Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions

TL;DR

This paper develops a method to transform nonlinear, non-autonomous PDEs into a normal form in infinite-dimensional spaces, enabling clearer analysis of invariant manifolds and approximate solutions.

Contribution

It introduces a normal form approach for nonlinear, non-autonomous PDEs viewed as ODEs in Fréchet spaces, facilitating the study of invariant and approximate center manifolds.

Findings

Normal form transformation separates center, stable, and unstable coordinates.

Method applies to a class of nonlinear, non-autonomous PDEs.

Enables analysis of approximate center manifolds with controlled precision.

Abstract

We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. In particular, our method empowers us to study approximate centre manifolds, given by solutions of ODEs that are central up to a desired, possibly nonzero precision. The main motivation is the case where the Fr\'echet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way.

Equations

$$\dot{x}(t)=Ax(t)+f(t,x(t))$$

$$\dot{X}(t)=AX(t)+F_p(t,x(t))$$

$${\xi }_p:I\times V\to V$$

$$(f\circ g)(t,v):=f(t,g(t,v)),(f\circ h)(t,v):=f(t,h(v)),$$

$$\Vert u{\Vert }_{W^{k,p}}:=\sum _{\alpha \in {\mathbb{N}}_0^n;|\alpha |\le k}{\Vert \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}\Vert }_{L^p}.$$

$$V_l\hookrightarrow V_k\stackrel{A}{\to }{V}_l$$

$$\Vert f_{V_{\infty }}^{\prime }(t,v){\Vert }_{\mathcal{B}(V_k,V_l)}=O(\Vert v{\Vert }_{V_k}),$$

$$\dot{x}(t)=Ax(t)+f(t,x(t))\text{for all }t\in I.$$

$$J_c$$

$$J_s$$

$$J_u$$

$$v^q:=\prod _{j=1}^{\infty }(e_j,v{)}_{V_1}^{q_j}.$$

$$F_c(t,v)$$

$$F_s(t,v)$$

$$F_u(t,v)$$

$$\dot{X}(t)=AX(t)+F(t,X(t)),$$

$$D_{\tilde{\mu }}\cap (I\times V_a)$$

$$x(t)=\xi (t,X(t)),$$

$$E_{a}:=\bigl{\{}(t,\xi(t,v)):t\in I_{v},v\in V_{a},(t,v)\in D_{\tilde{\mu}}\bigr{\}}\subset\mathbb{R}\times V_{\infty}.$$

$$\frac{d}{dt}({\xi }_t^{-1}\circ x)(t)=A({\xi }_t^{-1}\circ x)(t)+F(t,({\xi }_t^{-1}\circ x)(t)).$$

$$\Vert \phi (\text{relbar},v{)}^{(l)}{\Vert }_{V_k}:I\to [0,\infty )$$

$${\int }_a^{\infty }{e}^{-\mu t}f(t)dt,$$

$$x(t)={\xi }_p(t,X(t))$$

$$\dot{X}(t)=AX(t)+F_p(t,X(t))$$

$$\dot{x}(t)=Ax(t)+f(t,x(t))+R_p(t,X(t)).$$

$$D_1,\dots ,D_s:C_c^{\infty }(\Omega ;{\mathbb{R}}^m)\to C_c^{\infty }(\Omega ;\mathbb{R})$$

$$(Du{)}^q:=(D_{1}u{)}^{q_1}\cdots (D_{s}u{)}^{q_s}.$$

$$f_j(t,u)=\sum _{q\in {\mathbb{N}}_0^s,|q|\le s}{a}_q^j(t)(Du{)}^q$$

$$W^{\infty }(\Omega ;{\mathbb{R}}^m):=\bigcap _{k=1}^{\infty }{W}^{k-1,k+1}(\Omega ;{\mathbb{R}}^m).$$

$$F_p,{\xi }_p,R_p:\mathbb{R}\times W^{\infty }(\Omega ;{\mathbb{R}}^m)\to W^{\infty }(\Omega ;{\mathbb{R}}^m),$$

$$\dot{x}(t)=Ax(t)+f(t,x(t))+R_p(t,X(t))$$

$$\Vert R_p(t,v){\Vert }_{W^{l-1,l+1}}=O(\Vert v{\Vert }_{W^{k-1,k+1}}^p)$$

$$\underset{h\to 0}{\lim }\frac{\Vert \phi (h){\Vert }_W}{\Vert h{\Vert }_V}=0,$$

Full text

1. 1 Introduction

2. 1.1 Background and motivation

3. 1.2 Results

4. 1.3 Ingredients of the proof

5. 1.4 Outline of this paper

6. 1.5 Notation and conventions

7. 2 Preliminaries and results

8. 2.1 Nested sequences of Banach spaces

9. 2.2 Setup and goal

10. 2.3 Dynamics in a normal form

11. 2.4 Invariant manifolds

12. 2.5 Main result: an approximate normal form

13. 2.6 A general class of PDEs in bounded domains

14. 3 Derivatives and polynomials

15. 3.1 First order derivatives

16. 3.2 Higher order derivatives

17. 3.3 Example: Burgers’ equation

18. 3.4 Bounded polynomial maps

19. 3.5 Standard monomials

20. 4 Compact derivatives and polynomials

21. 4.1 Compact multilinear maps

22. 4.2 Compact polynomial maps

23. 4.3 Compactly differentiable maps

24. 4.4 A class of compactly differentiable maps

25. 5 Polynomial and differentiable maps on graded Fréchet spaces

26. 5.1 Properties of nested sequences of Banach spaces

27. 5.2 Comparable sequences of Banach spaces

28. 5.3 Sequences of Banach spaces comparable to sequences of Hilbert spaces

29. 5.4 Example: Sobolev spaces and $C^{k}$-spaces

30. 6 A coordinate transform

31. 6.1 A residual

32. 6.2 Construction of the coordinate transform

33. 6.3 Proof of Proposition 6.4

34. 7 Centre, stable and unstable coordinates

35. 7.1 Centre, stable and unstable components of $\hat{F}^{q}$

36. 7.2 Update terms for $\xi_{p}$ and $F_{p}$

37. 7.3 Proof of Theorem 2.17

38. 8 Dynamics of the normal form equation

39. 9 Example: a non-autonomous version of Burgers’ equation

40. 9.1 Centre manifold via Theorem 2.17

41. 9.2 Invariant manifolds via direct computations

42. A Compact and finite-rank operators into Banach spaces

43. B Computer algebra code for Burgers example computation

1 Introduction

1.1 Background and motivation

Various invariant manifolds are central to many areas of dynamical systems, including using centre manifolds to construct and justify reduced low-dimensional models of high-dimensional dynamics [29, e.g.]. Many dynamical systems involve pdes in infinite dimensional state spaces of functions, and some applications require infinite dimensional centre manifolds [7, 28, e.g.]. In general we also want to cater for non-autonomous systems, with an aim to subsequently generalise to stochastic/rough dynamics [14, e.g.]. Further, encompassing unstable dynamics with both centre and stable is necessary for application to Saint Venant-like, cylindrical, problems [18, 19, 20, e.g.], and to deriving boundary conditions for approximate pdes [27, e.g.]. Consequently, here we address the general challenge of constructing and justifying various infinite dimensional invariant manifolds for non-autonomous dynamical systems which have stable, unstable and centre modes. A crucial novel feature of the approach is that we further develop a backward theory recently initiated for finite dimensional systems [30]: analogous backward theory has been very useful in other domains [16, e.g.].

Applications of the extant forward theory in such a general setting is often confounded by impractical preconditions. Non-autonomous invariant manifold theories typically require bounded operators, and Lipschitz and/or uniformly bounded nonlinearities, [3, 4, 5, 10, 17, e.g.]. The extant boundedness requirement [17, Hypothesis 2.1(i) and 3.8(i), e.g.] arises from the general necessity of both forward and backward time convolutions with the semigroup (e.g., $e^{At}$ for systems that linearise to $\dot{x}=Ax$), convolutions that must be continuous in extant forward theory, but cannot be continuous with unbounded operators. Despite many interesting specific scenarios having rigorous invariant manifolds established via strongly continuous semigroup operators and by mollifying nonlinearity [9, 32, e.g.], extant non-autonomous forward theory fails to rigorously apply in many practical cases.

Our main motivation for studying odes in infinite-dimensional vector spaces is their possible application to analysing invariant manifolds of pdes in finite space-time dimensions. In that setting, the infinite-dimensional vector space in question is a space of functions, and the maps occurring in the ode are differential operators. The linear part of such an ode is a linear partial differential operator, which typically is unbounded in applications. Such an operator can be viewed as a bounded operator between different Banach spaces, with norms adapted to make the operator bounded. (For example, the operator $\frac{d}{dx}$ is unbounded on $L^{2}(\mathbb{R})$, but becomes bounded if we take its domain to be a first-order Sobolev space.) Centre manifold theory in this setting was developed by several authors [17, 21, 33, e.g.], and applied to pdes.

However, to achieve our goal of developing the desired backward theory, and robustly constructing invariant manifolds via coordinate transformations to approximate normal forms, we need to go beyond this setting. This essentially boils down to the fact that a bounded operator on a single Banach space can be iterated to yield new bounded operators, whereas this is not possible for a bounded operator between different Banach spaces. The necessity of iterating operators in our constructions leads us outside the setting of Banach spaces, to graded Fréchet spaces: intersections of infinite sequences of Banach spaces connected by bounded inclusion maps. These include spaces relevant to the study of pdes, such as spaces of smooth functions.

1.2 Results

The first step in the proposed backward theory is to establish an approximate conjugacy between a given system and a ‘nearby’ system for which we know its invariant manifolds, by its construction. Figure 1 illustrates what this article achieves. Planned future research will then provide novel finite domain and error bounds as illustrated in Figure 1. That is, instead of proving that there exists a reduced dimensional manifold for a specified system, which is then approximately constructed, our main results, Theorems 2.17 and 2.22, establish that there is a system which is both ‘close’ to the specified system, and also has a reduced dimensional manifold which we know exactly. In essence, we invoke an (extended) normal form coordinate transform—related to Hartman–Grobman theory [3, 4, 5, e.g.]—and use it from a new point of view.

An intuitive formulation of our main result on the existence of such a normal form is the following Theorem 1.1. Theorem 2.17 is a precise formulation, and Corollary 2.22 is a special case that applies to nonlinear, non-autonomous pdes. (In that special case, the space $V$ is a space of functions, and the maps that occur are partial differential operators.)

Theorem 1.1** (Normal form theorem, intuitive formulation).**

Let $I$ be an interval in $t$, and $V$ a possibly infinite-dimensional vector space. Consider a nonlinear, non-autonomous ode

[TABLE]

for $x\colon I\to V$, where $A$ is a linear operator on $V$ (independent of $t$), and $f\colon I\times V\to V$ and its derivative vanish on $I\times\{0\}$. For each $p\geq 2$, there are both an ode

[TABLE]

such that

• •

if $X(t)$ satisfies 1.2a, then setting $x(t)=\xi_{p}(t,X(t))$ defines a solution of 1.1 up to a residual term that vanishes to order $p$;

• •

the map $F_{p}\colon I\times V\to V$ is of order $2$ in its second entry $X$;

• •

*the component of a solution to 1.2a in the stable subspace for $A$ decays exponentially quickly to zero as $t$ increases in $I$. Its component in the unstable subspace for $A$ decays exponentially quickly to zero as $t$ decreases in $I$. If the solution starts out in either the centre-stable or the centre-unstable subspace for $A$, then its component in the central subspace for $A$ is bounded by a constant for all $t\in I$, or at worst by a specified, small exponential growth rate. *

The three most important things to make more precise in this intuitive formulation is what ‘order $p$’ means, the related question what topology on $V$ is used, and what kinds of maps $A$, $f$, $\xi_{p}$ and $F_{p}$ are.

