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Slow Manifolds for Infinite-Dimensional Evolution Equations

We extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinite-dimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds $S_{ε,ζ}$ under more restrictive assumptions on the linear part of the slow equation. The second parameter $ζ$ does not appear in the finite-dimensional setting and describes a certain splitting of the slow variable space in a fast decaying part and its complement. The finite-dimensional setting is contained as a special case in which $S_{ε,ζ}$ does not depend on $ζ$. Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.