Slow Manifolds for Infinite-Dimensional Evolution Equations

TL;DR

This paper extends Fenichel theory to infinite-dimensional systems, showing slow manifolds approximate solutions well and constructing a two-parameter family of such manifolds, with applications to PDE systems.

Contribution

It generalizes classical Fenichel theory to infinite dimensions and introduces a novel two-parameter family of slow manifolds, including a new splitting parameter for the slow variable space.

Findings

Slow manifolds approximate solutions effectively in infinite-dimensional systems.

A two-parameter family of slow manifolds $S_{\epsilon,\zeta}$ is constructed.

Applications demonstrated on three PDE examples.

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 $S_{\epsilon,\zeta}$ under more restrictive assumptions on the linear part of the slow equation. The second parameter $\zeta$ 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 $S_{\epsilon,\zeta}$ does not depend on $\zeta$. Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.