The last point in Theorem 1.1 means that the centre, stable and unstable manifolds (in this case, linear subspaces) of 1.2a are exactly the centre, stable and unstable spaces of $A$, respectively. (And similarly for the centre-stable and centre-unstable subspaces.) The centre, stable, unstable, centre-stable and centre-unstable subspaces for the dynamics in $x$ described by 1.2a and $x(t)=\xi_{p}(t,X(t))$ (which becomes an ode in $x$ if $\xi_{p}(t,\relbar)$ is invertible for all $t$) are then obtained from these spaces via an application of the coordinate transform $\xi_{p}$. In this way, we show that any (non-autonomous) system of the form 1.1 is arbitrarily close to a system with robustly defined invariant manifolds.

This definition of these key invariant manifolds is a crucial reformation of the backward theory proposed. Classic definitions of un/stable and centre manifolds require the existence of limits as time goes to $\pm\infty$ [2, 6, 17, 26, e.g.]. This consequently requires solutions of the dynamical system to be well-behaved for all time, which requires constraints that in applications are often not found, or are hard to establish. For example, in stochastic systems very rare events will eventually happen over the infinite time requiring global Lipschitz and boundedness that are oppressive in applications. By modifying definitions we establish results for finite times, which are useful in many applications, and for a wider range of non-autonomous systems.

1.3 Ingredients of the proof

The key ingredients of the proofs of our main results are nested sequences of Banach spaces, whose intersections are graded Fréchet spaces; compact polynomial maps between Banach spaces and graded Fréchet spaces; and compactly differentiable maps between such spaces.

1.3.1 Sequences of Banach spaces

It is important to specify what is meant by a nearby infinite-dimensional mathematical system in Figure 1. Intuitively, we mean by this that solutions of the nearby system 1.2 are solutions of the original system 1.1 up to any desired order in the magnitude of such a solution, as in Theorem 1.1. To be more precise about what that this magnitude is, we need to specify norms or seminorms on spaces containing these solutions. Working with a single Banach space (i.e., a single norm) is too restrictive for applications. This is because in applications to pdes, the maps $A$ and $f$ in Theorem 1.1 are generally not continuous maps from a single Banach space to itself. This could be remedied by allowing maps between two different Banach spaces, but that would not let us iteratively apply maps involving $A$ and $f$, which we do in the proof of Theorem 1.1.

A type of space that is both general enough to apply to various nonlinear pdes, while being close enough to Banach spaces to allow us to define a meaningful notion of a solution of an equation up to a given order, is what are often called graded Fréchet spaces. These are intersections of sequences of Banach spaces, each with a bounded inclusion map into the next. The notion of an operator of a given order on a graded Fréchet space is then defined in terms of the norms on these Banach spaces, see Definition 2.4.

For several convergence questions, it would be useful if the Banach spaces that occur in the definition of a graded Fréchet space are Hilbert spaces. Then we can use orthogonality, for example. However, for applications to nonlinear pdes, it is not enough to use Hilbert spaces. For example, a nonlinear term $u\mapsto u^{2}$ is a well-defined (and differentiable) map from the Banach space $L^{4}(\mathbb{R})$ to the Hilbert space $L^{2}(\mathbb{R})$. To be able to use Hilbert space techniques in such settings, we use the notion of nested sequences of Banach space that are comparable to nested sequences of Hilbert spaces (see Definition 2.6). This effectively means that a graded Fréchet space that naturally occurs as an intersection of Banach spaces can equivalently be presented as an intersection of Hilbert spaces. Proving such a property in situations relevant to pdes involves the relevant Sobolev embedding theorems.

1.3.2 Compact polynomial maps

We construct coordinate transformations on graded Fréchet spaces to bring odes in such spaces into normal forms that allow us to define invariant manifolds directly and robustly. These transformations are polynomial maps, which we construct by adding (infinitely many) monomial terms with the right properties together. At the level of Banach spaces, a polynomial map can be naturally defined as a finite sum of restrictions to the diagonal of bounded multilinear maps. For example, Taylor polynomials of differentiable maps between normed vector spaces are polynomials of this type.

But not all such polynomial maps (for example, the identity map) can be approximated by sums of monomials. This leads us to define compact polynomial maps, which can be approximated in this way in settings relevant to us. The notion of a compact polynomial map that we use seems natural, but we have not been able to find it elsewhere in the literature. Different notions of compact polynomial maps were developed and studied by Gonzalo, Jaramillo and Pełczyńsky [15, 25].

1.3.3 Compactly differentiable maps

Our construction of the required coordinate transforms involves Taylor polynomials of differentiable maps between Banach spaces, and between graded Fréchet spaces. This construction is possible if those polynomials are compact in the sense just mentioned. That is the case for compactly differentiable maps, which we define for this purpose. We will see in Section 4.4 that, in applications to pdes, the relevant differentiable maps are indeed compactly differentiable. This follows from various Sobolev embedding theorems.

1.4 Outline of this paper

The main results of this paper, on normal forms and invariant manifolds of nonlinear, non-autonomous odes in Fréchet spaces, and of nonlinear, non-autonomous pdes in finite-dimensional spaces, are stated in Section 2. We illustrate our results by applying them to an example pde in Section 9.

In the rest of the paper, we prove our main results. We start by reviewing standard material on differentiable maps and polynomials on normed spaces in Section 3. In Sections 4 and 5, we develop technical tools we need for our proofs. Then in Sections 6 and 7, we use these tools to prove the main Theorems 2.17 and 2.21. We prove some properties of the normal form equation, which allow us to identify its invariant manifolds, in Section 8.

A key ingredient in the proof of a version Taylor’s theorem for compactly differentiable maps between Fréchet spaces, mentioned above, is the fact that a compact operator from a Banach space with the approximation property into another Banach space can be approximated by finite-rank operators in a suitable way. {ArXiv} This is reviewed in Appendix A.

1.5 Notation and conventions

We write $\mathbb{N}$ for the set of positive integers, and $\mathbb{N}_{0}$ for the set of nonnegative integers. We write $\mathbb{N}_{0}^{\infty}$ for the set of sequences in $\mathbb{N}_{0}$ with finitely many nonzero entries, interpreted as multi-indices. For $q\in\mathbb{N}_{0}^{\infty}$, or in $\mathbb{N}_{0}^{n}$, we denote the (finite) sum of its elements by $|q|$.

We denote spaces of bounded linear operators by the letter $\mathcal{B}$, and spaces of compact linear operators by the letter $\mathcal{K}$.

When we mention a normed vector space $V$, the implicitly given norm is denoted by $\|\cdot\|_{V}$. Similarly, if $V$ is an inner product space, then the inner product is denoted by $(\relbar,\relbar)_{V}$. Inner products on complex vector spaces are assumed to be linear in their second entries, and antilinear in their first entries. For maps $f,g\colon V\to\mathbb{R}$, when we write $f(v)=O(g(v))$, we implicitly mean that $f(v)=O(g(v))$ as $v\to 0$ in $V$.

If $V$ is a normed vector space, and $I$ is an open interval, and $f\colon I\to V$ and $f_{j}\colon I\to V$, for $j\in\mathbb{N}$, are maps, then we say that $f_{j}$ converges to $f$ if $f_{j}(t)$ converges to $f(t)$ in $V$ uniformly in $t$ in compact subsets of $I$. If $f$ and $f_{j}$ are smooth, then we say that $f_{j}$ converges to $f$ differentiably in $t$ if $f_{j}^{(n)}$ converges to $f^{(n)}$ for every $n\in\mathbb{N}_{0}$, in this sense.

For maps $f,g\colon I\times V\to V$ and $h\colon V\to V$, the maps $f\circ g\colon I\times V\to V$ and $f\circ h\colon I\times V\to V$ are defined by

[TABLE]

for $t\in I$ and $v\in V$.

If $\Omega$ is an open subset of $\mathbb{R}^{d}$ and $m\in\mathbb{N}$, then the Sobolev space of functions from $\Omega$ to $\mathbb{R}^{m}$ with weak derivatives up to order $k$ in $L^{p}$ is denoted by $W^{k,p}(\Omega;\mathbb{R}^{m})$. The norm on this space is

[TABLE]

If $m=1$, we write $W^{k,p}(\Omega)=W^{k,p}(\Omega;\mathbb{R})$.

2 Preliminaries and results

Our main result, Theorem 2.17, asserts that a broad class of nonlinear pdes and odes in infinite-dimensional vector spaces may be effectively approximated by normal form systems via well-chosen, time-dependent, coordinate transformations. In this normal form, the centre, stable and unstable components of the pde and ode are clearly separated, which allows us to define centre manifolds for this class of equations in a robust way (Definition 2.10).

We first state our result on normal forms, and the definition of centre manifolds, for odes in a class of abstract vector spaces (Section 2.5). Our main reason for developing this theory is to apply it to the study of pdes, for which the vector spaces used are spaces of functions, and the relevant maps between them are defined in terms of derivatives of functions. We discuss a relevant class of examples of such function spaces and maps in Section 2.6.

2.1 Nested sequences of Banach spaces

The normal form we obtain in Theorem 2.17 is approximate in the sense that functions satisfying an equation transformed into that form satisfy the original equation up to a residual term. An important point in Theorem 2.17 is that this residual vanishes up to a specified order. To make it precise what this vanishing up to a certain order means, we introduce the type of topological vector spaces we consider in this subsection. More details about these spaces and their properties are given in Section 5. A concrete class of examples of these spaces relevant to the study of pdes is given in Section 2.6.

Definition 2.1**.**

By a nested sequence of Banach spaces, we mean a sequence $\{V_{k}\}_{k=1}^{\infty}$ of Banach spaces such that

• •

for every $k$, $V_{k+1}\subset V_{k}$, where the inclusion map is bounded, and

• •

the intersection $V_{\infty}:=\bigcap_{l=1}^{\infty}V_{l}$ is dense in $V_{k}$ for every $k\in\mathbb{N}$.

We then consider $V_{\infty}$ as a Fréchet space111Much of what we write about Fréchet spaces of this form holds for more general projective limits of Banach spaces connected by bounded operators. But we do not need that degree of generality. with the seminorms (now actual norms) that are the restrictions of the norms on the spaces $V_{k}$.

A compactly nested sequence of Banach spaces is such a sequence such that for every $l\in\mathbb{N}$, there is a $k\geq l$ such that the inclusion $V_{k}\subset V_{l}$ is compact.

A Fréchet space $V_{\infty}$ as in this Definition 2.1 is often called a graded Fréchet space.

Let $\{V_{k}\}_{k=1}^{\infty}$ be a nested sequence of Banach spaces.

Definition 2.2**.**

The space $\mathcal{B}(V_{\infty})$ of bounded operators on $V_{\infty}$ consists of the linear maps $A\colon V_{\infty}\to V_{\infty}$ such that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that the linear map $A$ extends continuously to a map in $\mathcal{B}(V_{k},V_{l})$.

