Approximate inertial manifolds of exponential order
type: Item
instance of: scholarly article
title: Approximate inertial manifolds of exponential order
zbMATH Open document ID: 0883.34064
published in: Discrete and Continuous Dynamical Systems
publication date: 1998-03-16T00:00:00Z
review text: An approximate inertial manifold (AIM) is a smooth finite dimensional manifold of the phase space which attracts all orbits to a thin neighborhood of it in a finite time uniformly for initial conditions on a given bounded set. This neighborhood contains the global attractor. An AIM is said to have order \(\eta\) when its associated neighborhood has width \(\eta\). Let \(\lambda_n\) be the \(n\)th distinct eigenvalue of the Stokes operator. \textit{C. Foias} and \textit{R. Temam} [SIAM J. Math. Anal. 25, No. 5, 1269-1302 (1994; Zbl 0836.35118)] constructed a family \(\{{\mathcal M}_j\}^\infty_{j=0}\) of AIMs with order \(\eta\sim K(2^{-j} +e^{-\delta \lambda_{n+1}})\), \(\delta >0\). The author presents a construction of another family of AIM s which has also exponential order with respect to \(j\) and \(\lambda_n^{1- \alpha}\), \(0\leq \alpha \leq 1\) and converges even exponentially to the exact inertial manifold when the spectral gap condition is satisfied. The construction is based on a variation of the Lyapunov-Perron method. The presented construction can be applied for examples to the Navier-Stokes, Kuramoto-Sivashinsky and Cahn-Hilliard equations.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35-XX
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 1010976
zbMATH Keywords: approximate inertial manifold (AIM)
zbMATH Keywords: global attractor
zbMATH Keywords: family of AIM
zbMATH Keywords: converges exponentially to intertial manifold
zbMATH Keywords: Lyapunov-Perron method
zbMATH Keywords: Navier-Stokes
zbMATH Keywords: Kuramoto-Sivashinsky and Cahn-Hilliard equations
author: Ricardo M. S. Rosa
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: dcds.1995.1.421
OpenAlex ID: W1989497947
DOI: 10.3934/DCDS.1995.1.421
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description: scientific article; zbMATH DE number 1010976