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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 $\left(x,\theta ,t\right)\to \left({\Phi }\left(\theta ,t\right)x$, $\sigma \left(\theta ,t\right)\right)$ with the components given by a two- sided flow $\sigma$ on ${\Theta }$ and a family of linear selfmappings ${\Phi }\left(\theta ,t\right)$, $t\ge 0$, $\theta \in {\Theta }$ of $X$ that are strongly continuous in $t$, continuous in $\theta$, satisfying ${\Phi }\left(\theta ,0\right)=\text{Id}$ and ${\Phi }\left(\theta ,t+s\right)={\Phi }\left(\sigma \left(\theta ,t\right),s\right){\Phi }\left(\theta ,t\right)$.

Finite-dimensional examples arise if ${\Phi }\left(\theta ,t\right)$ 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 $P{\left(\theta \right)}_{\theta \in \stackrel{^}{{\Theta }}}$ onto the “stable” subbundle, satisfying $|{\Phi }\left(\theta ,t\right)P\left(\theta \right)|\le K{e}^{-\beta t}$ $\left(t\ge 0\right)$ and $|{\Phi }\left(\theta ,t\right)$ $\left(\text{Id}-P\left(\theta \right)\right)|\le K{e}^{-\beta t}$ $\left(t\le 0\right)$ for all $\theta$ in an invariant subset $\stackrel{^}{{\Theta }}\subset {\Theta }$, 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 ${\Phi }\left(\theta ,t\right)$ are set contractions for large $t$, in the weakly hyperbolic case (i.e. the only solution ${\Phi }\left(\theta ,t\right)x$ that exists and is bounded for $-\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 ${\Phi }\left(\theta ,t\right)$ 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)
##### 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