The Riemann problem for the planar motion of an elastic string
type: Item
instance of: scholarly article
title: The Riemann problem for the planar motion of an elastic string
zbMATH Open document ID: 0629.35073
DOI: 10.1016/0022-0396(86)90116-6
published in: Journal of Differential Equations
publication date: 1986-01-01T00:00:00Z
review text: The object of the paper is to provide a constructive proof of the following result: Let T satisfy (i) T is of class \(C^ 3\), (ii) \(T(\xi)>0\) for all \(\xi\), (iii) \(T(1)=0\), (iv) there exists \(\xi_ J>1\) such that \(sgn T''(\xi)=sgn(\xi-\xi_ J)\), (v) there are exactly two solutions \(\xi_ m<\xi_ M\) of the equation \(T(\xi)/\xi=T(\xi)\). For any given \(W_ L\), \(W_ R\) in \({\mathbb{R}}^ 4\), the initial value problem \(r_{tt} = [T(| r_ x|)r_ x/| r_ x|]_ x\); \((r_ x,r_ t)(x,0)=W_ L\) if \(x<0\), \((r_ x,r_ t)(x,0)=W_ R\) if \(x>0\) has a unique weak solution among those functions \((r_ x,r_ t)\) depending piecewise smoothly on \(x/t\) and whose jump discontinuities satisfy certain entrophy admissible condition.
reviewed by: Eutiquio C. Young
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 4023872
zbMATH Keywords: initial value problem
zbMATH Keywords: unique weak solution
zbMATH Keywords: jump discontinuities
author: Michael Shearer
MaRDI profile type: Publication
full work available at URL: 0022-0396(86)90116-6
OpenAlex ID: W2000012390
cites work: The Equations for Large Vibrations of Strings
cites work: Q5541936
cites work: Solutions in the large for nonlinear hyperbolic systems of equations
cites work: A system of non-strictly hyperbolic conservation laws arising in elasticity theory
cites work: Elementary Wave Solutions of the Equations Describing the Motion of an Elastic String
cites work: The Riemann problem for materials with nonconvex equations of state. I: Isentropic flow
description: scientific article; zbMATH DE number 4023872