Pointwise Asymptotic Behavior of Perturbed Viscous Shock Profiles
Peter Howard, Mohammadreza Raoofi
TL;DR
This work analyzes the pointwise long-time behavior of perturbations to viscous shock profiles in systems with parabolic or partially parabolic regularization. It combines a refined Green-function framework with diffusion-wave corrections and a shock-tracking mechanism to derive explicit space-time decay estimates for the perturbation and its derivatives, extending Raoofi’s L^p results to pointwise bounds and aligning with Liu’s diffusion-wave picture for weak shocks. The main results demonstrate that, under spectral and hyperbolic stability and transversality, perturbations decompose into a translated viscous shock plus diffusion waves, plus nonlinear interactions that decay at quantified rates in various norms. The methodology applies to a broad class of models, including Navier–Stokes and magnetohydrodynamics, and provides a robust, transparent bootstrapping approach for proving nonlinear stability with precise pointwise asymptotics.
Abstract
We consider the asymptotic behavior of perturbations of Lax and overcompressive type viscous shock profiles arising in systems of regularized conservation laws with strictly parabolic viscosity, and also in systems of conservation laws with partially parabolic regularizations such as arise in the case of the compressible Navier--Stokes equations and in the equations of magnetohydrodynamics. Under the necessary conditions of spectral and hyperbolic stability, together with transversality of the connecting profile, we establish detailed pointwise estimates on perturbations from a sum of the viscous shock profile under consideration and a family of diffusion waves which propagate perturbation signals along outgoing characteristics. Our approach combines the recent $L^p$-space analysis of Raoofi [$L^p$ Asympototic Behavior of Perturbed Viscous Shock Profiles, to appear J. Hyperbolic Differential Equations] with a straightforward bootstrapping argument that relies on a refined description of nonlinear signal interactions, which we develop through convolution estimates involving Green's functions for the linear evolutionary PDE that arises upon linearization of the regularized conservation law about the distinguished profile. Our estimates are similar to, though slightly weaker than, those developed by Liu in his landmark result on the case of weak Lax type profiles arising in the case of identity viscosity [Pointwise Convergence to Shock Waves for Viscous Conservation Laws, Comm. Pure Appl. Math. 50 (1997) 1113--1182].