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)


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
$$\lambda \left( {P\left( u \right)} \right) > \frac{{{\lambda _n} + {\lambda _{n + 1}}}}{2}{\text{ for all }}u$$
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:
$$\frac{d}{{dt}}|w\left( t \right){|^2}\left( {{k_4}\Lambda \left( t \right) - {k_7}} \right)|w{|^2} \leqslant 0.$$


Differential Equation Partial Differential Equation Tangent Space Cell Complex Integral Surface 
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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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