Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

Archive for Rational Mechanics and Analysis

Volumn 169, Issue 2, 2003, Pages 89-117

  Hanouzet, B ^a   Natalini, R ^b  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b CNR   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0141462542     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00205-003-0257-6     Document Type: Article

References (42)

1. AREGBA-DRIOLLET, D., NATALINI, R.: Convergence of relaxation schemes for conservation laws. Appl. Anal. 61, 163-193 (1996)
2. AREGBA-DRIOLLET, D., NATALINI, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973-2004 (2000) (electronic)
3. BIANCHINI, S.: A Glimm type functional for a special Jin-Xin relaxation model. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 19-42 (2001)
4. BOILLAT, G.: Sur l'Existence et la Recherche d'Équations de Conservation Supplémentaires pour les Systèmes Hyperboliques. C.R. Acad. Sci. Paris A 278, 909-912 (1974)
5. BOILLAT, G., RUGGERI, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rational Mech. Anal. 137, 305-320 (1997)
6. BOILLAT, G., RUGGERI, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 10, 285-292 (1998)
7. BOUCHUT, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation law. J. Statist. Phys. 113-170 (1999)
8. 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)
9. DAFERMOS, C.M.: A system of hyperbolic conservation laws with frictional damping. Z. Angew. Math. Phys. 46 (1995), no. Special Issue, S294-S307, Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids
10. DAFERMOS, C.M.: Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin, xvi+443 pp., 2000
11. FRIEDRICHS, K.O., LAX, P.D.: Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. U.S.A. 68, 1686-1688 (1971)
12. GIOVANGIGLI, V., MASSOT, M.: Asymptotic Stability of Equilibrium States for Multi-component Reacting Flows. Math. Models Meth. Appl. Sci. 2, 251-297 (1998)
13. GIOVANGIGLI, V., MASSOT, M.: The Local Cauchy Problem for Multicomponent Reactive Flows in Full Vibrational Nonequilibrium. Math. Meth. Appl. Sci. 1415-1439 (1998)
14. GODUNOV, S.K.: An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR 139, 521-523 (1961)
15. HANOUZET, B., HUYNH, Ph.: Approximation par relaxation d'un système de Maxwell non linéaire. C. R. Acad. Sci. Paris Sér. I Math. 330, 193-198 (2000)
16. HSIAO, L., LIU, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Physics 143, 599-605 (1992)
17. HSIAO, L., PAN, R.: Zero relaxation limit to centered rarefaction waves for a rate-type viscoelastic system. J. Differential Equations 157, 20-40 (1999)
18. HSIAO, L., SERRE, D.: Global existence of solutions for the system of compressible adiabatic flow through porous media. SIAM J. Math. Anal. 27, 70-77 (1996)
19. HUYNH, Ph.: Étude théorique et numérique de modèles de Kerr. Ph.D. thesis, Université Bordeaux I, 1999
20. JIN, S., XIN, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235-276 (1995)
21. JUDKINS, J.B., ZIOLKOWSKI, R-W.: Full wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a non linear Kerr medium exhibiting a finite response time. J. Opt. Soc. Am. B 45, 375-391 (1997)
22. KAWASHIMA, S.: Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. Roy. Soc. Edinburgh Sect. A 106, 169-194 (1987)
23. LI, T.T.: Global classical solutions for quasilinear hyperbolic systems. Masson, Paris; John Wiley & Sons, Ltd., Chichester, x+315 pp., 1994
24. LIU, T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153-175 (1987)
25. LI, H., PAN, R.: Zero relaxation limit for piecewise smooth solutions to a rate-type viscoelastic system in the presence of shocks. J. Math. Anal. Appl. 252, 298-324 (2000)
26. MAJDA, A.: Compressible fluid flow and systems of conservation laws in several space dimensions. Appl. Math. Sci., vol. 53, Springer-Verlag, 1978
27. MASSOT, M.: Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete and Continuous Dynamical Systems - Series B 2, 433-456 (2002)
28. MÜLLER, I., RUGGERI, T.: Rational extended thermodynamics. Second ed., Springer-Verlag, New York, 1998, With supplementary chapters by H. Struchtrup and Wolf Weiss
29. NATALINI, R.: Convergence to equilibrium for the relaxation approximation of conservation laws. Commun. Pure Appl. Math. 49, 795-823 (1996)
30. NATALINI, R.: A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Differential Equations 148, 292-317 (1998)
31. NATALINI, R.: Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws (Aachen, 1997), Chapman & Hall/CRC, Boca Raton, FL, pp. 128-198, 1999
32. NISHIDA, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics, Département de Mathématique, Université de Paris-Sud, Orsay, Publications Mathématiques d'Orsay, No. 78-02, 1978
33. RUGGERI, T., SERRE, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. To appear in Quart Appl Math.
34. SERRE, D.: Relaxations semi-linéaire et cinétique des systèmes de lois de conservation. Ann. Inst. H. Poincaré Anal Non Linéaire 17, 169-192 (2000)
35. SHIZUTA, Y. KAWASHIMA, S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249-275 (1985)
36. SIDERIS, T., THOMASES, B., WANG, D.: Decay and singularities of solutions of the three-dimensional Euler equations with damping. To appear in Comm. PDE.
37. SULICIU, I.: On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure. Internat. J. Engng. Sci. 28, 829-841 (1990)
38. TZAVARAS, A.E.: Materials with internal variables and relaxation to conservation laws. Arch. Rational. Mech. Anal. 146, 129-155 (1999)
39. WHITHAM, G.B.: Linear and nonlinear waves. A Wiley & Sons, New York, 1974
40. YONG, W.-A.: Basic aspects of hyperbolic relaxation systems. Advances in the theory of shock waves, Birkhäuser Boston, Boston, MA, pp. 259-305, 2001
41. YONG, W.-A.: An entropy condition and global existence result for hyperbolic balance laws. Preprint, December 2002
42. ZENG, Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Rational. Mech. Anal. 150, 225-279 (1999)