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Dichotomies for linear evolutionary equations in Banach spaces. (English) Zbl 0815.34049
Let $\Theta$ be a compact Hausdorff space and $X$ a Banach space. A linear evolutionary system is a mapping $(x, \theta, t) \to (\Phi (\theta,t) x$, $\sigma (\theta,t))$ with the components given by a two- sided flow $\sigma$ on $\Theta$ and a family of linear selfmappings $\Phi (\theta,t)$, $t \ge 0$, $\theta \in \Theta$ of $X$ that are strongly continuous in $t$, continuous in $\theta$, satisfying $\Phi (\theta,0) = \text{Id}$ and $\Phi (\theta,t + s) = \Phi (\sigma (\theta,t), s) \Phi (\theta,t)$. Finite-dimensional examples arise if $\Phi (\theta,t)$ is the fundamental operator solution of a system of linear ordinary differential equations and $\Theta$ is related to a space of time-dependent coefficients. For the finite-dimensional case, exponential dichotomies, i.e. continuous families of fibre-wise projectors $P (\theta)\sb{\theta \in \widehat \Theta}$ onto the “stable” subbundle, satisfying $\vert \Phi (\theta,t) P(\theta) \vert \le K e\sp{-\beta t}$ $(t \ge 0)$ and $\vert \Phi (\theta,t)$ $(\text{Id} - P(\theta)) \vert \le Ke\sp{-\beta t}$ $(t \le 0)$ for all $\theta$ in an invariant subset $\widehat \Theta \subset \Theta$, were constructed by the authors in a series of papers [{\it R. J. Sacker} and {\it G. R. Sell}, J. Differ. Equations 15, 429-458 (1974; Zbl 0294.58008); J. Differ. Equations 22, 478-496 (1976; Zbl 0338.58016 ); J. Differ. Equations 22, 497-522 (1976; Zbl 0339.58013)]. The present paper extends this theory to an infinite-dimensional setting, assuming that the $\Phi (\theta,t)$ are set contractions for large $t$, in the weakly hyperbolic case (i.e. the only solution $\Phi (\theta,t)x$ that exists and is bounded for $-\infty < t < \infty$ is the zero solution). Criteria for the existence of exponential dichotomies over subsets of $\Theta$ and over all $\Theta$ with constant codimension of the stable subbundle are given; if such a global dichotomy does not exist, a Morse decomposition of $\Theta$, indexed by increasing codimensions of the stable subbundle is shown to exist. The main difficulty in extending the theory to the infinite-dimensional case comes from the fact that $\Phi (\theta,t)$ need not be defined or unique for $t < 0$. Applications to nonlinear evolution equations and in particular to Navier-Stokes equations are sketched; in these cases, $\Theta$ is related to a compact attractor, and linearizing the equation gives rise to $\Phi$.
Reviewer: H.Engler (Bonn)

34G10Linear ODE in abstract spaces
37C75Stability theory
58D25Differential equations and evolution equations on spaces of mappings
35Q30Stokes and Navier-Stokes equations
47D06One-parameter semigroups and linear evolution equations
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