JournalscmhVol. 97, No. 1pp. 61–132

Slow manifolds for infinite-dimensional evolution equations

  • Felix Hummel

    Technical University of Munich, Garching bei München, Germany
  • Christian Kuehn

    Technical University of Munich, Garching bei München, Germany
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Abstract

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 Sϵ,ζS_{\epsilon,\zeta} under more restrictive assumptions on the linear part of the slow equation. The second parameter ζ\zeta 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 Sϵ,ζS_{\epsilon,\zeta} does not depend on ζ\zeta. 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. 61–132

DOI 10.4171/CMH/527