Weak stability in the global norm for systems of hyperbolic conservation laws
Author:
Blake Temple
Journal:
Trans. Amer. Math. Soc. 317 (1990), 673685
MSC:
Primary 35L65; Secondary 35B35
MathSciNet review:
948199
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Abstract: We prove that solutions for systems of two conservation laws which are generated by Glimm's method are weakly stable in the global norm. The method relies on a previous decay result of the author, together with a new estimate for the Lipschitz constant that relates solutions at different times. The estimate shows that this constant can be bounded by the supnorm of the solution, and is proved for any number of equations. The techniques do not rely on the existence of a family of entropies, and moreover the results would generalize immediately to more than two equations if one were to establish the stability of solutions in the supnorm for more than two equations.
 [1]
R.
Courant and K.
O. Friedrichs, Supersonic Flow and Shock Waves, Interscience
Publishers, Inc., New York, N. Y., 1948. MR 0029615
(10,637c)
 [2]
Ronald
J. DiPerna, Decay and asymptotic behavior of solutions to nonlinear
hyperbolic systems of conservation laws, Indiana Univ. Math. J.
24 (1974/75), no. 11, 1047–1071. MR 0410110
(53 #13860)
 [3]
James
Glimm, Solutions in the large for nonlinear hyperbolic systems of
equations, Comm. Pure Appl. Math. 18 (1965),
697–715. MR 0194770
(33 #2976)
 [4]
E. Isaacson, Global solution of a Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in enhanced oil recovery, J. Comput. Phys. (to appear).
 [5]
Eli
Isaacson and Blake
Temple, The structure of asymptotic states in a singular system of
conservation laws, Adv. in Appl. Math. 11 (1990),
no. 2, 205–219. MR 1053229
(91h:35205), http://dx.doi.org/10.1016/01968858(90)90009N
 [6]
E. Isaacson, D. Marchesin, D. Plohr, and B. Temple, The classification of solutions of quadratic Riemann problems (I), Joint MRC, PUC/RJ report, 1985.
 [7]
E. Isaacson and B. Temple, Examples and classification of nonstrictly hyperbolic systems of conservation laws, Abstracts Amer. Math. Soc. 6 (1985).
 [8]
Barbara
L. Keyfitz and Herbert
C. Kranzer, A system of nonstrictly hyperbolic conservation laws
arising in elasticity theory, Arch. Rational Mech. Anal.
72 (1979/80), no. 3, 219–241. MR 549642
(80k:35050), http://dx.doi.org/10.1007/BF00281590
 [9]
P.
D. Lax, Hyperbolic systems of conservation laws. II, Comm.
Pure Appl. Math. 10 (1957), 537–566. MR 0093653
(20 #176)
 [10]
Eduardo
H. Zarantonello (ed.), Contributions to nonlinear functional
analysis, Academic Press, New YorkLondon, 1971. Mathematics Research
Center, Publ. No. 27. MR 0366576
(51 #2823)
 [11]
Tai
Ping Liu, Invariants and asymptotic behavior of
solutions of a conservation law, Proc. Amer.
Math. Soc. 71 (1978), no. 2, 227–231. MR 500495
(81c:35085), http://dx.doi.org/10.1090/S00029939197805004957
 [12]
, Asymptotic behavior of solutions of general systems of nonlinear hyperbolic conservation laws, Indiana Univ. J. (to appear).
 [13]
, Decay to waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 585610.
 [14]
, Largetime behavior of solutions of initial and initialboundary value problems of general systems of hyperbolic conservation laws, Comm. Math. Phys. 57 (1977), 163177.
 [15]
D. G. Schaeffer and M. Shearer, The classification of systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, with Appendix by D. Marchesin, P. J. PaesLeme, D. G. Schaeffer, and M. Shearer, Duke University, preprint.
 [16]
D. Serre, Existence globale de solutions faibles sous une hypothese unilaterale pour un systeme hyperbolique non lineaire, Equipe d'Analyse Numerique, Lyon, SaintEtienne, July 1985.
 [17]
, Solutions a variation bornees pour certains systemes hyperboliques de lois de conservation, Equipe d'Analyse Numerique, Lyon, SaintEtienne, February 1985.
 [18]
M. Shearer, D. G. Schaeffer, D. Marchesin, and P. J. PaesLeme, Solution of the Riemann problem for a prototype system of nonstrictly hyperbolic conservation laws, Duke University, preprint.
 [19]
J. A. Smoller, Shock waves and reaction diffusion equations, SpringerVerlag, 1980.
 [20]
Blake
Temple, Global solution of the Cauchy problem for a class of
2×2\ nonstrictly hyperbolic conservation laws, Adv. in Appl.
