Young, Robin Sup-norm stability for Glimm’s scheme. (English) Zbl 0792.35120 Commun. Pure Appl. Math. 46, No. 6, 903-948 (1993). We consider the Cauchy problem for a general \(N \times N\) system of conservation laws. Existence of solutions was proved by Glimm using his celebrated random choice scheme. In this paper, we obtain a third order interaction estimate analogous to that obtained by Glimm for \(2\times 2\) systems. By using this estimate, and identifying a global cancellation effect, we obtain \(L^ \infty\)-stability for solutions generated by Glimm’s scheme. As an immediate consequence we have \(L^ 1\)-stability and \(L^ \infty\)-decay, obtained by Temple for \(2\times 2\) systems. Reviewer: R.Young (New York) Cited in 11 Documents MSC: 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems 35L67 Shocks and singularities for hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) Keywords:general \(N \times N\) system of conservation laws; cancellation effect; Glimm’s scheme PDFBibTeX XMLCite \textit{R. Young}, Commun. Pure Appl. Math. 46, No. 6, 903--948 (1993; Zbl 0792.35120) Full Text: DOI References: [1] and , Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1948. · Zbl 0041.11302 [2] Glimm, Comm. Pure Appl. Math. 18 pp 697– (1965) [3] Glimm, Memoirs AMS 101 (1970) [4] Lax, Comm. Pure Appl. Math. 10 pp 537– (1957) [5] Liu, Comm. Math. Phys. 57 pp 135– (1977) [6] Schochet, Commun. PDE 16 pp 1423– (1991) [7] Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1982. [8] Temple, Trans. AMS 298 pp 43– (1986) [9] Temple, J. Diff. Eqns. 83 pp 79– (1990) [10] Temple, Trans. AMS 317 pp 673– (1990) [11] An Extension of Glimm’s Method to Third Order in Wave Interactions, Doctoral Dissertation, University of California at Davis, 1991. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.