Advances in Differential Equations

Pointwise asymptotic behavior of perturbed viscous shock profiles

Peter Howard and Mohammadreza Raoofi

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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 [33] 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 [21].

Article information

Adv. Differential Equations, Volume 11, Number 9 (2006), 1031-1080.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L65: Conservation laws
Secondary: 35B25: Singular perturbations 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations


Howard, Peter; Raoofi, Mohammadreza. Pointwise asymptotic behavior of perturbed viscous shock profiles. Adv. Differential Equations 11 (2006), no. 9, 1031--1080.

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