Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid
type: Item
instance of: scholarly article
title: Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid
zbMATH Open document ID: 1182.35048
author: Caidi Zhao
author: Yongsheng Li
author: Shengfan Zhou
published in: Journal of Differential Equations
publication date: 2009-11-05T00:00:00Z
review text: The first goal of this article is to prove the regularity of trajectory attractor and to reveal the trajectory smoothing effect for the following 2D incompressible non-Newtonian fluid: \[ \frac{\partial{u}}{\partial{t}}+(u\cdot\nabla)u+\nabla{p}=\nabla\cdot\tau(e(u))+g(x),\;x=(x_{1},x_{2})\in{\mathbb{D}},\qquad \nabla\cdot{u}=0, \] where \(\mathbb{D}\) is a smooth bounded domain of \(\mathbb{R}^{2}\), \(g(x)=(g^{(1)},g^{(2)})\) is the time-independent external force function, and \(\tau(e(u))=(\tau_{ij}(e(u)))_{2\times2}\), which is usually called the extra stress tensor of the fluid, is a matrix of order \(2\times2\) and \[ \tau_{ij}(e(u))=2\mu_{0}(\varepsilon+|e|^{2})^{-\alpha/2}e_{ij}-2\mu_{1}\Delta{e}_{ij},\quad i,j=1,2, \] \[ e_{ij}=e_{ij}(u)=\frac{1}{2}\left(\frac{\partial{u}_{i}}{\partial{x}_{j}}+\frac{\partial{u}_{j}}{\partial{x}_{i}}\right),\quad |e|^{2}=\sum_{i,j=1}^{2}|e_{ij}|^{2}, \] and \(\mu_{0}\), \(\mu_{1}\), \(\alpha\), \(\varepsilon\) are parameters which generally depend on the temperature and pressure. The second goal is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from \(\Omega_m\) to \(\Omega=\mathbb{R}\times(-L,L)\), where \(\{\Omega_m\}_{m=1}^\infty\) is an expanding sequence of simply connected, bounded and smooth subdomains of \(\Omega\) such that \(\Omega_m\rightarrow\Omega\) as \(m\rightarrow+\infty\).
reviewed by: Titus Petrila
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5625444
zbMATH Keywords: incompressible non-Newtonian fluid
zbMATH Keywords: trajectory attractor
zbMATH Keywords: global attractor
zbMATH Keywords: regularity
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: varying domains
zbMATH Keywords: trajectory smoothing effect
MaRDI profile type: Publication
full work available at URL: j.jde.2009.07.031
OpenAlex ID: W2025666088
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DOI: 10.1016/J.JDE.2009.07.031
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