Pointwise Green’s Function Approach to Stability for Scalar Conservation Laws

We study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law, obtaining an estimate independent of shock strength. We find that for a perturbation with initial data decaying algebraically or slower, the perturbation decays in time at the rate of decay of the integrated initial data in any Lp norm, p ≥ 1. Stability in any Lp norm is a direct consequence. The approach taken is that of obtaining pointwise estimates on the perturbation through a Duhamel’s principle argument that employs recently developed pointwise estimates on the Green’s function for the linearized equation. c © 1999 John Wiley & Sons, Inc.