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)


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])
$$\frac{1}{2}\frac{d}{{dt}}|{\upsilon _1}\left( t \right){|^2} + \left( {Tr A\left( t \right)P\left( t \right)} \right)|{\upsilon _1}\left( t \right) \wedge \cdots \wedge {\upsilon _m}\left( t \right){|^2} = 0,$$
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
$$\mathop {\lim }\limits_{t \to \infty } \operatorname{int} \frac{1}{t}\int_o^t {Tr} \left( {A\left( s \right)P\left( s \right)} \right)ds > 0.$$


Volume Element Tangent Space Local Parametrization Hausdorff Dimension Global Attractor 
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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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