Math. 3 (1982), no. 3, 335–375. MR 673246
(84f:35091), http://dx.doi.org/10.1016/S01968858(82)800109
 [21]
, Systems of conservation laws with coinciding shock and rarefaction curves, Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983.
 [22]
Blake
Temple, Decay with a rate for noncompactly
supported solutions of conservation laws, Trans. Amer. Math. Soc. 298 (1986), no. 1, 43–82. MR 857433
(87k:35163), http://dx.doi.org/10.1090/S00029947198608574336
 [23]
Blake
Temple, Degenerate systems of conservation laws, Nonstrictly
hyperbolic conservation laws (Anaheim, Calif., 1985) Contemp. Math.,
vol. 60, Amer. Math. Soc., Providence, RI, 1987,
pp. 125–133. MR 873538
(88b:35131), http://dx.doi.org/10.1090/conm/060/873538
 [24]
, Supnorm estimates in Glimm's method, preprint.
 [25]
, No contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc., 288 (1985).
 [1]
 R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Wiley, New York, 1948. MR 0029615 (10:637c)
 [2]
 R. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24 (1975), 10471071. MR 0410110 (53:13860)
 [3]
 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715. MR 0194770 (33:2976)
 [4]
 E. Isaacson, Global solution of a Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in enhanced oil recovery, J. Comput. Phys. (to appear).
 [5]
 E. Isaacson and B. Temple, The structure of asymptotic states in a singular system of conservation laws, preprint. MR 1053229 (91h:35205)
 [6]
 E. Isaacson, D. Marchesin, D. Plohr, and B. Temple, The classification of solutions of quadratic Riemann problems (I), Joint MRC, PUC/RJ report, 1985.
 [7]
 E. Isaacson and B. Temple, Examples and classification of nonstrictly hyperbolic systems of conservation laws, Abstracts Amer. Math. Soc. 6 (1985).
 [8]
 B. Keyfitz and H. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1980). MR 549642 (80k:35050)
 [9]
 P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 19 (1957), 537566. MR 0093653 (20:176)
 [10]
 , Shock waves and entropy, Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, editor, Academic Press, New York, 1971, pp. 634643. MR 0366576 (51:2823)
 [11]
 T.P. Liu, Invariants and asymptotic behavior of solutions of a conservation law, preprint. MR 500495 (81c:35085)
 [12]
 , Asymptotic behavior of solutions of general systems of nonlinear hyperbolic conservation laws, Indiana Univ. J. (to appear).
 [13]
 , Decay to waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 585610.
 [14]
 , Largetime behavior of solutions of initial and initialboundary value problems of general systems of hyperbolic conservation laws, Comm. Math. Phys. 57 (1977), 163177.
 [15]
 D. G. Schaeffer and M. Shearer, The classification of systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, with Appendix by D. Marchesin, P. J. PaesLeme, D. G. Schaeffer, and M. Shearer, Duke University, preprint.
 [16]
 D. Serre, Existence globale de solutions faibles sous une hypothese unilaterale pour un systeme hyperbolique non lineaire, Equipe d'Analyse Numerique, Lyon, SaintEtienne, July 1985.
 [17]
 , Solutions a variation bornees pour certains systemes hyperboliques de lois de conservation, Equipe d'Analyse Numerique, Lyon, SaintEtienne, February 1985.
 [18]
 M. Shearer, D. G. Schaeffer, D. Marchesin, and P. J. PaesLeme, Solution of the Riemann problem for a prototype system of nonstrictly hyperbolic conservation laws, Duke University, preprint.
 [19]
 J. A. Smoller, Shock waves and reaction diffusion equations, SpringerVerlag, 1980.
 [20]
 B. Temple, Global solution of the Cauchy problem for a class of nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3 (1982), 335375. MR 673246 (84f:35091)
 [21]
 , Systems of conservation laws with coinciding shock and rarefaction curves, Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983.
 [22]
 , Decay with a rate for noncompactly supported solutions of conservation laws, Trans. Amer. Math. Soc. 298 (1986), 4382. MR 857433 (87k:35163)
 [23]
 , Degenerate systems of conservation laws, Contemp. Math., vol. 60, Amer. Math. Soc., Providence, R.I., 1987. MR 873538 (88b:35131)
 [24]
 , Supnorm estimates in Glimm's method, preprint.
 [25]
 , No contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc., 288 (1985).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009481990
PII:
S 00029947(1990)09481990
Article copyright:
© Copyright 1990
American Mathematical Society
