L^p asymptotic behavior of perturbed viscous shock profiles

2004/08/17 by Mohammadreza Raoofi, Raoofi, Mohammadreza
Mathematics · #35L65 (35B25 35B30 35K65) #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B30 #msc:35L65

paper · pdf · doi:10.48550/arxiv.math/0408227

59 pp

arxiv created 2004/08/17 · arxiv updated 2009/12/01

Abstract

We investigate the L^p asymptotic behavior (1≤ p ≤ ∞) of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or have real viscosity matrix (partially parabolic, e.g., compressible Navier--Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e., the solutions to a viscous Burger's equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in L^p.

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