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Hyperbolic sets for semilinear parabolic equations

Hyperbolic sets for semilinear parabolic equations In this note we considerC r semiflows on Banach spaces, roughly speakingC r flows defined only for positive values of time. Such semiflows arise as the “general solution” of a large class of partial differential equations that includes the Navier-Stokes equation. Our main result (Proposition B) is that under certain assumptions on the P.D.E. (satisfield by the Navier-Stokes equation) a hyperbolic set for the corresponding semiflow (hyperbolicity is defined following closely the finite dimensional case) is always ε-equivalent to a hyperbolic set for an ordinary differential equation that can be easily deduced from the P.D.E. As an example we consider the P.D.E. $$\frac{{\partial u}}{{\partial t}} = - \Delta u + \varepsilon F(x,u,u')$$ where u:M → ℝ k andM is a closed smooth Riemannian manifold. Applying normal hyperbolicity techniques the phase portrait of (0) can be analyzed proving that every example of hyperbolic set for O.D.E. can appear as a hyperbolic set for the semiflow generated by (0). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

Hyperbolic sets for semilinear parabolic equations

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 1975 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/BF02584781
Publisher site
See Article on Publisher Site

Abstract

In this note we considerC r semiflows on Banach spaces, roughly speakingC r flows defined only for positive values of time. Such semiflows arise as the “general solution” of a large class of partial differential equations that includes the Navier-Stokes equation. Our main result (Proposition B) is that under certain assumptions on the P.D.E. (satisfield by the Navier-Stokes equation) a hyperbolic set for the corresponding semiflow (hyperbolicity is defined following closely the finite dimensional case) is always ε-equivalent to a hyperbolic set for an ordinary differential equation that can be easily deduced from the P.D.E. As an example we consider the P.D.E. $$\frac{{\partial u}}{{\partial t}} = - \Delta u + \varepsilon F(x,u,u')$$ where u:M → ℝ k andM is a closed smooth Riemannian manifold. Applying normal hyperbolicity techniques the phase portrait of (0) can be analyzed proving that every example of hyperbolic set for O.D.E. can appear as a hyperbolic set for the semiflow generated by (0).

Journal

Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals

Published: Mar 4, 2007

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