## $L_1$ stability for $2 \times 2$ systems of hyperbolic conservation laws

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- by Tai-Ping Liu and Tong Yang
- J. Amer. Math. Soc.
**12**(1999), 729-774 - DOI: https://doi.org/10.1090/S0894-0347-99-00292-1
- Published electronically: April 13, 1999
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## 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.## References

- 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 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 10.1016/0022-247X(72)90114-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 10.1007/BFb0093706 - Dunham Jackson,
*A class of orthogonal functions on plane curves*, Ann. of Math. (2)**40**(1939), 521–532. MR**80**, DOI 10.2307/1968936 - 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 10.1016/0022-0396(76)90102-9 - James Glimm,
*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**194770**, DOI 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 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 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) Publ. Math. Res. Center Univ. Wisconsin, No. 27, 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 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**, DOI 10.1007/BF01625772 - 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 10.1016/0022-0396(76)90114-5 - 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 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**, DOI 10.1007/978-1-4684-0152-3 - 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 10.1090/S0002-9947-1985-0776388-5

## Bibliographic 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 - © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1646841