In this paper, we study the evolution of the L 1 L_1 distance of solutions for systems of 2 × 2 2× 2 hyperbolic conservation laws. For the approximate solutions constructed by Glimm’s scheme with the aid of the wave tracing method, we introduce a nonlinear functional which is equivalent to the L 1 L_1 distance between solutions, nonincreasing in time, and expressed explicitly in terms of the wave patterns of the solutions. This functional reveals the nonlinear mechanism of wave interactions and coupling which affect the L 1 L_1 topology.
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Liu et al. (1999) studied this question.