stability for systems

of hyperbolic conservation laws

Authors:
Tai-Ping Liu and Tong Yang

Journal:
J. Amer. Math. Soc. **12** (1999), 729-774

MSC (1991):
Primary 35L67, 76L05; Secondary 35L65, 35A05

DOI:
https://doi.org/10.1090/S0894-0347-99-00292-1

Published electronically:
April 13, 1999

MathSciNet review:
1646841

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the evolution of the distance of solutions for systems of hyperbolic conservation laws. For the approximate solutions constructed by Glimm's scheme with the aid of the wave tracing method, we introduce a nonlinear functional which is equivalent to the distance between solutions, nonincreasing in time, and expressed explicitly in terms of the wave patterns of the solutions. This functional reveals the nonlinear mechanism of wave interactions and coupling which affect the topology.

**[1]**A. Bressan, R.M. Colombo, The semigroup generated by conservation laws, Arch. Rational Mech. Anal. 133 (1995), 1-75. MR**96m:35198****[2]**A. Bressan, A locally contractive metric for systems of conservation laws, Estratto dagli Annali Della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche-serie IV, vol. XXII. Fasc. 1 (1995). MR**96k:35113****[3]**A. Bressan, Lecture Notes on System of Conservation Laws, S.I.S.S.A., Trieste 1995.**[4]**A. Bressan, P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, preprint S.I.S.S.A., Trieste 1996.**[5]**C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33-41. MR**46:2210****[6]**C.M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, Lecture Notes in Mathematics, Editor: T. Ruggeri, Montecatini Terme, 1994, Springer. MR**99a:35169****[7]**R. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), 137-188. MR**80:35119****[8]**R. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Diff. Equa. 20 (1976), 187-212. MR**53:8672****[9]**J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR**33:2976****[10]**J. Glimm, P. Lax, Decay of solutions of systems of hyperbolic conservation laws, Memoirs Amer. Math. Soc., 101, 1970. MR**34:4662****[11]**P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR**20:176****[12]**P.D. Lax, Shock waves and entropy, Contribution to Nonlinear Functional Analysis, ed. E. Zarantonello, Academic Press, N.Y. 1971, 603-634. MR**52:14677****[13]**P. LeFloch, Z. P. Xin, Uniqueness via the adjoint problems for systems of conservation laws, Comm. Pure Appl. Math., Vol. XLVI, 1499-1533 (1993). MR**94h:35153****[14]**T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1975), 135-148. MR**57:10259****[15]**T.-P. Liu, Uniqueness of weak solutions of the Cauchy problem for general 2x2 conservation laws, J. Diff. Equa. 20 (1976), 369-388. MR**52:14678****[16]**T.-P. Liu, T. Yang, Uniform boundedness of solutions of hyperbolic conservation laws, Methods and Appl. Anal. 4 (1997), 339-355. MR**98m:35127****[17]**T.-P. Liu, T. Yang, A generalised enropy for scalar conservation laws, Comm. Pure Appl. Math. (to appear).**[18]**T.-P. Liu, T. Yang, stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. Jour.**[19]**O. Oleinik, On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics, Usp. Mat. Nauk (N.S.) 12 (1957), 169-176. (Russian) MR**20:1057****[20]**J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1982. MR**84d:35002****[21]**B. Temple, No -contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 471-480. MR**86h:35084**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
35L67,
76L05,
35L65,
35A05

Retrieve articles in all journals with MSC (1991): 35L67, 76L05, 35L65, 35A05

Additional Information

**Tai-Ping Liu**

Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305-2060

Email:
liu@math.stanford.edu

**Tong Yang**

Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Email:
matyang@cityu.edu.hk

DOI:
https://doi.org/10.1090/S0894-0347-99-00292-1

Received by editor(s):
March 8, 1998

Received by editor(s) in revised form:
September 9, 1998

Published electronically:
April 13, 1999

Additional Notes:
The first author’s research was supported in part by NSF Grant DMS-9623025

The second author’s research was supported in part by the RGC Competitive Earmarked Research Grant 9040290

Article copyright:
© Copyright 1999
American Mathematical Society