Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions
Abstract
In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.
1. Introduction
The perturbation theory of normally hyperbolic invariant manifolds introduced by Fenichel Reference 4Reference 9 has proved to be a useful tool in the theory of dynamical systems. One important consequence of Fenichel’s works is that they provide a suitable framework for the treatment of fast-slow systems Reference 5Reference 7 of the form
where is a small parameter, are matrices, and are differentiable nonlinearities. The unknown functions and are called fast and slow variable, respectively. System Equation 1.1 is already written in a variant of (local) Fenichel normal form Reference 5Reference 7 separating matrices and the nonlinearities which is also a quite natural form in the PDE context to be considered below. For the classical finite-dimensional case, Fenichel’s techniques are also known as geometric singular perturbation theory. The main result is that – under suitable assumptions – for all , small enough there is a manifold which is locally invariant under the flow generated by Equation 1.1 and which can be written as a graph over the slow variable. More precisely, one may write
where and are the finite-dimensional vector spaces and respectively, take values in, and , is a Lipschitz continuous function. These manifolds, which are called slow manifolds, are over compact subsets in -close to the critical manifold
where denotes the unique solution of
Moreover, the flow on converges to the slow flow on which is defined to be the flow which is generated by the singular limit of Equation 1.1 as that is ,
The existence of such slow manifolds is usually taken as a formal justification for the intuitive idea, that after a short initial time the dynamics of Equation 1.1 only evolve on the slow time scale and are described well by the slow subsystem Equation 1.2. Since
is supposed to have the unique solution one may rewrite ,Equation 1.2 as
Altogether, we can then reduce Equation 1.1 to Equation 1.3. The advantage of Equation 1.3 is that the fast variable is now uniquely determined by the slow variable, i.e., the dimension of the dynamical problem Equation 1.1 has been reduced.
It has been an open problem for a few decades, how to generalize Fenichel theory to the infinite-dimensional setting, with fast-slow systems of partial differential equations as an important application. Even though persistence of normally hyperbolic invariant manifolds in Banach spaces was derived by Bates, Lu and Zeng in Reference 2 for bounded semiflow perturbations, the existence of slow manifolds for PDEs, involving spatial differential operators in the slow variable equations, had only been shown in very special cases such as for the Maxwell-Bloch equations Reference 8. Recently, there have been two new attempts to provide techniques for a geometric singular perturbation theory in infinite dimensions. In Reference 3, slow manifolds in infinite dimensions were approximated by finite-dimensional slow manifolds within a Galerkin procedure, paving the way for an extension of geometric blow-up from ODEs to PDEs. A more direct approach was taken in Reference 6, where a two-parameter family of slow manifolds was constructed via a Lyapunov-Perron argument. The main ingredient of the latter procedure is a splitting of the slow variable space into a quickly decaying part and a part on which the linear dynamics are invertible. We will introduce both approaches in Section 2 and provide a precise comparison result in Section 3, relating the two types of slow manifolds to each other via estimates for their distance and its decay in Finally, in Section .4, we exemplify this main result at the hand of a slow-fast PDE with fast reaction-diffusion dynamics, also discussing intricacies of the Galerkin limits.
2. The two approaches
2.1. Assumptions
In the following, we discuss in detail the assumptions for the subsequent statements. It is, in fact, one of the main difficulties in infinite-dimensional geometric singular perturbation theory to find conditions, which allow for the construction of slow manifolds and are, at the same time, satisfied in many important applications. Although the list of assumptions we impose is quite long, it has already been demonstrated in Reference 6 that the conditions are satisfied for a large class of PDEs, e.g. reaction-diffusion systems; in comparison to Reference 6, we add a few assumptions which allow us to trade regularity for better estimates. Moreover, we also add a splitting in the fast variable space so that we can define an appropriate Galerkin approximation.
For the formulation of the assumptions, we use the notion of an interpolation-extrapolation scale, for which we refer the reader to Reference 1, Chapter V. In the following, let .
2.1.1. Assumption
We consider the fast-slow system Equation 1.1 on Banach spaces and supplemented by the initial conditions ,
where are elements of the interpolation-extrapolation scales introduced hereafter (see also Reference 1, Chapter V) and we have for Assume further that the nonlinearities satisfy . and Then the following conditions ensure that .Equation 1.1 together with Equation 2.1 has a unique solution which is approximated well by the slow flow in a sense which we will make precise later.
- (i)
Generation of semigroups: the closed linear operator generates an exponentially stable -semigroup on the Banach space The closed linear operator . is the generator of a -semigroup on the Banach space .
- (ii)
Generation of Banach scales: the interpolation-extrapolation scales generated by and (see Reference 1, Chapter V) are – up to uniform equivalence of norms for each fixed and all – given by and If . then , shall be equivalent to the interpolation-extrapolation scale generated by for some .
- (iii)
Bounded Fréchet derivatives: let if is holomorphic and otherwise. In addition, we choose Let further . if is holomorphic and otherwise. The nonlinearities and are continuously differentiable and there are constants (which may depend on such that )
- (iv)
Bounds for semigroups: we choose constants (which may depend on as well as ) and (which do not depend on such that for all )
and
- (v)
Relation of constants: we define if and take if Moreover, we assume .
The conditions of Assumption are almost identical to the ones in Reference 6, Section 4. Here they are slightly simplified in the sense that the differentiability of the nonlinearities, which is assumed here, is not necessary for all results in Reference 6.
2.1.2. Assumption
This assumption is sufficient for obtaining a two-parameter family of slow manifolds Reference 6, in particular specifying the role of the second parameter we assume that for each small : there is a splitting independently from , into a fast part , and a slow part such that the projections and commute with on .
The crucial characterization of the fast part is that contains the parts of that decay under the semigroup almost as fast as the space under analogously, the slow space ; contains the parts of which do not decay or which only decay slowly under the semigroup compared to under This idea is expressed in point .v of the following assumptions:
- (i)
Closed subspaces: the spaces and are closed in for all and will be endowed with the norms .
- (ii)
Lipschitz bound: using the notation and the nonlinearity , satisfies
- (iii)
Group in slow subspace: the realization of in i.e. ,
with
generates a -group which satisfies on for For the sake of readability, we will still write . instead of .
- (iv)
Invertability in fast subspace: the realization of in i.e. ,
with
has in its resolvent set. For the sake of readability, we will still write instead of .
- (v)
Speed of decay in and : there are constants such that for all small enough there are constants such that for all , and we have the estimates
- (vi)
Estimate for contraction property in Lyapunov-Perron argument: the parameters and constants introduced above satisfy
where denotes the gamma function.
Assumption is identical to the conditions in Reference 6, Section 5 except for the fact that in Reference 6, Section 5 it is only assumed for Here, we make use of additional regularity in certain estimates and therefore formulate the assumption for . .
2.1.3. Assumption
If we want to use a Galerkin approximation in both the slow and the fast variable, then it is useful to also impose similar conditions on i.e. that there is a splitting , such that the conditions i–v in Assumption hold with and being replaced by and respectively. ,
2.1.4. Assumption
This assumption will enable us to trade regularity for additional decay behavior. We assume that for there is a constant such that, for all we have the estimates ,