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Exponential Decay of Volume Elements and the Dimension of the Global Attractor

  • P. Constantin
  • C. Foias
  • B. Nicolaenko
  • R. Teman
Part of the Applied Mathematical Sciences book series (AMS, volume 70)

Abstract

Let Σ0 be an m-dimensional smooth manifold in θY for some fixed θ ∈ [1, ∞), let u0 ∈Σ0, and let u = ϕ(α) be a local parametrization of Σ0 in a neighborhood of u0, where a α =(α1,…,α m ) runs over a neighborhood of 0 in ℝ m and u0 = ϕ(0). The infinitesimal volume element of S(t0 at S(t)u0 is |υ 1 (t) ^ … ^ υ m (t)| where υ i (ts) evolve according to (2.3) and υ i (0) = ∂ϕ(α)/∂α i |α=o. Using (2.7) and (2.9) we deduce the equation (see [CF1])
$$\frac{1}{2}\frac{d}{{dt}}|{\upsilon _1}\left( t \right){|^2} + \left( {Tr A\left( t \right)P\left( t \right)} \right)|{\upsilon _1}\left( t \right) \wedge \cdots \wedge {\upsilon _m}\left( t \right){|^2} = 0,$$
(8.1)
where P(t) is the projector on the tangent space to S(t0 at S(t)u0. Thus the volume element will decay exponentially if
$$\mathop {\lim }\limits_{t \to \infty } \operatorname{int} \frac{1}{t}\int_o^t {Tr} \left( {A\left( s \right)P\left( s \right)} \right)ds > 0.$$
(8.2)

Keywords

Volume Element Tangent Space Local Parametrization Hausdorff Dimension Global Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • P. Constantin
    • 1
  • C. Foias
    • 2
  • B. Nicolaenko
    • 3
  • R. Teman
    • 4
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  4. 4.Department de MathematiquesUniversité de Paris-SudOrsayFrance

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