SHARP POINTWISE BOUNDS FOR PERTURBED VISCOUS SHOCK WAVES

Abstract

Refining previous work of Zumbrun, Mascia–Zumbrun, Raoofi, Howard–Zumbrun and Howard–Raoofi, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known results of Liu obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program initiated by Zumbrun and Howard. As pointed out by Liu, the key to obtaining sharp bounds is to take account of cancellation associated with the property that the solution decays faster along characteristic than in other directions. Thus, we must here estimate characteristic derivatives for the entire nonlinear perturbation, rather than judicially chosen parts as in the work of Raoofi and Howard–Raoofi, a requirement that greatly complicates the analysis.