Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to N-waves

The goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws. General convex flux, f″(u) 0, is considered with an assumption limu→0 uf′(u)/f(u)=γ>1. We show that, under suitable conditions on the initial value, its solution converges to an N-wave in L1 norm with the optimal convergence order of O(1/t). The technique we use in this article is to enclose the solution with two rarefaction waves. We also show a uniform convergence order in the sense of graphs. A numerical example of this phenomenon is included.