Remark 2.3**.**

In Definition 2.2, if $k\leq l$, then the composition

[TABLE]

is a bounded operator on $V_{l}$. So we may always take $k\geq l$ in this context, but this does not need to be assumed a priori. Similar remarks apply in analogous situations, such as Definitions 2.4 and 2.5 below.

Definition 2.4**.**

A map $f\colon V_{\infty}\to V_{\infty}$ is of order $n$, written as $f=\mathcal{O}(n)$, if for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $\|f(v)\|_{V_{l}}=O(\|v\|_{V_{k}}^{n})$ as $v\to 0$ in $V_{k}$.

If $I$ is an open interval, a map $f\colon I\times V_{\infty}\to V_{\infty}$ is of order $n$, written as $f=\mathcal{O}(n)$, if for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $\|f(t,v)\|_{V_{l}}=O(\|v\|_{V_{k}}^{n})$ as $v\to 0$ in $V_{k}$, uniformly in $t$ in compact subsets of $I$.

Definition 2.5**.**

An $n$ times differentiable map from $V_{\infty}$ to itself is a map $f\colon V_{\infty}\to V_{\infty}$ such that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $f$ extends to an $n$ times differentiable map from $V_{k}$ to $V_{l}$. If a map is $n$ times differentiable for every $n\in\mathbb{N}$, then it is infinitely differentiable.

Basic material on differentiable maps between normed vector spaces is reviewed in Section 3.

Definition 2.6**.**

Two nested sequences $\{V_{k}\}_{k=1}^{\infty}$ and $\{W_{k}\}_{k=1}^{\infty}$ of Banach spaces are comparable if for every $k\in\mathbb{N}$, there are $l_{1},l_{2},l_{3},l_{4}\in\mathbb{N}$ such that we have bounded inclusions $V_{l_{1}}\subset W_{k}\subset V_{l_{2}}$ and $W_{l_{3}}\subset V_{k}\subset W_{l_{4}}$.

In the setting of this definition, $V_{\infty}=W_{\infty}$.

2.2 Setup and goal

Let $\{V_{k}\}_{k=1}^{\infty}$ be a compactly nested sequence of Banach spaces, such that $V_{1}$ is a Hilbert space. Let $A\in\mathcal{B}(V_{\infty})$. Let $I\subset\mathbb{R}$ be an open interval in $t$ containing $t=0$, and let $f\colon I\times V_{\infty}\to V_{\infty}$ be infinitely differentiable with respect to $V_{\infty}$ and $I$. Suppose that $f=\mathcal{O}(2)$, and that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $f\colon V_{k}\to V_{l}$ is differentiable, and

[TABLE]

uniformly in $t$ in compact subsets of $I$.

Suppose that $\{e_{j}\}_{j=1}^{\infty}\subset V_{\infty}$ is a set of eigenvectors of $A$ which is a Hilbert basis of $V_{1}$ (an orthonormal set that spans a dense subspace of $V_{1}$). We assume that the sequence $\{V_{k}\}_{k=1}^{\infty}$ is comparable to a nested sequence of separable Hilbert spaces in which the vectors $e_{j}$ are orthogonal. However, we will see in Remark 6.1 that we may equivalently make the seemingly stronger but more concrete assumption that the spaces $V_{k}$ themselves are separable Hilbert spaces. The assumption that $\{V_{k}\}_{k=1}^{\infty}$ is comparable to a nested sequence of separable Hilbert spaces is easier to check in practice than the condition that every space $V_{k}$ can be chosen to be a separable Hilbert space itself. For example, Section 2.6 discusses a class of relevant cases where the spaces $V_{k}$ are not Hilbert spaces for $k\geq 2$. In this sense, the notion of comparable sequences of Banach spaces is a tool that makes it easier to check the conditions of Theorem 2.17.

We study smooth maps $x\colon I\to V_{\infty}$ satisfying the non-autonomous dynamical system differential equation

[TABLE]

Since the nonlinearity $f$ satisfies 2.1, $x=0$ is an equilibrium of the system 2.2. We provide a novel backward approach to establish invariant manifolds in a finite domain about the equilibrium $x=0$. For these invariant manifolds to be useful in applications, the time interval $I$ will be long enough for transient dynamics to decay to insignificance in the context of the application. The proofs of our main results simplify considerably if the time interval $I$ is short, or bounded. But we emphasise that we only aim this theory to support the many applications where the time interval $I$ is long enough, or unbounded, so that the theorems are non-trivially useful in the application.

2.3 Dynamics in a normal form

We define invariant manifolds, or sets, for dynamical systems in a particular normal form, and show that this definition captures the essence of such manifolds. In Section 2.5, we show that a very general class of odes of the form 2.2 can be brought into this normal form, modulo residuals that vanish to a desired order.

Spectral gap in an exponential trichotomy

Let $\alpha,\beta,\gamma,\tilde{\mu}$ be such that $0\leq\alpha<\tilde{\mu}<\min(\beta,\gamma)$, and no eigenvalues of $A$ have real parts in the intervals $(-\beta,-\alpha)$ and $(\alpha,\gamma)$ (depending upon the circumstances, $\beta$ or $\gamma$ could be $\infty$, and/or $\alpha$ may be zero). For every $j\in\mathbb{N}$, let $\alpha_{j}$ be the eigenvalue of $A$ corresponding to $e_{j}$. With respect to the parameters $\alpha$, $\beta$ and $\gamma$, we define the sets of indices of central, stable and unstable eigenvalues and eigenvectors, respectively, as

[TABLE]

For $a=c,s,u$, let $V_{a}$ be the closure in $V_{1}$ of the span of the eigenvectors $e_{j}$, for $j\in J_{a}$. For any map $g$ into $V_{1}$ and $a\in\{c,s,u\}$, we write $g_{a}$ for its components in $V_{a}$. The sets $V_{c},V_{s},V_{u}$ are respectively called the centre/stable/unstable subspaces. Further, we define the centre-stable subspace $V_{cs}:=V_{c}\oplus V_{s}$, and the centre-unstable subspace $V_{cu}:=V_{c}\oplus V_{u}$.

For $v\in V_{\infty}$ and a multi-index $q\in\mathbb{N}_{0}^{\infty}$, we set

[TABLE]

For $q\in\mathbb{N}_{0}^{\infty}$, write $q=q^{c}+q^{s}+q^{u}$, for $q^{c},q^{s},q^{u}\in\mathbb{N}_{0}^{\infty}$ such that $q^{c}_{j}=0$ if $j\not\in J_{c}$, $q^{s}_{j}=0$ if $j\not\in J_{s}$ and $q^{u}_{j}=0$ if $j\not\in J_{u}$.

Normal form dynamics

Definition 2.7**.**

A smooth map $F\colon I\times V_{\infty}\to V_{\infty}$ separates invariant subspaces if the components of $F$ in $V_{c}$, $V_{s}$ and $V_{u}$ are of the forms

[TABLE]

for all $t\in I$ and $v\in V_{\infty}$, for smooth maps $F^{q}\colon I\to V_{\infty}$, where the series converge in $\operatorname{Pol}(V_{\infty})$, differentiably in $t$.

Consider a polynomial map $F\colon I\times V_{\infty}\to V_{\infty}$ that separates invariant subspaces, and the ode

[TABLE]

in smooth maps $X\colon I\to V_{\infty}$. Because $F$ separates invariant subspaces, this ode has very explicit invariant manifolds, by Lemma 2.8 and Proposition 2.9 below.

Lemma 2.8**.**

Suppose that $X\colon I\to V_{\infty}$ satisfies 2.4, where $F$ separates invariant subspaces. Let $a\in\{c,s,u\}$. If there exists a $t\in I$ such that $X(t)\in V_{a}$, then $X(t)\in V_{a}$ for all $t\in I$.

Proposition 2.9**.**

There is a neighbourhood $D_{\tilde{\mu}}$ of $I\times\{0\}$ in $I\times V_{\infty}$, with the following property. Let $X\colon I\to V_{\infty}$ be a solution of 2.4, for some open interval $I$ containing [math], and where $F$ sepearates invariant subspaces. Write $X=X_{c}+X_{s}+X_{u}$, with $X_{a}\in V_{a}$ for $a=c,s,u$.

• •

If $(t,X(t))\in D_{\tilde{\mu}}$ for all $t\in I$ with $t\geq 0$, then for all such $t$, $\|X_{s}(t)\|_{V_{1}}\leq\|X_{s}(0)\|_{V_{1}}e^{-(\beta-\tilde{\mu})t}$.

• •

If $(t,X(t))\in D_{\tilde{\mu}}$ for all $t\in I$ with $t\leq 0$, then for all such $t$, $\|X_{u}(t)\|_{V_{1}}\leq\|X_{u}(0)\|_{V_{1}}e^{(\gamma-\tilde{\mu})t}$.

• •

Suppose that $X_{s}(0)=0$ or $X_{u}(0)=0$. If $(t,X(t))\in D_{\tilde{\mu}}$ for all $t\in I$, then for all $t\in I$, $\|X_{c}(t)\|_{V_{1}}\leq\|X_{c}(0)\|_{V_{1}}e^{(\alpha+\tilde{\mu})|t|}$.

Since $\beta-\tilde{\mu}$ and $\gamma-\tilde{\mu}$ are positive, this proposition in particular states that stable solutions decrease to zero exponentially quickly as $t$ increases in $I$, while unstable solutions decrease to zero exponentially quickly as $t$ decreases in $I$. The numbers $\alpha$ and $\tilde{\mu}$ represent bounds on what one takes to be relatively small real parts of eigenvalues of $A$ (classically, these numbers are zero), so that the third point in Proposition 2.9 intuitively states that central solutions, at worst, only grow relatively slowly as $|t|$ increases.

Lemmas 2.8 and 2.9 are proved in Section 8. The specific form of the set $D_{\tilde{\mu}}$ is also specified there, see 8.7.

2.4 Invariant manifolds

Lemmas 2.8 and 2.9 show that, for every $a=c,s,u$, the set

[TABLE]

is a centre, stable or unstable submanifold of $I\times V_{\infty}$ for 2.4, respectively. Furthermore, for $a=cs$ and $a=cu$, we obtain centre-stable and centre-unstable manifolds, respectively. (Here we use the cases of the third point in Proposition 2.9 where $X_{u}(0)=0$ and $X_{s}(0)=0$, respectively.) This motivates Definition 2.10 of invariant subspaces of dynamical systems of a certain form. To state it precisely, we incorporate existence of solutions of 2.4.

For $v\in V_{\infty}$, we write $a_{v}$ for the infimum of the set of all $a>0$ such that there is a solution $X:(-a,0]\to V_{\infty}$ of 2.4, with $X(0)=v$. Similarly, $b_{v}$ is the supremum of the set of all $b>0$ such that there is a solution $X:[0,b)\to V_{\infty}$ of 2.4, with $X(0)=v$. If such $a$ and $b$ exist, we set $I_{v}:=(-a_{v},b_{v})$. (In particular, $I_{v}=\mathbb{R}$ if such a solution exists for all $a,b>0$.) If such an $a$ exists but no $b$, we set $I_{v}:=(-a_{v},0)$, and if such a $b$ exists but no $a$, we set $I_{v}:=(0,b_{v})$. If there are no such $a,b>0$, we set $I_{v}:=\emptyset$.

