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By Serge Alinhac (auth.), Ferruccio Colombini, Nicolas Lerner (eds.)

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By the continuity of the functionals o; and the uniform convergence u~ _ uO, for all /) 48 P. Baiti and A. 8) () E [a, bl. 1) is proved. Completion of the proof To complete the proof of Theorem 2, we need to construct suitable functionals 4>0 satisfying the conditions (Cl), (C2). Roughly speaking, 4>o{w) should measure the weighted distance between wand uO. Since we do not have an explicit formula for this distance, we resort to a suitable approximation. Let c > 0 be given. For any 0 E la, b[, let Xl < ...

Bressan, G. Crasta and B. , Trieste, 1996. [6J A. Bressan and A. Marson, A variational calculus for discontinuous solutions of systems of conservation laws, Comm. Part. Diff. Equat. 20 (1995), 1491-1552. [7] M. Crandall, The semigroup approach to first-order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108-132. [8J K. Deimling, "Nonlinear Functional Analysis", Springer-Verlag, Berlin, 1985. [9] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions", CRC Press, 1992.

5) Given two states u-, u+ sufficiently close to the origin, by the implicit function theorem there exist unique wave sizes a1, ... 6) The solution of the Riemann problem with data (u-, u+) thus consists ofn+ 1 constant states Wo = U-,W1, ... ,Wn = u+, where each couple of states Wi-1,Wi is connected by an i-wave of size ai (a shock or a rarefaction, depending on the sign of ai). :.. - ai, for the size of the i-wave determined by the Riemann data (u-, u+). 9) 35 Weighted path length Let us now recall the basic interaction estimates [10, 13].

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