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2006 Pointwise asymptotic behavior of perturbed viscous shock profiles
Peter Howard, Mohammadreza Raoofi
Adv. Differential Equations 11(9): 1031-1080 (2006).

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 [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].

Citation

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Peter Howard. Mohammadreza Raoofi. "Pointwise asymptotic behavior of perturbed viscous shock profiles." Adv. Differential Equations 11 (9) 1031 - 1080, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1162.35051
MathSciNet: MR2263670

Subjects:
Primary: 35L65
Secondary: 35B25, 35B40, 35K55

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.11 • No. 9 • 2006
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