Trajectory attractor and global attractor for Keller-Segel-Stokes model with arbitrary porous medium diffusion
type: Item
instance of: scholarly article
title: Trajectory attractor and global attractor for Keller-Segel-Stokes model with arbitrary porous medium diffusion
zbMATH Open document ID: 1421.35036
DOI: 10.12775/TMNA.2017.024
author: Wenlong Sun
author: Yeping Li
published in: Topological Methods in Nonlinear Analysis
publication date: 2018-02-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmna/1506477632
review text: The authors analyze the longtime behavior of solutions to the Keller-Segel-Stokes model \[ \begin{aligned} n_{t}+u\cdot \nabla n & =\Delta (n^m) -\nabla\cdot(n\chi(c)\nabla c),\cr c_{t}+u\cdot \nabla c & =\Delta c- nf(c),\cr u_t+\nabla p & =\eta \Delta u-n\nabla \phi,\cr \nabla\cdot u & =0, \end{aligned} \] in a bounded smooth domain \(\Omega\subset \mathbb{R}^2\), where \(n\) denotes the bacterial density, \(c\) the oxygen concentration and \(u\) the velocity field. Moreover, \(m>1\) is the adiabatic exponent, \(\chi\) the chemostatic sensitivity, \(f\) the oxygen consumption rate, \(p\) the pressure, \(\eta\) the viscosity, and \(\nabla \phi\) the gravitation force of the fluid. The following boundary conditions are assumed \[ \partial_\nu n|_{\partial \Omega}=\partial_\nu c|_{\partial \Omega}=u|_{\partial \Omega}=0, \] the initial densities \(n_0\), \(c_0\) are non-negative and \[ n_0\in L^\infty(\Omega),\qquad c_0\in W^{1,\infty}(\Omega),\qquad u_0\in D(A^\theta), \quad \theta>\frac{1}{2}, \] where \(A^\theta\) denotes the fractional power space associated with the Stokes operator. Under suitable assumptions on \(\chi, f\) and \(\phi\), the existence and boundedness of global weak solutions was previously shown in [\textit{Y. Tao}, and \textit{M. Winkler}, Discrete Contin. Dyn. Syst. 32, No. 5, 1901--1914 (2012; Zbl 1276.35105)]. The uniqueness of solutions remained open, and hence, the authors consider the trajectory attractor of the system, which is a suitable concept for problems without uniqueness. The existence of the trajectory attractor is proven, as well as the existence of a global attractor in a generalized sense.
reviewed by: Stefanie Sonner
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 6836834
zbMATH Keywords: Keller-Segel-Stokes model
zbMATH Keywords: trajectory attractor
zbMATH Keywords: non-uniqueness
zbMATH Keywords: global attractor
zbMATH Keywords: translation semigroup
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6836834