Generalized characteristics in hyperbolic systems of conservation laws. (English) Zbl 0714.35046

In the paper the Cauchy problem for a strictly hyperbolic nonlinear system of conservation laws is considered. Because the solutions of this problem are generally discontinuous in a finite time, they may exist in the large as weak solutions. The author’s goal is to make a first step in directly studying those. It is known that behaviour of the solutions in question can be described in terms of their generalized characteristics and entropy estimates. In particular, any generalized characteristic is proved to propagate either with appropriate classical characteristic speed or with appropriate shock speed. In addition, the estimates of the second derivatives of entropy with respect to the Riemann invariants are obtained for the genuinely nonlinear system of two conservation laws. References, 17 in number, fully cover the stated problem.
Reviewer: V.Chernyatin


35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
35L67 Shocks and singularities for hyperbolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A30 Geometric theory, characteristics, transformations in context of PDEs
Full Text: DOI


[1] Dafermos, C. M., Characteristics in hyperbolic conservation laws. Non-linear Analysis and Mechanics, Vol. I, ed. R. J. Knops, London: Pitman 1977, pp. 1-58.
[2] Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26 (1977), 1097-1119. · Zbl 0377.35051
[3] Dafermos, C. M., Regularity and large time behavior of solutions of a conservation law without convexity. Proc. Royal Soc. Edinb. 99A (1985), 201-239. · Zbl 0616.35054
[4] DiPerna, R. J., Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws. Indiana Univ. Math. J. 24 (1975), 1047-1071. · Zbl 0309.35050
[5] Diperna, R. J., Singularities of solutions of nonlinear hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 60 (1975), 75-100. · Zbl 0324.35062
[6] DiPerna, R. J., Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27-70. · Zbl 0519.35054
[7] Filippov, A. F., Differential equations with discontinuous right-hand side. Mat. Sb. (N.S.) 51 (1960), 99-128. English transl. Amer. Math. Soc. Transl., Ser. 2, 42, 199-231. · Zbl 0138.32204
[8] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 697-715. · Zbl 0141.28902
[9] Glimm, J., & P. D. Lax, Decay of solutions of nonlinear hyperbolic conservation laws. Memoirs Amer. Math. Soc. 101 (1970). · Zbl 0204.11304
[10] Greenberg, J. M., Asymptotic behavior of solutions to the quasilinear wave equation. Springer Lecture Notes in Math. 446 (1975), 198-246. · Zbl 0307.35064
[11] Lax, P. D., Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803
[12] Lax, P. D., Shock waves and entropy. Contributions to Functional Analysis, ed. E. A. Zarantonello, New York: Academic Press 1971, pp. 603-634.
[13] Liu, T.-P., Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math. 30 (1977), 585-610. · Zbl 0357.35059
[14] Liu, T.-P., Pointwise convergence to N-waves for solutions of hyperbolic conservation laws. (To appear.)
[15] Temple, B., Decay with a rate for noncompactly supported solutions of conservation laws. Trans. Amer. Math. Soc. 298 (1986), 43-82. · Zbl 0618.65076
[16] Volpert, A. I., The space BV and quasilinear equations. Mat. Sb. (N.S.) 73 (1967), 255-302. English transl. Math. USSR Sbornik 2 (1967), 225-267.
[17] Wu, Z.-Q., The ordinary differential equations with discontinuous right members and the discontinuous solutions of the quasilinear partial differential equations. Scientia Sinica 13 (1964), 1901-1917. · Zbl 0154.10901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.