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Dichotomies for linear evolutionary equations in Banach spaces. (English) Zbl 0815.34049

Let Θ be a compact Hausdorff space and X a Banach space. A linear evolutionary system is a mapping (x,θ,t)(Φ(θ,t)x, σ(θ,t)) with the components given by a two- sided flow σ on Θ and a family of linear selfmappings Φ(θ,t), t0, θΘ of X that are strongly continuous in t, continuous in θ, satisfying Φ(θ,0)=Id and Φ(θ,t+s)=Φ(σ(θ,t),s)Φ(θ,t).

Finite-dimensional examples arise if Φ(θ,t) is the fundamental operator solution of a system of linear ordinary differential equations and Θ 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(θ) θΘ ^ onto the “stable” subbundle, satisfying |Φ(θ,t)P(θ)|Ke -βt (t0) and |Φ(θ,t) (Id-P(θ))|Ke -βt (t0) for all θ in an invariant subset Θ ^Θ, 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 Φ(θ,t) are set contractions for large t, in the weakly hyperbolic case (i.e. the only solution Φ(θ,t)x that exists and is bounded for -<t< is the zero solution). Criteria for the existence of exponential dichotomies over subsets of Θ and over all Θ with constant codimension of the stable subbundle are given; if such a global dichotomy does not exist, a Morse decomposition of Θ, 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 Φ(θ,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, Θ is related to a compact attractor, and linearizing the equation gives rise to Φ.

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