Existence of global stochastic flow and attractors for Navier-Stokes equations
type: Item
instance of: scholarly article
title: Existence of global stochastic flow and attractors for Navier-Stokes equations
zbMATH Open document ID: 0938.35202
author: Nigel J. Cutland
author: Marek Capiński
publication date: 2000-06-09T00:00:00Z
review text: The authors prove the existence of a random attractor for the flow given by the solutions of a 2D stochastic Navier-Stokes equation on a torus. A random attractor for a flow \(\varphi\) is a random compact set \(A\) such that \[ \varphi(t,A(\omega),\omega)=A(\theta_t\omega),\quad t\geq 0,\qquad \lim_{t\to\infty} \text{dist}(\varphi(t,x,\theta_{-t}\omega),A(\omega))=0 \] (\(\theta_t\) is a group of measure preserving maps which plays the role of adapting the events to the time instant), where the first property means the invariance of the attractor with respect to the flow, while the second means the asymptotic behaviour. The forcing term in the equation is a multiplicative noise. There are some rather technical hypotheses on this term, one of them being a sort of orthogonality. This property seems crucial in order to conclude some energy estimates and get rid of the random part, which is fully stochastic. This means that the problem of finding an attractor cannot be solved pathwise. The main tools used in the paper are from nonstandard analysis. The problem is put in a nonstandard framework and the random attractor is the standard part of the set of points reached by the flow at infinity. In particular, the compactness of the attractor is assured by a fundamental result of nonstandard analysis.
reviewed by: Marco Romito
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 26E35
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 37H10
zbMATH DE Number: 1371489
zbMATH Keywords: stochastic attractors
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: existence of a random attractor
zbMATH Keywords: stochastic Navier-Stokes equation on a torus
zbMATH Keywords: compactness of the attractor
MaRDI profile type: Publication
full work available at URL: s004400050238
OpenAlex ID: W2075047140
DOI: 10.1007/S004400050238
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1371489