An existence theorem for the Dirichlet problem in the elastodynamics of incompressible materials
type: Item
instance of: scholarly article
title: An existence theorem for the Dirichlet problem in the elastodynamics of incompressible materials
zbMATH Open document ID: 0659.73011
DOI: 10.1007/BF00251494
author: William J. Hrusa
author: Michael Renardy
published in: Archive for Rational Mechanics and Analysis
publication date: 1988-01-01T00:00:00Z
review text: The authors consider initial-boundary value problems. The incompressibility condition causes difficulties in dealing with boundary conditions and several authors attempted various approximations of such problems by solutions for compressible materials. The authors by-pass energy estimates in terms of spatial derivatives by replacing them by derivatives with respect to time. Using the notation \(F^ i_{\alpha}=\partial y^ i/\partial x^{\alpha}\), where y denotes the position of a point inside the material, the equations of motion are \[ \ddot y^ i=(-\partial p/\partial x^{\alpha})(\partial x^{\alpha}/\partial y^ i)+(\partial^ 2W(F)/\partial F^ i_{\alpha}\partial F^ i_{\beta})(\partial^ 2y^ j/\partial x^{\alpha}\partial x^{\beta})+f^ i, \] where \(W=W(F)\) is the stored energy function, \(p=p(x,t)\) is an unknown pressure, and \(f=f(x,t)\) is a prescribed body force. The motion must satisfy the constraint det F\(=1\). The initial- boundary conditions are: \(y(x,0)=y_ 0(x)\), \(\dot y(x,0)=y_ 1(x)\) in \(\Omega\), and \(y(x,t)=x\) for all \(t>0\), \(x\in \partial \Omega\). To force uniqueness the authors normalize the pressure distribution: \(\int_{\Omega}p(x,t)dx=0.\) The authors make several assumptions concerning smoothness and compatibility of data (such as vanishing of second and third order time derivatives on the boundary), which permit them to prove the existence and regularity of q and p (of order 4-k on [0,T] and k in \(\Omega\), where k is the order of smoothness of f(x,t): \(f\in \cap^{2}_{k=0}C^ k([0,T];H^{2-k}(\Omega))\). The technique of proof is outlined. It requires solvability of an auxiliary linear elliptic system. Under additional assumptions they prove uniqueness of solutions.
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 74G30
Mathematics Subject Classification ID: 74H25
Mathematics Subject Classification ID: 74J99
zbMATH DE Number: 4078905
zbMATH Keywords: energy methods
zbMATH Keywords: local (in time) existence of classical solutions to Dirichlet problem
zbMATH Keywords: limits of solutions for compressible materials
zbMATH Keywords: differentiation of equations of motion
zbMATH Keywords: analogues of energy for sufficiently high derivatives of solution
zbMATH Keywords: a priori bounds
zbMATH Keywords: initial- boundary value problems
zbMATH Keywords: derivatives with respect to time
zbMATH Keywords: smoothness
zbMATH Keywords: compatibility of data
zbMATH Keywords: regularity
reviewed by: Vadim Komkov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00251494
OpenAlex ID: W2008942145
description: scientific article; zbMATH DE number 4078905