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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)

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)
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