*(English)*Zbl 0815.34049

Let ${\Theta}$ be a compact Hausdorff space and $X$ a Banach space. A linear evolutionary system is a mapping $(x,\theta ,t)\to \left({\Phi}\right(\theta ,t)x$, $\sigma (\theta ,t))$ 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 ${\Phi}(\theta ,t+s)={\Phi}\left(\sigma \right(\theta ,t),s){\Phi}(\theta ,t)$.

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 $P{\left(\theta \right)}_{\theta \in \widehat{{\Theta}}}$ onto the “stable” subbundle, satisfying $\left|{\Phi}(\theta ,t)P\left(\theta \right)\right|\le K{e}^{-\beta t}$ $(t\ge 0)$ and $\left|{\Phi}\right(\theta ,t)$ $(\text{Id}-P\left(\theta \right))|\le K{e}^{-\beta t}$ $(t\le 0)$ for all $\theta $ in an invariant subset $\widehat{{\Theta}}\subset {\Theta}$, were constructed by the authors in a series of papers [*R. J. Sacker* and *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 ${\Phi}$.

##### MSC:

34G10 | Linear ODE in abstract spaces |

37C75 | Stability theory |

58D25 | Differential equations and evolution equations on spaces of mappings |

35Q30 | Stokes and Navier-Stokes equations |

47D06 | One-parameter semigroups and linear evolution equations |