stability for systems of hyperbolic conservation laws
Authors:
TaiPing Liu and Tong Yang
Journal:
J. Amer. Math. Soc. 12 (1999), 729774
MSC (1991):
Primary 35L67, 76L05; Secondary 35L65, 35A05
Published electronically:
April 13, 1999
MathSciNet review:
1646841
Fulltext PDF Free Access
Abstract 
References 
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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]
Alberto
Bressan and Rinaldo
M. Colombo, The semigroup generated by 2×2 conservation
laws, Arch. Rational Mech. Anal. 133 (1995),
no. 1, 1–75. MR 1367356
(96m:35198), http://dx.doi.org/10.1007/BF00375350
 [2]
Alberto
Bressan, A locally contractive metric for systems of conservation
laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22
(1995), no. 1, 109–135. MR 1315352
(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]
Constantine
M. Dafermos, Polygonal approximations of solutions of the initial
value problem for a conservation law, J. Math. Anal. Appl.
38 (1972), 33–41. MR 0303068
(46 #2210)
 [6]
C.
M. Dafermos, Entropy and the stability of classical solutions of
hyperbolic systems of conservation laws, Recent mathematical methods
in nonlinear wave propagation (Montecatini Terme, 1994) Lecture Notes in
Math., vol. 1640, Springer, Berlin, 1996, pp. 48–69. MR 1600904
(99a:35169), http://dx.doi.org/10.1007/BFb0093706
 [7]
R. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), 137188. MR 80:35119
 [8]
Ronald
J. DiPerna, Global existence of solutions to nonlinear hyperbolic
systems of conservation laws, J. Differential Equations
20 (1976), no. 1, 187–212. MR 0404872
(53 #8672)
 [9]
James
Glimm, Solutions in the large for nonlinear hyperbolic systems of
equations, Comm. Pure Appl. Math. 18 (1965),
697–715. MR 0194770
(33 #2976)
 [10]
J.
Glimm and P.
D. Lax, Decay of solutions of systems of
hyperbolic conservation laws, Bull. Amer. Math.
Soc. 73 (1967), 105.
MR
0204826 (34 #4662), http://dx.doi.org/10.1090/S000299041967116665
 [11]
P.
D. Lax, Hyperbolic systems of conservation laws. II, Comm.
Pure Appl. Math. 10 (1957), 537–566. MR 0093653
(20 #176)
 [12]
Peter
Lax, Shock waves and entropy, Contributions to nonlinear
functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin,
Madison, Wis., 1971) Academic Press, New York, 1971,
pp. 603–634. MR 0393870
(52 #14677)
 [13]
Philippe
LeFloch and Zhou
Ping Xin, Uniqueness via the adjoint problems for systems of
conservation laws, Comm. Pure Appl. Math. 46 (1993),
no. 11, 1499–1533. MR 1239319
(94h:35153), http://dx.doi.org/10.1002/cpa.3160461103
 [14]
Tai
Ping Liu, The deterministic version of the Glimm scheme, Comm.
Math. Phys. 57 (1977), no. 2, 135–148. MR 0470508
(57 #10259)
 [15]
Tai
Ping Liu, Uniqueness of weak solutions of the Cauchy problem for
general 2×2 conservation laws, J. Differential Equations
20 (1976), no. 2, 369–388. MR 0393871
(52 #14678)
 [16]
TaiPing
Liu and Tong
Yang, Uniform 𝐿₁ boundedness of solutions of
hyperbolic conservation laws, Methods Appl. Anal. 4
(1997), no. 3, 339–355. MR 1487836
(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.
A. Oleĭnik, On the uniqueness of the generalized solution of
the Cauchy problem for a nonlinear system of equations occurring in
mechanics, Uspehi Mat. Nauk (N.S.) 12 (1957),
no. 6(78), 169–176 (Russian). MR 0094543
(20 #1057)
 [20]
Joel
Smoller, Shock waves and reactiondiffusion equations,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Science], vol. 258, SpringerVerlag, New YorkBerlin,
1983. MR
688146 (84d:35002)
 [21]
Blake
Temple, No 𝐿₁contractive
metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), no. 2, 471–480. MR 776388
(86h:35084), http://dx.doi.org/10.1090/S00029947198507763885
 [1]
 A. Bressan, R.M. Colombo, The semigroup generated by conservation laws, Arch. Rational Mech. Anal. 133 (1995), 175. 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 Matematicheserie 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), 3341. 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), 137188. MR 80:35119
 [8]
 R. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Diff. Equa. 20 (1976), 187212. MR 53:8672
 [9]
 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715. 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), 537566. MR 20:176
 [12]
 P.D. Lax, Shock waves and entropy, Contribution to Nonlinear Functional Analysis, ed. E. Zarantonello, Academic Press, N.Y. 1971, 603634. 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, 14991533 (1993). MR 94h:35153
 [14]
 T.P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1975), 135148. 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), 369388. MR 52:14678
 [16]
 T.P. Liu, T. Yang, Uniform boundedness of solutions of hyperbolic conservation laws, Methods and Appl. Anal. 4 (1997), 339355. 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), 169176. (Russian) MR 20:1057
 [20]
 J. Smoller, Shock waves and reactiondiffusion equations, SpringerVerlag, New York, 1982. MR 84d:35002
 [21]
 B. Temple, No contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 471480. MR 86h:35084
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Additional Information
TaiPing Liu
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 943052060
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:
http://dx.doi.org/10.1090/S0894034799002921
PII:
S 08940347(99)002921
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 DMS9623025
The second author’s research was supported in part by the RGC Competitive Earmarked Research Grant 9040290
Article copyright:
© Copyright 1999
American Mathematical Society
