Attractors for a non-linear parabolic equation modelling suspension flows
type: Item
instance of: scholarly article
title: Attractors for a non-linear parabolic equation modelling suspension flows
zbMATH Open document ID: 1172.35326
author: José M. Amigó
author: I. Catto
author: Ángel Giménez
author: J. Valero
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2009-02-05T00:00:00Z
review text: The authors study the following scalar parabolic equation \[ \begin{aligned} & \partial_tp - D(p(t))\partial_{\sigma\sigma}^2p + \frac{1}{T_0}\mathbf 1_{\mathbb{R}\setminus [-1,1]}(\sigma)p = \frac{D(p(t))}{\alpha}\delta_0(\sigma),\\ & p(0,\sigma) = p_0(\sigma), \quad p \geq 0 \end{aligned} \] that describes the evolution of mesoscopic blocks in suspensions by means of finding the probability distribution density \(p(t,\sigma)\) which represents the distribution of stress in the assembly of blocks in time. Here \(\mathbf 1_{\mathbb{R}\setminus [-1,1]}\) denotes the characteristic function of the open set \(\mathbb{R}\setminus [-1,1]\), \(\sigma \in \mathbb{R}\) and \[ D(p,t) = \frac{\alpha}{T_0}\int_{|\sigma > 1}p(t,\sigma)d\sigma. \] The main aim is to prove the existence of a global attractor with respect to the weak topology of a suitable Banach space. From mathematical point of view, the above-mentioned problem presents several difficulties when studying the large time behavior. First, in view of the fact that the uniqueness theorem has not been obtained in a suitable phase space, the authors work with a multivalued semiflow rather than with a semigroup operators. Second, due to the terms \(D(p(t))\) and \(\mathbf 1_{\mathbb{R}\setminus [-1,1]}\) on the left-hand side of the equation under consideration, the dissipative mechanism of this equation is weakened, and it is not possible to obtain absorbing term. Nevertheless, the authors prove that the solutions starting at a bounded set of a suitable phase space remain uniformly bounded in time.
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 58C06
Mathematics Subject Classification ID: 76A05
zbMATH DE Number: 5503075
zbMATH Keywords: non-Newtonian fluids
zbMATH Keywords: set-valued dynamical system
zbMATH Keywords: global attractor
MaRDI profile type: Publication
full work available at URL: dcdsb.2009.11.205
OpenAlex ID: W2049005117
DOI: 10.3934/DCDSB.2009.11.205
description: scientific article; zbMATH DE number 5503075