Definition 2.10**.**

Let $\xi\colon I\times V_{\infty}\to V_{\infty}$ be a smooth map, and let $F\colon I\times V_{\infty}\to V_{\infty}$ be a polynomial map that separates invariant subspaces. Consider the dynamical system for smooth maps $x\colon I\to V_{\infty}$ determined by

[TABLE]

for $t\in I$, for a smooth map $X\colon I\to V_{\infty}$ satisfying 2.4. Let $D_{\tilde{\mu}}$ be as in Proposition 2.9. For every $a=c,s,u,cs,cu$, set

[TABLE]

The set $E_{c}$ is a centre subset of the dynamical system in $x$; the set $E_{s}$ is a stable subset of the system; and the set $E_{u}$ is an unstable subset of the system. The set $E_{cs}$ is a centre-stable subset, and $E_{cu,p}$ is a centre-unstable subset of the system. Such spaces are invariant or integral subsets of the dynamical system in $x$.

Remark 2.11**.**

If the map $\xi_{t}:=\xi(t,\relbar)$ in Definition 2.10 is invertible for all $t\in I$ (on a suitable domain), then the dynamical system in $x$ in that definition is equivalent to the ode

[TABLE]

Remark 2.12**.**

In general, existence and uniqueness of solutions of 2.4 is not guaranteed, hence the careful definition of $I_{v}$. Existence and uniqueness of solutions is an assumption in previous definitions [17, Theorem 2.9, e.g.]; see Hypothesis 2.7 in that reference. There are existence and uniqueness results if $f$ satisfies a local Lipschitz condition, but that is not the case in many applications to pdes. Under additional assumptions, Vanderbauwhede & Iooss [33, proof of Theorem 3] showed such a local Lipschitz condition holds.

Remark 2.13**.**

Invariant subsets or submanifolds are not unique in general; here this non-uniqueness is due to various possibilities for $I_{v}$, $D_{\tilde{\mu}}$ and $\xi_{p}$, and is reflected in the use of the indefinite article in Definition 2.10.

Example 2.14**.**

For one example of the non-uniqueness engendered via $\xi$, consider the classic example system of $\dot{x}=-x^{2}$ and $\dot{y}=-y$ in the role of 2.2 (and let the step function $H(x):=1$ when $x>0$, and $H(x):=0$ when $x\leq 0$). This ode system may be given, for every $C$, as the coordinate transformation, 2.5, $x=X$ and $y=Y+CH(X)e^{-1/X}$ together with the system, 2.4, $\dot{X}=-X^{2}$ and $\dot{Y}=-Y$ (by design, here symbolically identical to the original $xy$-system). Lemma 2.8 identifies $Y=0$ as the centre subspace of this $XY$-system. Definition 2.10 then gives the classic non-uniqueness that, for every $C$, $y=CH(x)e^{-1/x}$ are centre manifolds for the $xy$-system.

Remark 2.15**.**

In the setting of Definition 2.10, if $\xi$ is a local diffeomorphism in the Fréchet manifold sense, then the subsets $E_{j}$ in Definition 2.10 are Fréchet manifolds. This would justify the more specific terminology invariant submanifolds rather than just invariant subsets.

2.5 Main result: an approximate normal form

Our main result, Theorem 2.17, states that for an ode of the form 2.2, there is a dynamical system in the normal form used to define invariant manifolds in Definition 2.10, such that solutions of the normal form system satisfy 2.2 up to a residual term that vanishes to any desired order. In this sense, 2.2 is arbitrarily close to a dynamical system with clearly and robustly defined invariant manifolds.

Definition 2.16**.**

A function $f\colon I\to\mathbb{R}$ grows at most polynomially if there are $C,r>0$ such that for all $t\in I$, $|f(t)|\leq C(1+|t|^{r})$. (This condition holds for all bounded functions if $I$ is bounded.)

An infinitely differentiable map $\varphi\colon I\times V_{\infty}\to V_{\infty}$ has polynomial growth if for every $v\in V_{\infty}$, every $k\in\mathbb{N}$, and every $l\in\mathbb{N}_{0}$, the function

[TABLE]

grows at most polynomially.

We use the term $\tilde{\mu}$-regular integral for an integral of the form

[TABLE]

where $\Re(\mu)>\tilde{\mu}$ and $f$ grows at most polynomially. The larger $\tilde{\mu}$, the better the convergence properties of $\tilde{\mu}$-regular integrals.

Theorem 2.17**.**

Let $p\in\mathbb{N}$ be such that $p\geq 2$, $\beta-(p+1)\alpha>\tilde{\mu}$ and $\gamma-(p+1)\alpha>\tilde{\mu}$. Suppose that $f$ has polynomial growth. Then there are three infinitely differentiable maps $F_{p},\xi_{p},R_{p}\colon I\times V_{\infty}\to V_{\infty}$, such that

• •

$F_{p}=\mathcal{O}(2)$* and $F_{p}$ separates invariant manifolds;*

• •

$R_{p}=\mathcal{O}(p)$,

and if a smooth map $x\colon I\to V_{\infty}$ is given by

[TABLE]

for all $t\in I$, for a smooth map $X\colon I\to V_{\infty}$ satisfying

[TABLE]

for all $t\in I$, then for all $t\in I$,

[TABLE]

Finally, there is a construction of the map $\xi_{p}$ in which all integrals over $I$ that occur are $\tilde{\mu}$-regular.

We prove this theorem in Sections 3, 4, 5, 6 and 7; see in particular Section 7.3.

Theorem 7.9 shows that the maps $F_{p}$ and $\xi_{p}$ can be chosen to be polynomials of a certain type. The conclusion that the integrals occurring are $\tilde{\mu}$-regular is more than just convenient: this is clear in the classical case where $\alpha=0$ (see Remark 2.18).

Remark 2.18**.**

In cases where the centre eigenvalue bound $\alpha$ equals zero, we can always choose $\tilde{\mu}$ so small that the conditions on $\alpha,\beta,\gamma,\tilde{\mu}$ and $p$ in Theorem 2.17 are satisfied. In these cases, the residual $R_{p}$ can be made to vanish to arbitrarily large order $p$. Furthermore, the integrals that occur in the construction of $\xi_{p}$ (see Definition 7.2) are $\tilde{\mu}$-regular for some $\tilde{\mu}>0$ precisely if they converge. Hence $\tilde{\mu}$-regularity for some $\tilde{\mu}>0$ is a necessary condition for the construction to make sense.

Choosing the centre eigenvalue bound $\alpha$ positive, which imposes a positive lower bound on $\tilde{\mu}$, restricts the vanishing order $p$ of $R_{p}$, but also makes the construction of the coordinate transform $\xi_{p}$ more robust, in the sense that the integrals over $I$ in its construction are $\tilde{\mu}$-regular. Many researchers choose to phrase problems as singular perturbations [8, 24, 34, e.g.]. In such cases the bounds on the hyperbolic rates $\beta,\gamma\propto\frac{1}{\varepsilon}\to\infty$ as the perturbation parameter $\varepsilon\to 0$. Consequently, choosing $\tilde{\mu},p\propto 1/\sqrt{\varepsilon}$ (say) then the residual $R_{p}$ again can be made to vanish to arbitrarily large order for small enough $\varepsilon$.

However, in applications we generally require an invariant manifold in some chosen domain of interest that resolve phenomena on chosen time scales of interest. Such subjective choices, informed by the governing equations, generally dictate the chosen bound $\alpha$ separated by a big enough gap from the bounds $\beta,\gamma$ so that the centre manifold evolution, constructed to a valid order $p$, provides a useful model over the chosen domain for the desired phenomena.

Remark 2.19**.**

The derivative at $0\in V_{\infty}$ of the coordinate transformation $\xi_{p}$ is the identity map, and hence invertible. If a suitable generalisation of the inverse function theorem applies to $\xi_{p}$, such as a version of the Nash–Moser theorem, then it follows that $\xi_{p}$ is a local diffeomorphism at zero. Then it would be justified to call the invariant subsets of Definition 2.10 invariant submanifolds in this setting (at least in a neighbourhood of zero), see Remark 2.15.

Remark 2.20**.**

In the proof of Theorem 2.17, explicit constructions of the maps $F_{p}$ and $\xi_{p}$ are given. In practice, however, it can be easier to determine these maps in more direct ways. This is illustrated in an example in Section 9. Theorem 2.17 implies that one can always find these maps. We prove this by giving a construction that always leads to an answer, even though more direct constructions may exist in specific situations.

Similarly, the domain $D_{\mu}$ in Proposition 2.9, defined in 8.7, is guaranteed to have the properties in Proposition 2.9. In practice, these properties often hold on much larger domains.

2.6 A general class of PDEs in bounded domains

Because Theorem 2.17 applies to abstract Banach spaces $V_{k}$, it gives one the flexibility to choose these spaces such that, for specific pde applications,

the residual $R_{p}$ is of order $p$ with respect to norms relevant to the problem, and 2. 2.

the spaces $V_{k}$ incorporate the relevant boundary conditions.

This subsection explores a class of nonlinear pdes to which Theorem 2.17, and hence Definition 2.10, apply.

Let $d\in\mathbb{N}$ be the dimension of the domain of the pdes to be considered. Let $\Omega$ be a bounded, open subset of $\mathbb{R}^{d}$, or of a $d$-dimensional manifold, with $C^{1}$ boundary. Let $m\in\mathbb{N}$, and let $1\leq p<\infty$. For $k\in\mathbb{N}$, let $V_{k}$ be the Sobolev space $W^{k-1,k+1}(\Omega,\mathbb{R}^{m})$.

Let $A\colon C_{c}^{\infty}(\Omega;\mathbb{R}^{m})\to C_{c}^{\infty}(\Omega;\mathbb{R}^{m})$ be a linear partial differential operator. (Here the subscript $c$ denotes compactly supported functions.) Let $s\in\mathbb{N}$, with $s\geq 2$, be the ‘polynomial’ order of the nonlinearities in the pdes. Let

[TABLE]

be linear partial differential operators. For index-vector $q\in\mathbb{N}_{0}^{s}$ and $u\in C^{\infty}_{c}(\Omega,\mathbb{R}^{m})$, we set

[TABLE]

Let $\alpha$, $\beta$, $\gamma$ and $\tilde{\mu}$ be as in Section 2.5. Fix smooth functions222The real line may be replaced by a smaller open interval. $a_{q}^{j}\colon\mathbb{R}\to\mathbb{C}$, for $q\in\mathbb{N}_{0}^{s}$, with $|q|\leq s$, such that these functions and all their derivatives grow at most polynomially. Define $f\colon\mathbb{R}\times C_{c}^{\infty}(\Omega;\mathbb{R}^{m})\to C_{c}^{\infty}(\Omega;\mathbb{R}^{m})$ by $f(t,u):=(f_{1}(t,u),\ldots,f_{m}(t,u))$, where for each $j$,

[TABLE]

for $t\in\mathbb{R}$ and $u\in C_{c}^{\infty}(\Omega;\mathbb{R}^{m})$. Suppose that $f=\mathcal{O}(2)$.

