On the Oldroyd model of a viscoelastic fluid
type: Item
instance of: scholarly article
title: On the Oldroyd model of a viscoelastic fluid
zbMATH Open document ID: 0943.76007
DOI: 10.1007/BF02465149
publication date: 2000-04-12T00:00:00Z
review text: The author studies the following initial-boundary value problem: \(v_t(t, x)+ D(v)- \gamma_1\Delta v(t,x)- \gamma_2\Delta \int^t_0\exp(\alpha(s- t)) v(s, x) ds+ \nabla p(t,x)= f(t,x)\), \(\text{div }v(t, x)= 0\), \((t,x)\in Q\), \(\int_\Omega p(t,x) dx= 0\), \(t\geq 0\), \(v(0,x)= v_0(x)\), \(x\in\Omega\), \(v(t,x)= 0\), \(t\geq 0\), \(x\in\partial\Omega\), where \(v(t,x)\) is the velocity, \(p(t,x)\) is the pressure, \(x\in\mathbb{R}^n\), \(\gamma_1> 0\), \(\gamma_2\), \(\alpha\in\mathbb{R}\), \(D(v)= \sum^n_{k=1} v_k(\partial v/\partial x_k)\), \(Q= [0,\infty)\times \Omega\), \(\Omega\in\mathbb{R}^n\) is a bounded domain, \(n\geq 2\), and \(\partial\Omega\) is the boundary of \(\Omega\) of class \(C^2\). This problem describes the motion of viscoelastic Oldroyd fluid. The author studies the solvability in Sobolev classes under necessary conditions for small initial and boundary data for arbitrary \(n\geq 2\). Thus the paper generalizes some previous results obtained for \(n= 2,3\). The main result of the paper establishes conditions under which the problem has a unique solution. The proof consists in reducing the problem to a sequence of auxiliary problems, the solutions of which can be easily found. Additionally, the author describes some properties of the solution obtained.
reviewed by: Vasile Ionescu
Mathematics Subject Classification ID: 76A10
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 1407620
zbMATH Keywords: viscoelastic Oldroyd fluid
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: initial-boundary value problem
author: V. P. Orlov
MaRDI profile type: Publication
cites work: Q3216305
cites work: Stability of zero solution of mathematical model of multidimensional barotropic viscoelastic medium
full work available at URL: https://doi.org/10.1007/bf02465149
OpenAlex ID: W2038639337
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1407620