L-p stability for entropy solutions of scalar conservation laws with strict convex flux

Here we consider the scalar convex conservation laws in one space dimension with strictly convex flux which is in C-1. Existence, uniqueness and L-1 contractivity were proved by Kruzkov [14]. Using the relative entropy method, Leger showed that for a uniformly convex flux and for the shock wave solutions, the L-2 norm of a perturbed solution relative to the shock wave is bounded by the L-2 norm of the initial perturbation. Here we generalize the result to L-p norm for all 1 <= p < infinity. Also we show that for the non-shock wave solution, L-p norm of the perturbed solution relative to the modified N-wave is bounded by the L-p norm of the initial perturbation for all 1 <= p < infinity. (C) 2014 Elsevier Inc. All rights reserved.