LpLp 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 C1C1. Existence, uniqueness and L1L1 contractivity were proved by Kružkov [14]. Using the relative entropy method, Leger showed that for a uniformly convex flux and for the shock wave solutions, the L2L2 norm of a perturbed solution relative to the shock wave is bounded by the L2L2 norm of the initial perturbation. Here we generalize the result to LpLp norm for all 1⩽p<∞1⩽p<∞. Also we show that for the non-shock wave solution, LpLp norm of the perturbed solution relative to the modified N  -wave is bounded by the LpLp norm of the initial perturbation for all 1⩽p<∞1⩽p<∞.