Pointwise estimates and stability for dispersive-diffusion shock waves

We study the stability and pointwise behavior of perturbed viscous shock wa ves for a general scalar conservation law with constant diffusion and dispe rsion. Along with the usual Lax shocks, such equations are known to admit u ndercompressive shocks. We unify the treatment of these two cases by introd ucing a new wave-tracking method based on "instantaneous projection", givin g improved estimates even in the Lax case. Another important feature connec ted with the introduction of dispersion is the treatment of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all L-P norms, p greater than or equal to 1 for the Lax case and p > 1 for the undercompressive case. Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for extension to systems.