Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation
type: Item
instance of: scholarly article
title: Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation
zbMATH Open document ID: 0952.35118
DOI: 10.1023/A:1009002920348
author: Zoran Grujić
published in: Journal of Dynamics and Differential Equations
publication date: 2000-09-04T00:00:00Z
review text: The author considers the Kuramoto-Sivashinsky equation \(u_t+u_{xxxx}+u_{xx}+uu_x=0\), where \(u=u(x,t)\) is \(L-\)periodic in \(x\), \(x\in \mathbb{R}\), \(t>0\), \(\int_0^Lu(x,t)dx =0\). The phase space of the equation is the Hilbert space. The main statement is that the radius of special analyticity on the global attractor is a lower-semicontinuous function at the stationary solutions. The existence is proved of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter \(L\) is proved. As an application of the main result the author proves that the number of rapid spatial oscillations of functions belonging to this neighborhood is, up to a logarithmic correction, at most linear in \(L\).
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35K35
zbMATH DE Number: 1453231
zbMATH Keywords: Kuramoto-Sivashinsky equation
zbMATH Keywords: attractors
zbMATH Keywords: analyticity in the space variable
zbMATH Keywords: spatial chaos
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: a:1009002920348
OpenAlex ID: W123387736
description: scientific article; zbMATH DE number 1453231