We write

[TABLE]

Then $C^{\infty}_{c}(\Omega;\mathbb{R}^{m})\subset W^{\infty}(\Omega;\mathbb{R}^{m})\subset C^{\infty}(\Omega;\mathbb{R}^{m})$. The maps $A$ and $f$ extend continuously to $W^{\infty}(\Omega;\mathbb{R}^{m})$. Suppose that that the eigenfunctions $\{e_{j}\}_{j=1}^{\infty}$ of this extension of $A$ form a Hilbert basis of $L^{2}(\Omega,\mathbb{R}^{m})$.

Theorem 2.21**.**

The spaces $V_{k}$ and the maps $A$ and $f$ satisfy the hypotheses of Theorem 2.17.

We prove Theorem 2.21 in Section 4.4. Together with Theorem 2.17, it has the following immediate consequence.

Corollary 2.22**.**

Let $p\in\mathbb{N}$ be such that $p\geq 2$. Suppose that $\alpha$, $\beta$, $\gamma$ and $\tilde{\mu}$ satisfy $\beta-(p+1)\alpha>\tilde{\mu}$ and $\gamma-(p+1)\alpha>\tilde{\mu}$ (as in Theorem 2.17). Then there are infinitely differentiable maps

[TABLE]

where $F_{p}$ is a polynomial map that separates invariant subspaces, such that if $X$ and $x$ are as in 2.7 and 2.6, then

[TABLE]

for all $t\in I$. Further, for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that for all $u\in W^{\infty}(\Omega;\mathbb{R}^{m})$,

[TABLE]

as $v\to 0$ in $W^{k-1,k+1}(\Omega,\mathbb{R}^{m})$. There is a construction of the map $\xi_{p}$ in which all integrals over $I$ that occur are $\tilde{\mu}$-regular.

This corollary shows that any pde of the form 2.2, with $A$ and $f$ as in this subsection, is equivalent up to a residual of order $p$ to a pde with clear invariant manifolds, as in Definition 2.10.

Example 2.23**.**

Suppose that $\Omega=S^{1}$, the circle. This amounts to imposing periodic boundary conditions. Take $m=1$, and let $A\colon C^{\infty}(S^{1})\to C^{\infty}(S^{1})$ by any linear partial differential operator with constant coefficients. Its eigenfunctions, $e_{j}(\theta)=e^{ij\theta}$ for $j\in\mathbb{Z}$ and $\theta\in\mathbb{R}/2\pi\mathbb{Z}$, are orthogonal in the Sobolev spaces $W^{k,2}(S^{1})$. For a map $f$ as in Theorem 2.21, that is, a polynomial expression in derivatives of functions, whose polynomial coefficients increase at most polynomially, Theorem 2.21 implies that the conditions of Theorem 2.17 are satisfied in this case, so Corollary 2.22 applies. This generalises directly to cases where $\Omega$ is a higher-dimensional torus; that is, to problems in $\mathbb{R}^{d}$ with periodic boundary conditions. Here we used the case where the domain $\Omega$ is a manifold, rather than an open subset of $\mathbb{R}^{d}$.

Most of the rest of this paper is devoted to proofs of Theorems 2.17 and 2.21, and developing the tools used in these proofs. In Section 8, we prove Lemmas 2.8 and 2.9. In Section 9 we illustrate Corollary 2.22 by working out an example.

3 Derivatives and polynomials

In this section we review standard material on derivatives of maps between normed vector spaces. We also briefly discuss polynomial maps between normed vector spaces. Throughout this section, $(V,\|\cdot\|_{V})$ and $(W,\|\cdot\|_{W})$ are normed vector spaces, possibly infinite-dimensional. Let $U\subset V$ be an open subset, and let $f\colon U\to W$ be a map. We fix an element $u\in U$.

3.1 First order derivatives

This subsection and the next contain some standard definitions and facts about derivatives of maps between normed vector spaces. Details and proofs can be found in various textbooks [35, e.g.].

For a map $\varphi\colon V\supset\operatorname{dom}(\varphi)\to W$, we use the notation $\varphi(h)=o(h)$ for the statement

[TABLE]

were $h$ runs over $\operatorname{dom}(\varphi)\setminus\{0\}$.

Definition 3.1**.**

The map $f:U\to W$ is differentiable at $u$, if there is an operator $f^{\prime}(u)\in\mathcal{B}(V,W)$ such that

[TABLE]

Then $f^{\prime}(u)$ is the derivative of $f$ at $u$. If $f$ is differentiable at every point in $U$, then we say that $f$ is differentiable. In that case, the derivative of $f$ is the map

[TABLE]

mapping $u\in U$ to $f^{\prime}(u)$.

The derivative of a map at a point is unique, if it exists.

Lemma 3.2** (Chain rule).**

Let $(X,\|\cdot\|_{X})$ be a third normed vector space. Let $A\subset W$ be an open subset containing $f(U)$. If $g\colon A\to X$ is differentiable at $f(u)$ and $f$ is differentiable at $u$, then $g\circ f$ is differentiable at $u$, and

[TABLE]

{ArXiv}

Proof.

For all $h\in V$ such that $u+h\in U$, differentiability of $f$ at $u$ and of $g$ at $f(u)$ imply that

[TABLE]

Since $g^{\prime}(f(u))$ and $f^{\prime}(u)$ are bounded operators, the second term on the right-hand side equals $g^{\prime}(f(u))f^{\prime}(u)h+o(h)$, while the last term is $o(h)$. ∎

Definition 3.3**.**

The map $f$ is a near-identity at $u$ if the map 3.1 is continuous in a neighbourhood of $u$, and

[TABLE]

3.2 Higher order derivatives

Fix a positive integer $n\in\mathbb{N}$. We write $\mathcal{B}^{n}(V,W)$ for the space of multilinear maps $\lambda\colon V^{n}\to W$ for which the norm

[TABLE]

is finite. There is a natural isometric isomorphism

[TABLE]

mapping an operator $T$ in the left-hand side to the operator $\lambda\in\mathcal{B}^{n}(V,W)$ given by

[TABLE]

for $v_{1},\ldots,v_{n}\in V$.

Suppose $f:U\to W$ is differentiable. The map $f$ is twice differentiable at $u$ if the map 3.1 is differentiable at $u$. Then we write

[TABLE]

Inductively, for $n\geq 2$, $f$ is defined to be $n$ times differentiable at $u$ if it is $n-1$ times differentiable, and the map

[TABLE]

is differentiable at $u$. We then set

[TABLE]

In this case, we write

[TABLE]

As before, we say that $f$ is $n$ times differentiable if it is $n$ times differentiable at every point in $U$. And infinitely differentiable means $n$ times differentiable for every $n\in\mathbb{N}$.

Theorem 3.4** (Taylor’s theorem).**

Suppose $f$ is $n+1$ times differentiable. Suppose that $\|f^{(n+1)}(\xi)\|\leq M$ for all $\xi$ in a closed ball around $u$ contained in $U$. Then for every $h$ in this ball,

[TABLE]

{ArXiv}

3.3 Example: Burgers’ equation

An example of a map to which we would like to apply the material in this section and the next is the nonlinear term $u_{x}u$ in Burgers’ equation

[TABLE]

Let $\Omega\subset\mathbb{R}$ be a bounded, open interval in $x$. For every $k\in\mathbb{N}_{0}$, consider the $k$th $L^{2}$-Sobolev space $W^{k,2}(\Omega)$, with the inner product

[TABLE]

Consider the map $f\colon W^{1,2}(\Omega)\to L^{1}(\Omega)$ given by

[TABLE]

First of all, for $u\in C^{\infty}_{c}(\Omega)$, the Cauchy–Schwartz inequality for $L^{2}(\Omega)$ (or Hölder’s inequality) implies that

[TABLE]

So $f$ indeed maps $W^{1,2}(\Omega)$ into $L^{1}(\Omega)$.

We claim that $f$ is infinitely differentiable. Indeed, for $u,h\in W^{1,2}(\Omega)$,

[TABLE]

And by 3.6, $\|h^{\prime}h\|_{L^{1}}\leq\|h\|_{W^{1,2}}^{2},$ so

[TABLE]

The map $f^{\prime}(u)\colon W^{1,2}(\Omega)\to L^{1}(\Omega)$ is bounded, because, analogously to 3.6,

[TABLE]

If $u,h_{1},h_{2}\in W^{1,2}(\Omega)$, then

[TABLE]

So $f^{(2)}(u)$ is the operator in $\mathcal{B}^{2}(W^{1,2}(\Omega),L^{1}(\Omega))$ given by

[TABLE]

The term $o(h_{2})$ in the definition of the derivative is zero in this case, and that $f^{(2)}(u)$ does not depend on $u$. This implies that for every $n\geq 3$, $f^{(n)}(u)=0$. So $f$ is indeed infinitely differentiable.

3.4 Bounded polynomial maps

An operator in $\mathcal{B}^{n}(V,W)$ is said to be symmetric if it is invariant under permutations of its arguments. Let $S\mathcal{B}^{n}(V,W)$ be the subspace of symmetric operators in $\mathcal{B}^{n}(V,W)$. An example of such a symmetric operator is the $n$th derivative of a map.

Lemma 3.5**.**

If $f$ is $n$ times differentiable at $u$, then $f^{(n)}(u)$ is symmetric.

We denote the permutation group of $\{1,\ldots,n\}$ by $\Sigma_{n}$. {ArXiv}

Lemma 3.6**.**

The subspace $S\mathcal{B}^{n}(V,W)\subset\mathcal{B}^{n}(V,W)$ is closed.

Proof.

If $T\in\overline{S\mathcal{B}^{n}(V,W)}\setminus S\mathcal{B}^{n}(V,W)$, and $v_{1},\ldots,v_{n}\in V$ and $\sigma\in\Sigma_{n}$ are such that $T(v_{1},\ldots,v_{n})\not=T(v_{\sigma(1)},\ldots,v_{\sigma(n)}),$ set

[TABLE]

Let $\tilde{T}\in S\mathcal{B}^{n}(V,W)$ be such that $\|\tilde{T}-T\|<\varepsilon/2$, for the norm 3.2. Then symmetry of $S$ and the triangle inequality imply that

[TABLE]

a contradiction. ∎

By this lemma, $S\mathcal{B}^{n}(V,W)$ is a Banach space if $V$ and $W$ are.

Let $S\colon\mathcal{B}^{n}(V,W)\to S\mathcal{B}^{n}(V,W)$ be the symmetrisation operator: for every $\lambda\in\mathcal{B}^{n}(V,W)$ and $v_{1},\ldots,v_{n}\in V$,

[TABLE]

{ArXiv}

(An alternative proof of Lemma 3.6 is to show that $S$ is continuous, and to note that $S\mathcal{B}^{n}(V,W)$ is the zero level set of $S$ minus the identity.)

{journal} The operator $S$ is continuous, and that $S\mathcal{B}^{n}(V,W)$ is the zero level set of $S$ minus the identity, and hence closed in $\mathcal{B}^{n}(V,W)$. So $S\mathcal{B}^{n}(V,W)$ is a Banach space if $V$ and $W$ are.

An element $\lambda\in\mathcal{B}^{n}(V,W)$ defines a map $p_{\lambda}\colon V\to W$ by

[TABLE]

We have $p_{S\lambda}=p_{\lambda}$, and the map $\lambda\mapsto p_{\lambda}$ is injective on $S\mathcal{B}^{n}(V,W)$.

