Convergence rate approximate solutions to conservation laws with initial rarefactions

The authors address the question of local convergence rate of conservative Lip⁺-stable approximations u^epsilon(x,t) to the entropy solution u(x,t) of a genuinely nonlinear conservation law. This paper extends the previous results by including lip⁺ -unbounded initial data. Specifically, it is shown that for arbitrary L_INF intersection BV initial data, u and its derivatives may be recovered with an almost optimal, modulo a spurious log factor, error of O(ε/Inε/). This analysis relies on obtaining new Lip⁺-stability estimates for the speed a(u^ε), rather than for u^epsilon itself. This enables the establishment of an O(ε/Inε) convergence rate in W^-1,1, which, in turn, implies the above mentioned local convergence rate.