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An Optimality Condition for Approximate Inertial Manifolds

  • Conference paper
Turbulence in Fluid Flows

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 55))

Abstract

In this paper, we present a basic theoretical framework for interpreting essentially all known methods for constructing approximate inertial manifolds for any of a large class of nonlinear evolutionary equations. This class includes the 2D Navier-Stokes equations and many systems of reaction diffusion equations. We prove that, under reasonable assumptions, every approximate inertial manifold for a given equation is an actual inertial manifold of an approximate equation. This new theoretical framework allows one to introduce certain optimality conditions for approximate inertial manifolds. Finally we introduce a new method, the Gamma Method, for the construction of approximate inertial manifolds.

This research was supported in part by grants from the National Science Foundation, the Applied Mathematics and Computational Mathematics Program/DARPA, and the U. S. Army.

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Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

    George R. Sell

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  1. George R. Sell
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Editors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    George R. Sell

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405-4301, USA

    Ciprian Foias

  3. Laboratoire d’Analyse Numérique, Université Paris-Sud, Bât. 425, 91405, Orsay, France

    Roger Temam

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© 1993 Springer-Verlag New York, Inc.

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Sell, G.R. (1993). An Optimality Condition for Approximate Inertial Manifolds. In: Sell, G.R., Foias, C., Temam, R. (eds) Turbulence in Fluid Flows. The IMA Volumes in Mathematics and its Applications, vol 55. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4346-5_10

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  • DOI: https://doi.org/10.1007/978-1-4612-4346-5_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8743-8

  • Online ISBN: 978-1-4612-4346-5

  • eBook Packages: Springer Book Archive

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