Pointwise Asymptotic Behavior of Perturbed Viscous Shock Profiles
Pointwise Asymptotic Behavior of Perturbed Viscous Shock Profiles
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 Asymptotic Behavior of Perturbed
Viscous Shock Profiles, 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].
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