Weak stability in the global -norm for systems of hyperbolic conservation laws

Author:
Blake Temple

Journal:
Trans. Amer. Math. Soc. **317** (1990), 673-685

MSC:
Primary 35L65; Secondary 35B35

DOI:
https://doi.org/10.1090/S0002-9947-1990-0948199-0

MathSciNet review:
948199

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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****[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**410110**, https://doi.org/10.1512/iumj.1975.24.24088**[3]**James Glimm,*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**194770**, https://doi.org/10.1002/cpa.3160180408**[4]**E. Isaacson,*Global solution of a Riemann problem for a non-strictly 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**, https://doi.org/10.1016/0196-8858(90)90009-N**[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 non-strictly 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**, https://doi.org/10.1007/BF00281590**[9]**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**93653**, https://doi.org/10.1002/cpa.3160100406**[10]**Eduardo H. Zarantonello (ed.),*Contributions to nonlinear functional analysis*, Academic Press, New York-London, 1971. Mathematics Research Center, Publ. No. 27. MR**0366576****[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**, https://doi.org/10.1090/S0002-9939-1978-0500495-7**[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), 585-610.**[14]**-,*Large-time behavior of solutions of initial and initial-boundary value problems of general systems of hyperbolic conservation laws*, Comm. Math. Phys.**57**(1977), 163-177.**[15]**D. G. Schaeffer and M. Shearer,*The classification of**systems of non-strictly hyperbolic conservation laws, with application to oil recovery*, with Appendix by D. Marchesin, P. J. Paes-Leme, 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, Saint-Etienne, July 1985.**[17]**-,*Solutions a variation bornees pour certains systemes hyperboliques de lois de conservation*, Equipe d'Analyse Numerique, Lyon, Saint-Etienne, February 1985.**[18]**M. Shearer, D. G. Schaeffer, D. Marchesin, and P. J. Paes-Leme,*Solution of the Riemann problem for a prototype**system of non-strictly hyperbolic conservation laws*, Duke University, preprint.**[19]**J. A. Smoller,*Shock waves and reaction diffusion equations*, Springer-Verlag, 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**, https://doi.org/10.1016/S0196-8858(82)80010-9**[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**, https://doi.org/10.1090/S0002-9947-1986-0857433-6**[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**, https://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).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35L65,
35B35

Retrieve articles in all journals with MSC: 35L65, 35B35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0948199-0

Article copyright:
© Copyright 1990
American Mathematical Society