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 00ðuÞX0; is considered with an assumption limu-0 uf 0ðuÞ=f ðuÞ 1⁄4 g41: We show that, under suitable conditions on the initial value, its solution converges to an N-wave in L norm with the optimal convergence order of Oð1=tÞ: 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. r 2003 Elsevier Science (USA). All rights reserved.