Lyapunov functional for the Enskog-Boltzmann equation

Indiana University Mathematics Journal

Volumn 54, Issue 4, 2005, Pages 997-1014

  Ha, Seung Yeal ^a  

^a Seoul National University   (South Korea)

Author keywords

Enskog Boltzmann equation; L1 stability; Lyapunov functional

Indexed keywords

EID: 26444588729     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2005.54.2555     Document Type: Article

References (36)

1. LEIF ARKERYD, On the Enskog equation with large initial data, SIAM J. Math. Anal. 21 (1990), 631-646. MR1046792 (91e:76094)
2. 1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. Statist. Phys. 59 (1990), 845-867. MR1063185 (91m:82114)
3. _, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. Partial Differential Equations 14 (1989), 1071-1089. MR1017064 (91m:82113)
4. NICOLA BELLOMO and MIROSLAW LACHOWICZ, On the asymptotic equivalence between the Enskog and the Boltzmann equations, J. Statist. Phys. 51 (1988), 233-247. MR952754 (89h:82016)
5. _, Kinetic equations for dense gases - a review of physical and mathematical results, Internat. J. Modern Phys. B 1 (1987), 1193-1205. MR932350 (89b:82047)
6. NICOLA BELLOMO, MIROSLAW LACHOWICZ, JACEK POLEWCZAK, and GIUSEPPE TOSCANI, Mathematical Topics in Nonlinear Kinetic Theory. II, Series on Advances in Mathematics for Applied Sciences, vol. 1, World Scientific Publishing Co. Inc., Singapore, Teaneck, NJ, 1991, ISBN 981-02-0447-7. MR1119780 (92m:82110)
7. NICOLA BELLOMO, JACEK POLEWCZAK, and LUIGI PREZIOSI, Liapunov functionals and qualitative behaviour of the solution to the nonlinear Enskog equation, Developments in Partial Differential Equations and Applications to Mathematical Physics (Ferrara, 1991), Plenum, New York, 1992, pp. 1-13. MR1213918 (94g:82040)
8. JEAN-MICHEL BONY, Solutions globales bornèes pour les modèles discrets de l'équation de Boltzmann, en dimension 1 d'espace, Journées "Équations aux dérivées partielles" (Saint Jean de Monts, 1987), École Polytech., Palaiseau, 1987, p. Exp. No. XVI, 10. MR920011 (89b:35141) (French)
9. _, Existence globale à données de Cauchy petites pour les modèles discrets de l'équation de Boltzmann, Comm. Partial Differential Equations 16 (1991), 533-545. MR1113097 (92m:35263) (French)
10. 1 stability estimates for nxn conservation laws, Arch. Ration. Mech. Anal. 149 (1999), 1-22. MR1723032 (2000g:35139)
11. CARLO CERCIGNANI, Weak solutions of the Boltzmann equation and energy conservation, Appl. Math. Lett. 8 (1995), 53-59, See also: Errata, Appl. Math. Lett. 8 (1995), pp. 95-99. MR1357252 (97a:82072a)
12. _, A remarkable estimate for the solutions of the Boltzmann equation, Appl. Math. Lett. 5 (1992), 59-62. MR1345903 (96e:82073)
13. 1, J. Statist. Phys. 51 (1988), 291-297. MR952757 (89i:82032)
14. _, Existence of global solutions for the space inhomogeneous Enskog equation, Transport Theory Statist. Phys. 16 (1987), 213-221. MR898479 (88d:82066)
15. CARLO CERCIGNANI and REINHARD ILLNER, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions, Arch. Rational Mech. Anal. 134 (1996), 1-16. MR1392307 (97k:82037)
16. DAVID ENSKOG, Kinetiche Theorie der Wärmeleitung, Reibung und Selbstdiffusion in gewissen verdichteten Gasen und Fluβigkeiten, Kungl. Sv. Vetenskapsakademiens Handl. 63 (1922), 3-44,
17. English Transl. in Brush, S.G., Kinetic Theory, vol 3, Pergamon, New York 1972.
18. MARIA J. ESTEBAN and BENOÎT PERTH AME, On the modified Enskog equation for elastic and inelastic collisions. Models with spin, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 289-308. MR1127928 (92i:82094) (English, with French summary)
19. MIKHAIL FELDMAN and SEUNG-YEAL HA, Nonlinear functional of multi-D discrete velocity Boltzmann equations, J. Statist. Phys. 114 (2004), 1015-1033. MR2035636 (2004m:82121)
20. JAMES GLIMM, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR0194770 (33 #2976)
21. 1 stability of the Boltzmann equation for a hard sphere model, Arch. Rational Mech. Anal. 173 (2004), 279-296.
22. 1 stability of the Boltzmann equation for the Maxwellian molecules, Nonlinearity 18 (2005), 981-1001.
23. 1 stability of multi-dimensional discrete-velocity Boltzmann equations, Arch. Ration. Mech. Anal. 171 (2004), 25-42. MR2029529 (2005e:82096)
24. 1 stability estimate for a one-dimensional Boltzmann equation with inelastic collisions, J. Differential Equations 190 (2003), 621-642. MR1971148 (2004h:35035)
25. 1-stability for discrete velocity Boltzmann equations, Comm. Math. Phys. 239 (2003), 65-92. MR1997116 (2004g:82126)
26. DAVID K. HUFFMAN and CHARLES F. CURTISS, Kinetic theory of dense gases. IV. Transport virial coefficients, Phys. Fluids 8 (1965), 667-682. MR0195525 (33 #3725)
27. MIROSLAW LACHOWICZ, On the local existence and uniqueness of solution of initial value problem for the Enskog equation, Bull. Polish Acad. Sci. Math. 31 (1983), 89-97. MR717130 (84k:82101) (English, with Russian summary)
28. TAI-PING LIU and TONG YANG, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), 1553-1586. MR1711037 (2000m:35126)
29. 1 for the generalized Enskog equation, J. Statist. Phys. 59 (1990), 461-500. MR1049975 (91e:82059)
30. _, Global existence and asymptotic behavior for the nonlinear Enskog equation, SIAM J. Appl. Math. 49 (1989), 952-959. MR997931 (90g:35149)
31. 3, J. Statist. Phys. 56 (1989), 159-173. MR1003551 (90k:82050)
32. PIERRE RÉSIBOIS, H theorem for the (modified) nonlinear Enskog equation, Phys. Rev. Lett. 40 (1978), 1409-1411. MR0471846 (57 #11566)
33. PIERRE RÉSIBOIS and M. DE LEENER, Classical Kinetic Theory of Fluids, John Wiley, New York, 1977.
34. LUC C. TARTAR, Some existence theorems for semilinear hyperbolic systems in one space variable, MRC technical summary report, University of Wisconsin, 1980.
35. 3: some global existence, uniqueness and stability results, Comput. Math. Appl. 13 (1987), 851-859. MR897895 (88j:82022)
36. HENK VAN BEIJEREN and MATTHIEU H. ERNST, The modified Enskog equation, Physica 68 (1973), 437-456.