Definition 3.7**.**

A bounded homogeneous polynomial map of degree $n$ from $V$ to $W$ is a map of the form $p_{\lambda}$ as in 3.8. We write $\operatorname{Pol}^{n}(V,W)$ for the space of such maps. It inherits a norm from the space $S\mathcal{B}^{n}(V,W)$ via the linear isomorphism $\lambda\mapsto p_{\lambda}$. If $\lambda\not=0$, then the degree of  $p_{\lambda}$ is $n$.

A bounded polynomial map from $V$ to $W$ is a finite sum of bounded homogeneous polynomial maps. The degree of a bounded polynomial map is the degree of its highest-degree homogeneous term.

We write $\operatorname{Pol}(V,W)$ for the space of all bounded polynomial maps from $V$ to $W$. This is the algebraic direct sum of the spaces $\operatorname{Pol}^{n}(V,W)$.

{ArXiv}

By Lemma 3.6, $\operatorname{Pol}^{n}(V,W)$ is a Banach space if $V$ and $W$ are.

{journal} Because $S\mathcal{B}^{n}(V,W)$ is a Banach space if $V$ and $W$ are, so is $\operatorname{Pol}^{n}(V,W)$.

We could define $\operatorname{Pol}^{0}(V,W)$ as the space of constant maps into $W$, but we only consider homogeneous polynomials of order at least one.

If $f$ is $n$ times differentiable at $u$, then we have the map

[TABLE]

Lemma 3.8–3.11 below are basic facts showing that bounded polynomials and their orders and compositions behave as one would expect. {journal} Their proofs are short and straightforward.

Lemma 3.8**.**

Every bounded polynomial map is infinitely differentiable.

{ArXiv}

Proof.

Let $\lambda\in S\mathcal{B}^{n}(V,W)$, for some $n\geq 2$. Then for all $u,h\in V$,

[TABLE]

Hence $p_{\lambda}$ is differentiable, and

[TABLE]

where on the right-hand side, the operator $\lambda$ is applied to $n-1$ copies of $u$, to give an element of $\mathcal{B}(V,W)$. Hence $p_{\lambda}^{\prime}$ is a bounded polynomial map in $\operatorname{Pol}^{n-1}(V,\mathcal{B}(V,W))$. This proves the claim by induction. ∎

Lemma 3.9**.**

If $p\in\operatorname{Pol}^{n}(V,W)$, then there is a constant $C>0$ such that for all $v\in V$,

[TABLE]

{ArXiv}

Proof.

Let $\lambda\in\mathcal{B}^{n}(V,W)$. By boundedness and multilinearity of $\lambda$, we have for all nonzero $v\in V$,

[TABLE]

∎

Lemma 3.10**.**

If $p$ is a polynomial map from $V$ to $W$ of order lower than $n$, and

[TABLE]

as $v\to 0$ in $V$, then $p=0$.

{ArXiv}

Proof.

Let $m,n\in\mathbb{N}$. Let $\lambda\in S\mathcal{B}^{m}(V,W)$, and suppose that $\|p_{\lambda}(v)\|_{W}=O(\|v\|_{V}^{n})$, as $v\to 0$ in $V$. Then there is a $C>0$ such that for all $v\in V$ with unit norm and $s>0$ small enough,

[TABLE]

If $m<n$, this implies that $\|p_{\lambda}(v)\|_{W}=0$. ∎

Lemma 3.11**.**

If $p_{1}\in\operatorname{Pol}^{m}(U,V)$ and $p_{2}\in\operatorname{Pol}^{n}(V,W)$, then $p_{2}\circ p_{1}\in\operatorname{Pol}^{mn}(U,W)$.

{ArXiv}

Proof.

For $\lambda_{1}\in\mathcal{B}^{m}(U,V)$ and $\lambda_{2}\in\mathcal{B}^{n}(V,W)$, define $\lambda_{2}\circ\lambda_{1}\colon U^{mn}\to W$ by

[TABLE]

for $u_{jk}\in U$. Then one checks directly that $\lambda_{2}\circ\lambda_{1}\in\mathcal{B}^{mn}(U,V)$. This implies the claim about polynomials. ∎

3.5 Standard monomials

Let $V^{*}:=\mathcal{B}(V,\mathbb{C})$ be the continuous dual of $V$. We denote the pairing between $V^{*}$ and $V$ by $\langle\relbar,\relbar\rangle$. For every $j\in\mathbb{N}$, let $e^{j}\in V^{*}$ be given. What follows is most natural if $V$ is a Hilbert space and $e^{j}$ is given by taking inner products with an element $e_{j}$ of a Hilbert basis, but it applies more generally.

Consider a multi-index $q\in\mathbb{N}_{0}^{\infty}$. If $|q|=n$, and $m$ is the largest number for which $q_{m}\not=0$, then we define the element

[TABLE]

In other words, for all $v_{1},\ldots,v_{n}\in V$,

[TABLE]

We write $p^{q}:=p_{e^{q}}$ for the corresponding homogeneous polynomial. One could call this the standard $q$-monomial with respect to the set $\{e^{j}\}_{j=1}^{\infty}$. (If $V=\mathbb{C}^{k}$ and the elements $e^{j}$ are the standard coordinates, then the monomial functions in the usual sense are precisely the scalar multiples of the maps $p^{q}$.)

For $v\in V$, we write

[TABLE]

This product is finite (since $q$ has finitely many nonzero terms) and depends on the set $\{e^{j}\}$. The following lemma follows from the definition of the derivative.

Lemma 3.12**.**

The derivative of $p^{q}$ in 3.10 is given by

[TABLE]

for all $u,h\in V$.

4 Compact derivatives and polynomials

It is a nontrivial question in what sense differentiable maps between normed vector spaces can be approximated by polynomial maps [1, 11, 12, 13, 22, 23, e.g.]. In this section we discuss an approach to this problem that is suitable for our purposes. This discussion includes the further problem of approximating a polynomial by sums of the standard monomials of Section 3.5. The polynomials for which this is possible are the compact polynomials introduced in Section 4.2.

Section 4.3 introduces compactly differentiable maps. We combine these with Taylor’s theorem to express the lowest order parts of such maps in terms of standard monomials. We discuss a class of examples of compactly differentiable maps relevant to the study of pdes.

4.1 Compact multilinear maps

Let $V$ and $W$ be Banach spaces. Let $\mathcal{K}^{n}(V,W)\subset\mathcal{B}^{n}(V,W)$ be the image of the space

[TABLE]

under the isomorphism 3.3. {journal} Using induction on $n$, one can show that $\mathcal{K}^{n}(V,W)$ is closed in $\mathcal{B}^{n}(V,W)$, and hence a Banach space.

{ArXiv}

Lemma 4.1**.**

For every $n\in\mathbb{N}$ the space $\mathcal{K}^{n}(V,W)$ is closed in $\mathcal{B}^{n}(V,W)$.

Proof.

We use induction on $n$. For $n=1$ the claim is standard. Suppose the claim holds for $n$. Then

[TABLE]

which is a closed subspace of $\mathcal{B}(V,\mathcal{K}^{n}(V,W))$. And that space is closed in $\mathcal{B}^{n+1}(V,W)$ since $\mathcal{K}^{n}(V,W)$ is closed in $\mathcal{B}^{n}(V,W)$ by the induction hypothesis. ∎

By Lemma 4.1, $\mathcal{K}^{n}(V,W)$ is a Banach space.

Let $\{e^{j}\}_{j=1}^{\infty}\subset V^{*}$ and $\{f_{k}\}_{k=1}^{\infty}\subset W$ be countable subsets whose spans are dense. (So $V^{*}$ and $W$ are separable.) For any $\alpha\in\mathbb{N}^{n}$, consider the multilinear map

[TABLE]

A Banach space has the approximation property if every compact operator on the space is a norm-limit of finite-rank operators. This is always true for Hilbert spaces, but we need to consider more general Banach spaces for applications. {journal} The following result is standard in the case where $V$ and $W$ are Hilbert spaces.

Proposition 4.2**.**

If $V^{*}$ has the approximation property, then the space $\operatorname{span}\{e^{j}\otimes f_{k}:j,k\in\mathbb{N}\}$ is dense in $\mathcal{K}(V,W)$.

Proof.

Since $V^{*}$ has the approximation property, the space of finite-rank operators (linear operators with finite-dimensional images) is dense in $\mathcal{K}(V,W)$. See for example Proposition 4.12(b) in the book by Ryan [31]. The space $\operatorname{span}\{e^{j}\otimes f_{k}:j,k\in\mathbb{N}\}$ is dense in the space of finite-rank operators, so the claim follows. ∎

Lemma 4.3**.**

If $V^{*}$ has the approximation property, then for every $n\in\mathbb{N}$, the span of $\{e^{\alpha}\otimes f_{k}:\alpha\in\mathbb{N}^{n},k\in\mathbb{N}\}$ is dense in $\mathcal{K}^{n}(V,W)$.

Proof.

We prove this by induction on $n$. If $n=1$, then the claim is precisely Proposition A.1{ArXiv} in the appendix. Now suppose that the claim holds for a given $n$. By definition,

[TABLE]

By the induction hypothesis, the set $\{e^{\alpha}\otimes f_{k}:\alpha\in\mathbb{N}^{n},k\in\mathbb{N}\}$ has dense span in $\mathcal{K}^{n}(V,W)$. Therefore, Proposition A.1, with $W$ replaced by $\mathcal{K}^{n}(V,W)$, implies that the set

[TABLE]

has dense span in $\mathcal{K}^{n+1}(V,W)$. This is precisely the claim for $n+1$. ∎

A Schauder basis of a Banach space $V$ is a subset $\{e_{j}\}_{j=1}^{\infty}\subset V$ such that for each $v\in V$, there are unique complex numbers $\{v^{j}\}_{j\in\mathbb{N}}$ such that

[TABLE]

A space with a Schauder basis has the approximation property.

Lemma 4.4**.**

If $\{e^{j}\}_{j=1}^{\infty}$ is a Schauder basis of $V^{*}$ and $\{f_{k}\}_{k=1}^{\infty}$ is a Schauder basis of $W$, then for every $n\in N$, the set $\{e^{\alpha}\otimes f_{k}:\alpha\in\mathbb{N}^{n},k\in\mathbb{N}\}$ is a Schauder basis of $\mathcal{K}^{n}(V,W)$.

Proof.

Lemma 4.3 implies that $\{e^{\alpha}\otimes f_{k}:\alpha\in\mathbb{N}^{n},k\in\mathbb{N}\}$ has dense span. So it remains to show that if $a_{\alpha}^{k}\in\mathbb{C}$ are such that

[TABLE]

then $a_{\alpha}^{k}=0$ for all $\alpha$ and $k$. Since $\{f_{k}\}_{k=1}^{\infty}$ is a Schauder basis of $W$, this reduces to the case where $W=\mathbb{C}$. {journal} In that case, one can prove the claim by induction on $n$, using the fact that $\{e^{j}\}_{j=1}^{\infty}$ is a Schauder basis of $V^{*}$.

{ArXiv} We prove the claim in that case, by induction on $n$.

