Dynamics for the stochastic nonlocal Kuramoto-Sivashinsky equation
type: Item
instance of: scholarly article
title: Dynamics for the stochastic nonlocal Kuramoto-Sivashinsky equation
zbMATH Open document ID: 1120.60061
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-04-10T00:00:00Z
review text: The paper deals with \(du+\bigl(u_{xxxx}+u_{xx}+uu_x+\alpha H(u_{xxx})\bigr)\, dt =\sigma\, d W(t)\), \(u\) periodic on \((-L,L)\), \(L>0\). \(H\) denotes the Hilbert transform \(\psi\mapsto {1\over{2L}}\int_{-L}^L \cot{{\pi(x-y)}\over{2L}}\psi(y)\, dy\), \(\alpha\in (0,\infty)\), and \(W\) belongs to a special class of infinite-dimensional Wiener processes related to the eigenvectors of the periodic negative Laplacian on \((-L,L)\). The author establishes existence of a finite-dimensional random attractor. The work relies on a result by Debussche on the Hausdorff dimension of random attractors and a gauge function technique. Difficulties due to the presence of the Hilbert transform are circumvented by using Galerkin's method in proving existence of a weak solution.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37H99
Mathematics Subject Classification ID: 37L55
zbMATH DE Number: 5141600
zbMATH Keywords: Kuramoto-Sivashinsky equation
zbMATH Keywords: random attractor
zbMATH Keywords: Wiener process
zbMATH Keywords: Hilbert transform
author: Desheng Yang
reviewed by: Georg Hetzer
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.07.091
OpenAlex ID: W1973067931
cites work: Q4210478
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DOI: 10.1016/J.JMAA.2006.07.091
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description: scientific article; zbMATH DE number 5141600