Abstract
The aim of this paper is to give a simple proof of the classical Liu estimate on the decay of positive waves in a solution of a n×n system of conservation laws. In the first part, we transcribe the wave partition technique introduced in Comm. Math. Phys. 57 (1977), 135–148 (by means of the Glimm scheme) to the case of approximate solutions constructed by the wave front tracking scheme. Then, we use a decoupling argument on the characteristic speeds to establish the desired estimate.
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Ayad, S. and Heibig, A.: Global interaction of fields in a system of conservation laws, Comm. Partial Differential Equations 23(3-4) (1998), 701-725.
Bressan, A.: Global solutions of systems of conservation laws by wave front tracking, J. Math. Anal. Appl. 170 (1992), 414-432.
Bressan, A.: Lectures Notes on Systems of Conservation Laws, SISSA, Trieste, 1995.
Bressan, A. and Colombo, R. M.: Decay of positive waves in nonlinear systems of conservation laws, Ann. Scuola. Norm. Sup. Pisa. Sci. 26(1) (1996), 133-160.
Dafermos, C. M.: Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 107(2) (1989), 127-155.
Dafermos, C. M. and Geng, X.: Generalized characteristics uniqueness and regularity of solutions in a hyperbolic system of conservation laws, Ann. Inst. H. Poincaré, Analyse non linéaire 8(3/4) (1991).
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715.
Glimm, J. and Lax, P. D.: Decay of solutions of systems of nonlinear hyperbolic conservations laws, Mem. Amer. Math. Soc. 101 (1970).
Heibig, A.: Existence and uniqueness for some hyperbolic systems of conservation laws, Arch. Rational. Mech. Anal. 126 (1994), 79-101.
Heibig, A. and Sahel, A.: Une méthode des caractéristiques pour certains systèmes de lois de conservation, C.R. Acad. Sci. Paris Ser. Math. 322(1) (1996), 37-42.
Liu, T. P.: The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135-148.
Liu, T. P.: Admissible solutions of hyperbolic conservations laws, Mem. Amer. Math. Soc. 240 (1981).
. Oleinik, O. A.: Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk (N.S) 12 (1957), 3-73. English Transl. Amer. Math. Soc. Transl. Ser. 2(26), 95-172.
Serre, D.: Systèmes de lois de conservation I et II, Diderot éditeur, arts et sciences, 1996.
Smoller, J.: Shock Waves and Reaction Diffusion Equations, Springer, New York, 1983.
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Ayad, S., Heibig, A. Partition of Waves in a System of Conservation Laws. Acta Applicandae Mathematicae 66, 191–207 (2001). https://doi.org/10.1023/A:1010719628448
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DOI: https://doi.org/10.1023/A:1010719628448