If $n=1$, then the claim follows since $\{e^{j}\}_{j=1}^{\infty}$ is a Schauder basis of $V^{*}$. So suppose that the claim holds for a given $n$, and let $a_{\alpha}^{k}\in\mathbb{C}$ be such that

[TABLE]

Then for all $v_{1},\ldots,v_{n}\in V$,

[TABLE]

Since, $\{e^{j}\}_{j=1}^{\infty}$ is a Schauder basis of $V^{*}$, this implies that for every $j\in\mathbb{N}$,

[TABLE]

Because the sum

[TABLE]

converges in $\mathcal{B}^{n}(V,\mathbb{C})$, we find that the sum converges to zero in this space. By the induction hypothesis, this implies that $a_{j,\alpha}=0$ for every $j\in\mathbb{N}$ and $\alpha\in\mathbb{N}^{n}$.

∎

Remark 4.5**.**

In the induction step in the proof of the special case of Lemma 4.3 where $W$ is a Hillbert space, we still need the general version of Proposition A.1, where $W$ is a Banach space. This is because $\mathcal{K}^{n}(V,W)$ is only a Banach space, even if $W$ is a Hilbert space.

The subspace $S\mathcal{K}^{n}(V,W)$ of symmetric operators in $\mathcal{K}^{n}(V,W)$ is closed in $\mathcal{B}^{n}(V,W)$, since it is the intersection of the closed subspaces $S\mathcal{B}^{n}(V,W)$ and $\mathcal{K}^{n}(V,W)$ {ArXiv} (Lemmas 3.6 and 4.1).

Hence $S\mathcal{K}^{n}(V,W)$ is a Banach space with respect to the norm 3.2.

A Schauder basis $\{e_{j}\}_{j=1}^{\infty}$ of a Banach space $V$ is unconditional if there is a constant $C>0$ such that for all $a^{j},\varepsilon_{j}\in\mathbb{C}$ with $|\varepsilon_{j}|=1$, and all $n\in\mathbb{N}$,

[TABLE]

In that case, convergence of $\sum_{j=1}^{n}a^{j}e_{j}$ implies convergence of $\sum_{j\in A}a^{j}e_{j}$, for every $A\subset\mathbb{N}$.

Lemma 4.6**.**

Suppose that $V$ and $W$ are Hilbert spaces, and that $\{e_{j}\}_{j=1}^{\infty}$ and $\{f_{k}\}_{k=1}^{\infty}$ are orthogonal sets in $V$ and $W$ respectively, with dense spans. Let $e^{j}\in V^{*}$ be defined by taking inner products with $e_{j}$. Then $\{e^{\alpha}\otimes f_{k}:\alpha\in\mathbb{N}^{n},k\in\mathbb{N}\}$ is an unconditional Schauder basis of $\mathcal{K}^{n}(V,W)$.

Proof.

The set $\{e^{\alpha}\otimes f_{k}:\alpha\in\mathbb{N}^{n},k\in\mathbb{N}\}$ is a Schauder basis of $\mathcal{K}^{n}(V,W)$ by Lemma 4.4. It remains to show that it is unconditional. By rescaling the vectors $e_{j}$ and $f_{k}$, we reduce the proof to the case where $\{e_{j}\}_{j=1}^{\infty}$ and $\{f_{k}\}_{k=1}^{\infty}$ are Hilbert bases. In that case, for all finite subsets $A\subset\mathbb{N}_{0}^{n}\times\mathbb{N}$ and all $a_{\alpha}^{k}\in\mathbb{C}$,

[TABLE]

∎

Lemma 4.7**.**

Let $U$, $V$ and $W$ be normed vector spaces, and $n\in\mathbb{N}$. Let $\lambda\in\mathcal{B}^{n}(V,W)$, and $a_{1},\ldots,a_{n}\in\mathcal{K}(U,V)$. Define $\nu\colon U\times\cdots\times U\to W$ by

[TABLE]

for all $u_{1},\ldots,u_{n}\in U$. Then $\nu\in\mathcal{K}^{n}(U,W)$.

Proof.

We use induction on $n$. For $n=1$, $v\in\mathcal{K}^{1}(U,W)$ because the composition of a compact operator and a bounded operator is compact. Suppose that the claim holds for a given $n$. Let $\lambda\in\mathcal{B}^{n+1}(V,W)$, and $a_{1},\ldots,a_{n+1}\in\mathcal{K}(U,V)$. For a fixed $u\in U$, define $\nu_{u}\in\mathcal{B}^{n}(U,W)$ by

[TABLE]

for $u_{1},\ldots,u_{n}\in U$. For a fixed $v\in V$, define $\lambda_{v}\in\mathcal{B}^{n}(V,W)$ by

[TABLE]

for $v_{1},\ldots,v_{n}\in V$. Then for all $u_{1},\ldots,u_{n}\in U$,

[TABLE]

So by the induction hypothesis, $\nu_{u}\in\mathcal{K}^{n}(U,W)$. In this way, we obtain the map

[TABLE]

mapping $u\in U$ to $\nu_{u}$. It remains to show that $\tilde{\nu}$ is a compact operator.

Define $\tilde{\lambda}\in\mathcal{B}(V,\mathcal{K}^{n}(U,W))$ by

[TABLE]

for $v\in V$ and $u_{1},\ldots,u_{n}\in U$. (This map takes values in $\mathcal{K}^{n}(U,W)$ by the induction hypothesis.) Since $a_{n+1}$ is compact and $\tilde{\lambda}$ is bounded, we find that $\tilde{\nu}=\tilde{\lambda}\circ a_{n+1}$ is a compact operator. ∎

4.2 Compact polynomial maps

Definition 4.8**.**

A compact homogeneous polynomial map of degree $n$ from $V$ to $W$ is a map of the form $p_{\lambda}$ as in 3.8, for $\lambda\in\mathcal{K}^{n}(V,W)$. We write $\mathcal{K}\operatorname{Pol}^{n}(V,W)$ for the space of such maps. This space inherits a norm from the space $\operatorname{Pol}^{n}(V,W)$ it is contained in.

If $\lambda\not=0$, then the degree of $p_{\lambda}$ is $n$. A compact polynomial map from $V$ to $W$ is a finite sum of compact homogeneous polynomial maps. The degree of a compact polynomial map is the degree of its highest-degree homogeneous term. We write $\mathcal{K}\operatorname{Pol}(V,W)$ for the space of all compact polynomial maps between these spaces.

The isometric isomorphism $S\mathcal{B}^{n}(V,W)\cong\operatorname{Pol}^{n}(V,W)$ restricts to an isometric isomorphism $S\mathcal{K}^{n}(V,W)\cong\mathcal{K}\operatorname{Pol}^{n}(V,W)$. So $\mathcal{K}\operatorname{Pol}^{n}(V,W)$ is a closed subspace of the Banach space $\operatorname{Pol}^{n}(V,W)$, and hence is a Banach space itself.

For every $w\in W$, an operator of the form $e^{q}\otimes w$, with $e^{q}$ as in 3.9, is an element of $\mathcal{K}^{n}(V,W)$. Indeed, $e^{q}\otimes w$ is an iteration of rank-one operators, So $p^{q}\otimes w\in\mathcal{K}\operatorname{Pol}^{n}(V,W)$.

The following proposition is the reason why we are interested in compact polynomial maps.

Proposition 4.9**.**

*Suppose that $V$ and $W$ are Banach spaces, that $V^{*}$ has a Schauder basis $\{e^{j}\}_{j=1}^{\infty}$, and that $W$ has a Schauder basis $\{f_{k}\}_{k=1}^{\infty}$. Then the elements *

[TABLE]

where the multi-index $q$ ranges over the elements of $\mathbb{N}_{0}^{\infty}$ with $|q|=n$, and $k$ ranges over the positive integers, form a Schauder basis of $\mathcal{K}\operatorname{Pol}^{n}(V,W)$.

Proof.

Consider the space

[TABLE]

Lemma 4.4 implies that the set of $e^{\alpha}\otimes f_{k}$ where $\alpha_{1}\leq\cdots\leq\alpha_{n}$ is a Schauder basis of $X$. And $S\colon X\to S\mathcal{K}^{n}(V,W)$ is a bounded linear isomorphism with bounded inverse. Since such isomorphisms map Schauder bases to Schauder bases, we find that the set $Se^{\alpha}\otimes f_{k}$, for non-decreasing $\alpha$ as above, is a Schauder basis of $S\mathcal{K}^{n}(V,W)$.

For $\alpha\in\mathbb{N}$ with non-decreasing entries, define $q(\alpha)\in\mathbb{N}_{0}^{\infty}$ by

[TABLE]

Then $e^{\alpha}=e^{q(\alpha)}$. (Note that $e^{\alpha}$, for $\alpha\in\mathbb{N}^{n}$, and $e^{q}$, for $q\in\mathbb{N}_{0}^{\infty}$, are defined differently; compare 3.9 and 4.1.) Every sequence in $\mathbb{N}_{0}^{\infty}$ occurs in exactly one way as $q(\alpha)$, for alpha as above, so $Se^{q}\otimes f_{k}$, where $q\in\mathbb{N}^{\infty}_{0}$ and $k\in\mathbb{N}$, is a Schauder basis of $S\mathcal{K}^{n}(V,W)$. Since $p^{q}=p_{Se^{q}}$, the claim follows. ∎

A reformulation of Proposition 4.9 is that for every compact polynomial map $p\in\mathcal{K}\operatorname{Pol}^{n}(V,W)$, there are unique complex numbers $a^{k}_{q}$ such that

[TABLE]

where the sum converges in the norm on $\operatorname{Pol}^{n}(V,W)$. Conversely, all polynomial maps $p$ of this form are compact.

Lemma 4.10**.**

In the setting of Lemma 4.6, the set $\{p^{q}\otimes f_{k}:|q|=n,k\in\mathbb{N}\}$ is an unconditional Schauder basis of $\mathcal{K}\operatorname{Pol}^{n}(V,W)$.

Proof.

The proof is analogous to the proof of Proposition 4.9, where we now use Lemma 4.6 instead of Lemma 4.4, and we use the fact that bounded linear isomorphisms with bounded inverses map unconditional Schauder bases to unconditional Schauder bases. ∎

Lemma 4.11**.**

If $p_{1}\in\mathcal{K}\operatorname{Pol}^{m}(U,V)$ and $p_{2}\in\mathcal{K}\operatorname{Pol}^{n}(V,W)$, then $p_{2}\circ p_{1}\in\mathcal{K}\operatorname{Pol}^{mn}(U,W)$.

Proof.

The proof is similar to the proof of Lemma 3.11, with bounded multilinear maps replaced by compact ones. ∎

Remark 4.12**.**

Other notions of compact polynomial maps were studied by Gonzalo, Jaramillo and Pełczyńsky in [15, 25].

4.3 Compactly differentiable maps

Let $V$ and $W$ be normed vector spaces, let $U\subset V$ be an open subset containing a vector $u$, and let $f\colon U\to W$ be $n$ times differentiable at $u$.

Definition 4.13**.**

The map $f$ is $n$ times compactly differentiable at $u$ if

[TABLE]

If $f$ is $n$ times compactly differentiable at $u$, then by Lemma 3.5,

[TABLE]

Then the map $h\mapsto f^{(n)}(u)h^{n}$ of 3.4 is the compact polynomial map associated to $f^{(n)}(u)$. Together with Theorem 3.4 and Proposition 4.9, this leads to the following conclusion.

