Long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation
type: Item
instance of: scholarly article
title: Long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation
zbMATH Open document ID: 1024.35016
author: Bixiang Wang
author: Boling Guo
publication date: 2003-11-11T00:00:00Z
review text: The authors consider the following generalization of the Kolmogorov-Spiegel-Sivashinsky equation: \[ u_t+\alpha\Delta^2 u+\beta\Delta u + \nabla\cdot \vec f(u)+\Delta\phi(u)+\nu u+h(x)=0 \tag{1} \] in the three-dimensional bounded domain \(\Omega:=(0,L_1)\times(0,L_2)\times(0,L_3)\) endowed by periodic boundary conditions. Here \(\Delta\) is the Laplacian, \(\vec f\)\ and \(\phi\) are given nonlinear interaction functions, \(h\) are the external forces and \(\alpha,\beta,\nu>0\) are physical parameters which are assumed to satisfy the following relation: \[ 4\alpha\nu>\beta^2. \] The authors show that, under the appropriate assumptions on the functions \(\vec f\) and \(\phi\), the longtime behavior of solutions of (1) can be described in terms of a global attractor \(\mathcal A\) with finite Hausdorff and fractal dimensions. Moreover (under the additional assumptions on \(\vec f\), \(\phi\) and \(h\)), they prove the Gervey regularity of solutions of equation (1) and (based on this fact) construct the approximate inertial manifold with the exponential order of approximation for that problem.
reviewed by: S. V. Zelik
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 37L65
Mathematics Subject Classification ID: 35K35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B42
zbMATH DE Number: 1885857
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: approximate inertial manifolds
MaRDI profile type: Publication
full work available at URL: s10114-002-0195-5
OpenAlex ID: W2092021455
DOI: 10.1007/S10114-002-0195-5
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 1885857