zbMATH — the first resource for mathematics

Sup-norm stability for Glimm’s scheme. (English) Zbl 0792.35120
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)

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)
Full Text: DOI
[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. 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.