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Local Exponential Decay Toward Blocked Integral Surfaces

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

Abstract

Suppose Σ is an n-dimensional integral surface in Y, that is, an n-dimensional manifold without boundary that is positively invariant. Let, for each u ∈Σ P(u) denote the projector on the tangent space T u (Σ) to Σ at u. Let us assume that the surface is blocked in the sense that
$$\lambda \left( {P\left( u \right)} \right) > \frac{{{\lambda _n} + {\lambda _{n + 1}}}}{2}{\text{ for all }}u$$
(7.1)
and that λ n = Λ m which satisfies condition (3.13). Let us consider u o H and assume that the distance between uo and Σ is attained at some u1 ∈ Σ Then, clearly P(u1)(u1u1)= 0. Let us consider the trajectories S(t)uo, S(t)u1. Their difference w(t) = S(t)uoS(t)u1 satisfies (4.1). Denoting Λ(t) = (Aw(t), w(t))|w(t)|2, we have as in Chapter 4:
$$\frac{d}{{dt}}|w\left( t \right){|^2}\left( {{k_4}\Lambda \left( t \right) - {k_7}} \right)|w{|^2} \leqslant 0.$$
(7.2)

Keywords

Differential Equation Partial Differential Equation Tangent Space Cell Complex Integral Surface 
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.

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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