Rigorous numerics for dissipative partial differential equations. II: Periodic orbit for the Kuramoto-Sivashinsky PDE -- a computer-assisted proof
type: Item
instance of: scholarly article
title: Rigorous numerics for dissipative partial differential equations. II: Periodic orbit for the Kuramoto-Sivashinsky PDE -- a computer-assisted proof
zbMATH Open document ID: 1066.65105
published in: Foundations of Computational Mathematics
publication date: 2004-10-28T00:00:00Z
review text: [For part I see \textit{P. Zgliczynśki} and \textit{K. Mischaikow}, ibid. 1, No. 3, 255--288 (2001; Zbl 0984.65101)]. A method of self-consistent a priori bounds for dissipative partial differential equations is developed. These bounds allow studying rigorously the dynamic of dissipative partial differential equations. As an application of the theory a computer-assisted proof of the existence of a periodic orbit for the Kuramoto-Sivashinsky equation \(u_t= (u^2)_x- u_{xx}- vu_{xxxx}\), \(u(t,x)= u(t,x+ 2\pi)\), \(u(t,x)= -u(t, -x)\), where \(v= 0,127\) is presented. After the introduction the base of the method of self-consistent a priori bounds, which sets up the framework for treating all high-dimensional Galerkin projections as a single differential inclusion is presented. In section 3 the author discusses the notion of a Poincaré section and Poincaré return map in the context of self-consistent bounds. In section 4 as an application a result about the existence of periodic orbit of the Kuramoto-Sivashinsky equation is presented. In the section 5 the description of computer-assisted proof of the existence is described. In the last section a modification of the Lohner algorithm is presented, which was used in the rigorous integration of a differential inclusion in the proof of the existence of the orbit.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65P40
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 2111601
zbMATH Keywords: Kuramoto-Sivashinksy equation
zbMATH Keywords: dissipative partial differential equation
zbMATH Keywords: a priori bounds
zbMATH Keywords: periodic orbit
zbMATH Keywords: computer-assistent proof
zbMATH Keywords: Galerkin method
zbMATH Keywords: differential inclusion
zbMATH Keywords: Poincaré section
zbMATH Keywords: Poincaré return map
zbMATH Keywords: Lohner algorithm
author: Piotr Zgliczyński
reviewed by: Jiří Vaníček
MaRDI profile type: Publication
full work available at URL: s10208-002-0080-8
OpenAlex ID: W2100375085
DOI: 10.1007/S10208-002-0080-8
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description: scientific article; zbMATH DE number 2111601