\(L^p\)-decay rates for perturbations of degenerate scalar viscous shock waves
type: Item
instance of: scholarly article
title: \(L^p\)-decay rates for perturbations of degenerate scalar viscous shock waves
zbMATH Open document ID: 1227.35209
author: Ramón G. Plaza
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-07-18T00:00:00Z
review text: The author of this interesting paper considers a scalar conservation law with second-order viscosity in one dimension of form \[ u_t+f(u)_x=\big(b(u)u_x\big)_x, \] where \(f,b\in C^2\), \(b>0\), and \((x,t)\in \mathbb R\times [0,+\infty )\). It is supposed that the triple \((u_+,u_-,s)\in\mathbb R^3\), with \(u_-\neq u_+\). Without loss of generality, it is taken \(u_+<u_-\). Suppose also that the above stated triple \((u_+,u_-,s)\) constitutes a generalized shock front that is a weak solution of the form \[ u(x,t)=u_-, \quad x-st<0, \] \[ u(x,t)=u_+, \quad x-st>0 \] to the underlying inviscid conservation law \(u_t+f(u)_x=0\), which satisfies both Rankine-Hugoniot jump condition \(-s(u_+-u_-)+f(u_+)-f(u_-)=0\), and the generalized entropy condition \(-s(u-u_-)+ f(u)-f(u_-)<0\), for all \(u\in (u_+,u_-)\). A special interest here is the existence of traveling waves of form \(u(x,t)=\overline{u}(x-st)\) (shock profiles) for the initial problem stated above. Here \(\overline{u}\) satisfies the relations \[ (b(\overline{u})\overline{u}')'=f(\overline{u})'-s\overline{u}', \qquad \overline{u}(\xi )\to u_{\pm }, \] as \(\xi\to \pm \infty \). Here, \((\cdot )'=d/d\xi \) denotes differentiation w.r.t. the moving (Galilean) variable \(\xi =x-st\), and \(s\) is the shock speed. Assume that the flux function \(f(u)\) changes its convexity in \(u\in (u_{+},u_{-})\). The author considers sonic (or degenerate) shocks, namely, waves whose speed matches one of the characteristic speeds with \(s=f'(u_{+})\) or \(s=f'(u_{-})\). The main result is that sharper decay rates for zero-mass perturbations of viscous degenerate shocks can be obtained using energy methods and interpolation inequalities. The \(L^p\)-decay rates, with \(2\leq p\leq +\infty \), for zero mass perturbations of degenerate scalar viscous shock waves using energy methods are obtained. The Matsumura-Nishihara weight function is used. Basic estimates with interpolation inequalities of Nash-type are shown to get sharp d…
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76A02
Mathematics Subject Classification ID: 35L03
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 5925601
zbMATH Keywords: energy estimates
zbMATH Keywords: second-order viscosity
zbMATH Keywords: Rankine-Hugoniot jump condition
zbMATH Keywords: generalized entropy condition
zbMATH Keywords: Matsumura-Nishihara weight function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.04.091
OpenAlex ID: W2045327366
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DOI: 10.1016/J.JMAA.2011.04.091
description: scientific article; zbMATH DE number 5925601