Sharp pointwise bounds for perturbed viscous shock waves
  Refining previous work in \cite{Z.3, MaZ.3, Ra, HZ, HR}, 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 \cite{Liu97} obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program set out in \cite{ZH}. As pointed out in \cite{Liu91, Liu97}, 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 \cite{Ra, HR}. a requirement that greatly complicates the analysis. 
Peter Howard, Mohammadreza Raoofi, and Kevin Zumbrun