$L_1$ stability for $2 \times 2$ 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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the evolution of the $L_1$ distance of solutions for systems of $2\times 2$ 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 $L_1$ 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 $L_1$ topology.

- Alberto Bressan and Rinaldo M. Colombo,
*The semigroup generated by $2\times 2$ conservation laws*, Arch. Rational Mech. Anal.**133**(1995), no. 1, 1–75. MR**1367356**, DOI https://doi.org/10.1007/BF00375350 - 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** - A. Bressan, Lecture Notes on System of Conservation Laws, S.I.S.S.A., Trieste 1995.
- A. Bressan, P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, preprint S.I.S.S.A., Trieste 1996.
- 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**303068**, DOI https://doi.org/10.1016/0022-247X%2872%2990114-X - 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**, DOI https://doi.org/10.1007/BFb0093706 - R. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), 137-188.
- Ronald J. DiPerna,
*Global existence of solutions to nonlinear hyperbolic systems of conservation laws*, J. Differential Equations**20**(1976), no. 1, 187–212. MR**404872**, DOI https://doi.org/10.1016/0022-0396%2876%2990102-9 - James Glimm,
*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**194770**, DOI https://doi.org/10.1002/cpa.3160180408 - J. Glimm and P. D. Lax,
*Decay of solutions of systems of hyperbolic conservation laws*, Bull. Amer. Math. Soc.**73**(1967), 105. MR**204826**, DOI https://doi.org/10.1090/S0002-9904-1967-11666-5 - P. D. Lax,
*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**93653**, DOI https://doi.org/10.1002/cpa.3160100406 - 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** - 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**, DOI https://doi.org/10.1002/cpa.3160461103 - Tai Ping Liu,
*The deterministic version of the Glimm scheme*, Comm. Math. Phys.**57**(1977), no. 2, 135–148. MR**470508** - Tai Ping Liu,
*Uniqueness of weak solutions of the Cauchy problem for general $2\times 2$ conservation laws*, J. Differential Equations**20**(1976), no. 2, 369–388. MR**393871**, DOI https://doi.org/10.1016/0022-0396%2876%2990114-5 - Tai-Ping Liu and Tong Yang,
*Uniform $L_1$ boundedness of solutions of hyperbolic conservation laws*, Methods Appl. Anal.**4**(1997), no. 3, 339–355. MR**1487836**, DOI https://doi.org/10.4310/MAA.1997.v4.n3.a7 - T.-P. Liu, T. Yang, A generalised enropy for scalar conservation laws, Comm. Pure Appl. Math. (to appear).
- T.-P. Liu, T. Yang, $L_1$ stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. Jour.
- O. A. Oleĭnik,
*On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics*, Uspehi Mat. Nauk (N.S.)**12**(1957), no. 6(78), 169–176 (Russian). MR**0094543** - Joel Smoller,
*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146** - Blake Temple,
*No $L_1$-contractive metrics for systems of conservation laws*, Trans. Amer. Math. Soc.**288**(1985), no. 2, 471–480. MR**776388**, DOI https://doi.org/10.1090/S0002-9947-1985-0776388-5

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

MR Author ID:
303932

Email:
matyang@cityu.edu.hk

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