Nonlinear stability of rarefaction waves for a viscoelasticity
type: Item
instance of: scholarly article
title: Nonlinear stability of rarefaction waves for a viscoelasticity
zbMATH Open document ID: 0827.73019
author: Harumi Hattori
published in: Journal of Differential Equations
publication date: 1995-12-13T00:00:00Z
review text: Consider the problem \(V_t= U_x\), \(U_t= \rho\sigma (V)_x+ a* \sigma (V)_{xt}+ h\), which is obtained from the equation of viscoelasticity \(w_{tt}= \sigma (w_x)_x+ \int_{-\infty}^t a' (t- \tau) \varphi (w_x)_x d\tau+ f(x,t)\) by setting \(U= w_t\), \(V= w_x\). Here \(\sigma\) is assumed to be smooth, increasing, and convex, the kernel \(a\) is positive, decreasing and convex, and its derivatives up to the third order have finite moments up to the third order. Also, \(\rho>0\), such that the material under consideration is a solid. The author establishes the nonlinear stability of weak rarefaction waves of this problem, in the sense that solutions tend to rarefaction waves in the corresponding elastic solid \((a=0\), \(h=0)\), if the asymptotic states that are connected are sufficiently close, \(h\) is small, and the initial data are close to the elastic rarefaction waves. In particular, global solutions of the viscoelastic problem are found that are not small at \(x= \pm\infty\) but have finite limits there. The proof uses a smooth approximation of the initial data, an integral transformation of the second equation which is used to isolate \(\sigma (V)_x\), and an energy method.
Mathematics Subject Classification ID: 74Hxx
Mathematics Subject Classification ID: 74J10
Mathematics Subject Classification ID: 74H55
zbMATH DE Number: 635876
zbMATH Keywords: weak waves
zbMATH Keywords: kernel
zbMATH Keywords: elastic solid
zbMATH Keywords: asymptotic states
zbMATH Keywords: global solutions
zbMATH Keywords: smooth approximation
zbMATH Keywords: integral transformation
zbMATH Keywords: energy method
reviewed by: Hans Engler
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1073
OpenAlex ID: W2073148430
DOI: 10.1006/JDEQ.1994.1073
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description: scientific article; zbMATH DE number 635876