# Cone Invariance Properties

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

## Abstract

One of the features of a dissipative equation of type (2.1), (2.2) is the existence of compact absorbing sets. More precisely, there exists YH satisfying
$$Y{\text{ is convex, closed in }}H,{\text{ a bounded neighborhood or 0 in }}\mathcal{D}{\text{ (}}{A^{1/4}}{\text{) (in particular, }}Y{\text{ is compact in }}H).{\text{ }}$$
(5.1)
$${\text{For every }}\theta {\text{ }} \geqslant {\text{ 1 and any }}{u_o} \in \theta Y{\text{ the inequalities (3}}{\text{.7), (3}}{\text{.8), and (4}}{\text{.5a) are valid}}{\text{. The constants }}{k_{\text{1}}}{\text{, }}{k_{\text{2}}}{\text{, }}{k_{\text{3}}}{\text{, }}{k_{\text{5}}}{\text{, and }}{k_{\text{6}}}{\text{ depend on }}\theta {\text{ only}}{\text{. }}$$
(5.2)
$${\text{The set }}Y{\text{ is absorbing; i}}{\text{.e}}{\text{., if }}Z{\text{ is any bounded set in }}H{\text{, there exists a }}{{\text{t}}_{\text{o}}}{\text{ }} \geqslant {\text{ 0 (depending on }}Z{\text{) such that }}S{\text{(}}t{\text{)}}Z{\text{ }} \subset {\text{ }}Y{\text{ for }}t{\text{ }} \geqslant {\text{ }}{t_{\text{o}}}{\text{.}}$$
(5.3)

Manifold

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© 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