Large-time behavior of entropy solutions of conservation laws

Journal of Differential Equations

Volumn 152, Issue 2, 1999, Pages 308-357

  Chen, Gui Qiang ^a   Frid, Hermano ^b  

^a NORTHWESTERN UNIVERSITY   (United States)

^b FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

Author keywords

Compactness; Conservation laws; Discontinuous entropy solutions; Large time behavior; Riemann problem; Scaling sequence; Uniqueness

Indexed keywords

EID: 0033095936     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1998.3527     Document Type: Article

References (49)

1. Anzellotti G. Pairings between measures and functions and compensated compactness. Ann. Mat. Pura Appl. 135:1983;293-318.
2. Chen G.-Q. The method of quasidecoupling for discontinuous solutions to conservation laws. Arch. Rational Mech. Anal. 121:1992;131-185.
3. Chen G.-Q., Dafermos C. M. The vanishing viscosity method in one-dimensional thermoelasticity. Trans. Amer. Math. Soc. 347:1995;531-541.
4. Chen G.-Q., Frid H. Asymptotic decay of solutions of conservation laws. C. R. Acad. Sci. Paris Sér. I. 323:1996;257-262.
5. Chen G.-Q., Frid H. Asymptotic stability of Riemann waves for conservation laws. Z. Angew. Math. Phys. 48:1997;30-44.
6. G.-Q. Chen and H. Frid, Divergence-measure Fields and Conservation Laws, submitted for publication.
7. G.-Q. Chen and H. Frid, Asymptotic Stability and Decay of Solutions of Conservation Laws, preprint, Northwestern University, May, 1996.
8. Dafermos C. M. Generalized characteristics in hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 107:1989;127-155.
9. Dafermos C. M., Geng X. Generalized characteristics, uniqueness, and regularity of solutions in a hyperbolic system of conservation laws. Ann. Inst. H. Poincaré, Anal. Non Linéaire. 8:1991;231-269.
10. DiPerna R. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82:1983;27-70.
11. DiPerna R. Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88:1985;223-270.
12. DiPerna R. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28:1979;137-188.
13. Evans L. C., Gariepy R. F. Lecture Notes on Measure Theory and Fine Properties of Functions. 1992;CRC Press, Boca Raton.
14. Filippov A. F. Differential equations with discontinuous right-hand side. Mat. Sb. 51:1960;99-128.
15. Friedman A. Partial Differential Equations of Parabolic Type. 1964;Prentice-Hall.
16. Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18:1965;95-105.
17. Glimm J., Lax P. D. Decay of solutions of nonlinear hyperbolic conservation laws. Mem. Amer. Math. Soc. 101:1970.
18. Goodman J. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95:1986;325-344.
19. Heibig A. Existence and uniqueness of solutions for some hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 126:1994;79-101.
20. Heibig A. Régularité des solutions du problème de Riemann. Comm. Partial Differential Equations. 15:1990;693-709.
21. Hoff D., Smoller J. Global existence for systems of parabolic conservation laws in several space variables. J. Differential Equations. 68:1987;210-220.
22. Il'in A. M., Oleinik O. A. Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of time. Amer. Math. Soc. Transl. Ser. 2. 42:1964;19-23.
23. James F., Peng Y.-J., Perthame B. Kinetic formulation for chromatography and some other hyperbolic systems. J. Math. Pures Appl. 74:1995;367-385.
24. Kawashima S., Matsumura A. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys. 101:1985;97-127.
25. Kruzkov S. N. First-order quasilinear equations in several independent variables. Math. USSR Sb. 10:1970;217-243.
26. Ladyzenskaja O. A., Solonnikov V. A., Ural'tzeva N. N. Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Mono. 23:1968;Amer. Math. Soc. Providence.
27. Lax P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS. 11:1973;SIAM, Philadelphia.
28. Lax P. D. Shock waves and entropy. Contributions to Functional Analysis. 1971;Academic Press, New York. p. 603-634.
29. Leveque R. J., Temple B. Stability of Godunov's method for a class of 2×2 systems of conservation laws. Trans. Amer. Math. Soc. 288:1985;115-123.
30. Liu T.-P. Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 328:1986.
31. Liu T.-P. Asymptotic behavior of solutions of general systems of nonlinear hyperbolic conservation laws. Indiana Univ. Math. J. 27:1978;211-253.
32. Liu T.-P. Linear and nonlinear large time behavior of solutions of general systems of conservation laws. Comm. Pure Appl. Math. 30:1977;767-796.
33. Liu T.-P., Zeng Y. Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 599:1997.
34. Matsumura A., Nishihara K. On a stability of traveling wave solution of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 3:1986;1-13.
35. Matsumura A., Nishihara K. Global stability of the rarefacton wave of a one-dimensional model system for compressible viscous gas. Comm. Math. Phys. 144:1992;325-335.
36. Rhee H.-K., Aris R., Amundson N. R. On the theory of multicomponent chromatography. Philos. Trans. Roy. Soc. London Ser. A. 267:1970;419-455.
37. Schwartz L. Théorie des Distributions. Actualités Sci. Indust. 1091, 1122:1950-1951;Hermann, Paris.
38. Serre D. Richness and the classification of quasilinear hyperbolic systems. Glimm J., Majda A. J., Multidimensional Hyperbolic Problems and Computations. IMA. 29:1991;315-333 Springer-Verlag, New York.
39. Serre D. Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservations. J. Differential Equations. 68:1987;137-169.
40. Serre D., Xiao L. Asymptotic behavior of large weak entropy solutions of the dampedp. J. Partial Differential Equations. 10:1997;355-368.
41. Smoller J. Shock Waves and Reaction-Diffusion Equations. 1983;Springer-Verlag, New York.
42. Sobolev S. L. Partial Differential Equations of Mathematical Physics. 1964;Pergamon, Toronto.
43. Szepessy A., Xin Z. Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal. 122:1993;53-103.
44. Szepessy A., Zumbrun K. Stability of rarefaction waves in viscous media. Arch. Rational Mech. Anal. 1996.
45. Tartar L. Compensated compactness and applications to partial differential equations. Knops R. J., Research Notes in Mathematics, Nonlinear Analysis and Mechanics. 1979;136-211 Pitman Press, New York.
46. Temple B. Systems of conservation laws with invariant submanifolds. Trans. Amer. Math. Soc. 280:1983;781-795.
47. Volpert A. I. The space BV and quasilinear equations. Mat. Sb. (N.S.). 73:1967;255-302.
48. Weinberger H. F. Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire. 7:1990;407-425.
49. ∞ for multidimensional conversation laws, in "Advances in Nonlinear Partial Differential Equations and Related Areas" (G.-Q. Chen et al., Eds.), pp. 28-44, World Scientific, Singapore, 1998.