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 under more restrictive assumptions on the linear part of the slow equation. The second parameter 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 does not depend on . Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.
Cite this article
Felix Hummel, Christian Kuehn, Slow manifolds for infinite-dimensional evolution equations. Comment. Math. Helv. 97 (2022), no. 1, pp. 61–132DOI 10.4171/CMH/527