Decay estimate for a new measure of rarefaction waves in general systems of conservation laws

The sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations, cf. Bressan and Yang (2004) [10]. Such an estimate is well established for genuinely nonlinear system of conservation laws, cf. Bressan and Yang (2004) [9]. But similar result is not available for non-genuinely nonlinear system. In this paper, we give a new measure about the rarefaction waves. In addition, a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws.