Thermal layer solutions of the Boltzmann equation, Progr. Phys. 10:235-251 (1985).
8. C. Bardos, E. Caflish, and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard spere gas, Comm. Pure Appl. Math. 39:323-352 (1986).
9. Cercignani, C. Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1990).
10. L. Desvillettes, Sur quelques hypothèses nécessaires à l'obtention du développement de Chapman-Enskog, preprint 1994.
11. R. Esposito, J. L. Lebowitz, and R. Marra, Hydrodynamic limit of the stationary Boltzmann equation in a slab, Comm. Math. Phys. 160:49-80 (1994).
12. R. Esposito, J. L. Lebowitz, R. Marra, The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, J. Stat. Phys. 78:389-412 (1995).
13. R. Ellis and M. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pure. Appl. 54:125-156 (1975).
14. F. Golse and F. Poupaud, Stationary solutions of the linearized Boltzmann equation in a half space, Math. Meth. Appl. Sci. 11:483-502 (1989).
15. H. Grad, Asymptotic theory of the Boltzmann equation I, Phys. Fluids 6:147-181 (1963).
16. H. Grad, Asymptotic theory of the Boltzmann equation II, in Rarefied Gas Dynamics, (Acad. Press, NY, 1963), pp. 26-59.
17. H. Grad, High frequency sound recording according to Boltzmann equation, SIAM J. Appl. Math. 14:935-955 (1966).
18. J. P. Guiraud, Problème aux limites intérieur pour l'équation de Boltzmann en régime stationnaire, faiblement non linéaire, J. Mécanique 11:183-231 (1972).
19. A. Heintz, On solvability of boundary value problems for the non linear Boltzmann equation in a bounded domain, in Molecular Gas Dynamics, Aerodynamics of Rarefied Gases, Vol. 10, (Leningrad, 1980) (in Russian), pp. 16-24.
20. R. Illner, J. Struckmeier, Boundary value problems for the steady Boltzmann equation, J. Stat. Phys. 85:427-454 (1996).
21. M.N. Kogan, Rarefied Gas Dynamics, (Plenum, NY, 1969).
22. N. Maslova, The solvability of internal stationary problems for Boltzmann's equation at large Knudsen numbers, USSR Comp. Math. Math. Phys. 17:194-204 (1977).
23. N. Maslova, Nonlinear evolution equations, Kinetic approach, in Series on Advances in Mathematics for Applied Sciences, Vol. 10 (World Scientific, 1993).
24. Y. P. Pao, Boundary value problems for the linearized and weakly nonlinear Boltzmann equation, J. Math. Phys. 8:1893-1898 (1967).
25. Y. Sone, Kinetic Theory and Fluid Dynamics, (Birkhäuser Boston, 2002).
26. Y. Sone and T. Doi, Analytical study of bifurcation of a flow of a gas between coaxial circular cylinders with evaporation and condensation, Phys. Fluids 12:2639-2660 (2000).
27. Y. Sone and T. Doi, Bifurcation of a flow of a gas between rotating coaxial circular cylinders with evaporation and condensation, Rarefied Gas Dynamics, AIAA New York, 2002, 646-653.
28. Y. Sone and T. Doi, Bifurcation of and ghost effect on the temperature field in the Bénard problem of a gas in the continuum limit, Phys. Fluids 15:1405-1423 (2003).
29. S. Ukai and K. Asano, Steady solutions of the Boltzmann equation for a gas flow past an obstacle; I existence, Arch. Rat. Mech. Anal. 84:249-291 (1983).
30. S. Ukai, T. Yang, and S. Yu, Non-linear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 236:373-393 (2003).