Corollary 4.14**.**

Suppose that $V$ and $W$ are Banach spaces, that $V^{*}$ has a Schauder basis $\{e^{j}\}_{j=1}^{\infty}$, and that $W$ has a Schauder basis $\{f_{k}\}_{k=1}^{\infty}$. Suppose $f$ is $n+1$ times differentiable, and $k$ times compactly differentiable for every $k\leq n$. Then there are unique complex numbers $a^{k}_{q}$ such that

[TABLE]

where the part of the sum where $|q|=m$ converges as a function of $h$ in the norm on $\operatorname{Pol}^{m}(V,W)$, for $m=0,\ldots,n$.

(Note that, in 4.3, on the left-hand side $f$ is a map from $U$ to $W$, whereas on the right-hand side, $f_{k}$ is an element of $W$.)

Lemma 4.15**.**

A compact polynomial map is infinitely compactly differentiable.

Proof.

We show that the derivative of every homogeneous compact polynomial $p_{\lambda}\in\mathcal{K}\operatorname{Pol}^{n}(V,W)$, for $\lambda\in S\mathcal{K}^{n}(V,W)$, is a compact polynomial in $\mathcal{K}\operatorname{Pol}^{n-1}(V,\mathcal{K}(V,W))$. This implies the claim by induction on $n$. As in the proof of Lemma 3.8, $p^{\prime}_{\lambda}(u)h=n\lambda(h,u,\ldots,u)$ for all $u,h\in V$. In other words, $p^{\prime}_{\lambda}(u)=p_{\nu}$, with $\nu=n\lambda$, where we view $\lambda$ as an element of $\mathcal{K}^{n-1}(V,\mathcal{K}(V,W))$. This shows that $p^{\prime}_{\lambda}(u)\in\mathcal{K}\operatorname{Pol}^{n-1}(V,\mathcal{K}(V,W))$. ∎

Lemma 4.16**.**

Let $U$, $V$ and $W$ be normed vector spaces and let $f\colon U\to V$ and $g\colon V\to W$ be differentiable maps. If either $f$ or $g$ is compactly differentiable, then so is $g\circ f$.

Proof.

Lemma 3.2 implies that for all $u\in U$,

[TABLE]

If $f$ is compactly differentiable, then $f^{\prime}(u)\in\mathcal{K}(U,V)$. If $g$ is compactly differentiable, then $g^{\prime}(f(u))\in\mathcal{K}(V,W)$. In either case, we find that $(g\circ f)^{\prime}(u)\in\mathcal{K}(U,V)$. ∎

Lemma 4.17**.**

Let $U$, $V$ and $W$ be normed vector spaces, let $f\colon V\to W$ be $n$ times differentiable, and suppose that $U$ is a subspace of $V$, with compact inclusion map $j\colon U\hookrightarrow V$. Then $f\circ j$ is $n$ times compactly differentiable as a map from $U$ to $W$.

Proof.

Let $u,h_{1},\ldots,h_{n}\in U$. Then

[TABLE]

So the claim follows from Lemma 4.7. ∎

Remark 4.18**.**

It is possible for a map to be $n$ times compactly differentiable, but not $n-1$ times. For example, the first derivative of the identity operator on an infinite-dimensional Banach space $V$ is the identity map itself, and not a compact operator. But its higher-order derivatives are zero, and hence compact.

4.4 A class of compactly differentiable maps

We end this section by discussing a class of compactly differentiable maps (specifically, compact polynomials) that are relevant to the study of nonlinear pdes. These maps are polynomial expressions in derivatives of functions; see Proposition 4.20 below.

Let $\Omega\subset\mathbb{R}^{m}$ be a bounded open subset with $C^{1}$ boundary. For $k\in\mathbb{N}_{0}$ and $p>0$, consider the Sobolev space $W^{k,p}(\Omega)$.

Lemma 4.19**.**

Let $n\in\mathbb{N}$ and $p>1$. Pointwise multiplication of $n$ functions defines a map $\mu\in\mathcal{B}^{n}(W^{k,np}(\Omega),W^{k,p}(\Omega))$.

Proof.

Let $n\in\mathbb{N}$. By Hölder’s inequality, for all $u_{1},\ldots,u_{n}\in L^{np}(\Omega)$,

[TABLE]

For $\alpha\in\mathbb{N}_{0}^{m}$, there are combinatorial constants $c^{\alpha}_{\beta}$, for $\beta=(\beta^{(1)},\ldots,\beta^{(n)})$, with $\beta^{(1)},\ldots,\beta^{(n)}\in\mathbb{N}_{0}^{m}$ such that $|\beta^{(1)}|+\cdots+|\beta^{(n)}|\leq|\alpha|$, such that for all $u_{1},\ldots,u_{n}\in C^{\infty}_{c}(\Omega)$, we have the generalised Leibniz rule

[TABLE]

Together with 4.4, this implies that

[TABLE]

∎

Proposition 4.20**.**

Let $k,l,m_{1},m_{2},n\in\mathbb{N}$ and $p>1$, with $k\leq l$. Let

[TABLE]

be linear partial differential operators of orders smaller than $k$. Fix complex numbers $a_{q}^{j}$, for $q\in\mathbb{N}_{0}^{n}$ with $|q|\leq n$, and $j\in\{1,\ldots,m_{2}\}$. Define $f\colon C_{c}^{\infty}(\Omega;\mathbb{R}^{m_{1}})\to C_{c}^{\infty}(\Omega;\mathbb{R}^{m_{2}})$ by $f(u)=(f_{1}(u),\ldots,f_{m_{2}}(u))$, where for each $j$,

[TABLE]

for $u\in W^{l,np}(\Omega;\mathbb{R}^{m_{1}})$, and with $(Du)^{q}$ as in 2.9, defines a compact polynomial map

[TABLE]

Proof.

We first consider the case where $m_{2}=1$, and $a_{q}=1$ if $q=(1,\ldots,1)$, and zero otherwise. By Lemma 4.19, pointwise multiplication defines a map $\mu\in\mathcal{B}^{n}(W^{l-k,np}(\Omega),W^{l-k,p}(\Omega))$.

By Rellich’s lemma, boundedness of $\Omega$ implies that the maps $D_{1},\ldots,D_{n}$ extend to compact operators

[TABLE]

The map

[TABLE]

defined by

[TABLE]

for $u_{1},\ldots,u_{n}\in W^{l,np}(\Omega)$, is in $\mathcal{K}^{n}(W^{l,np}(\Omega),W^{l-k,p}(\Omega))$ by Lemma 4.7. Hence $p_{\nu}$ is an element of $\mathcal{K}\operatorname{Pol}^{n}(W^{l,np}(\Omega),W^{l-k,p}(\Omega))$.

Every component of a general map of the form 4.5 is a finite sum of maps of the form $p_{\nu}$ as above, applied to the components of $f$. Hence it is in $\mathcal{K}\operatorname{Pol}^{n}(W^{l,np}(\Omega;\mathbb{R}^{m_{1}}),W^{l-k,p}(\Omega;\mathbb{R}^{m_{2}}))$. ∎

Example 4.21**.**

{ArXiv}

Consider the map $f$ from Section 3.3, mapping $u$ to $u^{\prime}u$. We now view $f$ as a map from $W^{l,2p}(\Omega)$ to $W^{l-k,p}(\Omega)$, for $k\geq 1$, $l\geq k$ and $p>1$.

{journal} Let $k\geq 1$, $l\geq k$ and $p>1$ be integers. Let $\Omega\subset\mathbb{R}$ be a bounded open interval. Consider the map $f$ from $W^{l,2p}(\Omega)$ to $W^{l-k,p}(\Omega)$, mapping $u\in W^{l,2p}(\Omega)$ to $u^{\prime}u$.

Taking $m_{1}=m_{2}=1$, $n=2$, $D_{1}u=u^{\prime}$ and $D_{2}u=u$, for $u\in W^{l,2p}(\Omega)$, in Proposition 4.20, we find that $f$ is a compact polynomial in $\mathcal{K}\operatorname{Pol}^{2}(W^{l,2p}(\Omega),W^{l-k,p}(\Omega))$ for every $k>1$. Hence, by Lemma 4.15, $f$ is in particular infinitely compactly differentiable. For $k=1$, the map $f$ is only a bounded polynomial in $\operatorname{Pol}^{2}(W^{l,2p}(\Omega),W^{l-1,p}(\Omega))$.

Remark 4.22**.**

Proposition 4.20 extends directly to relatively compact open subsets $\Omega$ of manifolds. The latter extension is relevant, for example, if one uses periodic boundary conditions, so that one works with with functions on a torus.

5 Polynomial and differentiable maps on graded Fréchet spaces

Apart from polynomial and differentiable maps between normed vector spaces, we also use such maps between graded Fréchet spaces, defined in terms of nested sequences of Banach spaces, as in Definition 2.1. In this section, we discuss some further properties of such spaces, and in particular what it means for two sequences of Banach spaces defining such a space to be comparable.

5.1 Properties of nested sequences of Banach spaces

Let $\{V_{k}\}_{k=1}^{\infty}$ be a nested sequence of Banach spaces, as in Definition 2.1. Their intersection $V_{\infty}$ is a graded Fréchet space.

Definition 5.1**.**

The space $\mathcal{B}^{n}(V_{\infty})$ consists of the multilinear maps $\lambda\colon V_{\infty}\times\cdots\times V_{\infty}\to V_{\infty}$ (with $n$ factors $V_{\infty}$) such that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $\lambda$ extends continuously to a map in $\mathcal{B}^{n}(V_{k},V_{l})$.

Note that $\mathcal{B}(V_{\infty})=\mathcal{B}^{1}(V_{\infty})$.

Definition 5.2**.**

We write $\operatorname{Pol}^{n}(V_{\infty})$ for the space of all maps $p\colon V_{\infty}\to V_{\infty}$ such that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $p$ extends continuously to a polynomial in $\operatorname{Pol}^{n}(V_{k},V_{l})$. The space $\mathcal{K}\operatorname{Pol}^{n}(V_{\infty})$ is defined analogously.

A feature of the spaces $\operatorname{Pol}^{n}(V_{\infty})$ and $\mathcal{K}\operatorname{Pol}^{n}(V_{\infty})$ that is useful to us, is that they admit natural compositions. If $p_{1}\in\operatorname{Pol}^{m}(V_{\infty})$ and $p_{2}\in\operatorname{Pol}^{n}(V_{\infty})$, then Lemma 3.11 implies that $p_{2}\circ p_{1}$ lies in $\operatorname{Pol}^{mn}(V_{\infty})$. Similarly, Lemma 4.11 implies that $p_{2}\circ p_{1}\in\mathcal{K}\operatorname{Pol}^{mn}(V_{\infty})$ if $p_{1}\in\mathcal{K}\operatorname{Pol}^{m}(V_{\infty})$ and $p_{2}\in\mathcal{K}\operatorname{Pol}^{n}(V_{\infty})$.

Definition 5.3**.**

An $n$ times (compactly) differentiable map from $V_{\infty}$ to itself is a map $f\colon V_{\infty}\to V_{\infty}$ such that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $f$ extends to an $n$ times (compactly) differentiable map from $V_{k}$ to $V_{l}$.

Lemmas 3.2 and 4.16 imply that the spaces of differentiable and compactly differentiable maps from $V_{\infty}$ to itself are closed under composition. Lemma 4.17 implies that if $\{V_{k}\}_{k=1}^{\infty}$ is a compactly nested sequence of Banach spaces, then all $n$ times differentiable maps from $V_{\infty}$ to itself are $n$ times compactly differentiable.

It follows directly from Definition 2.4 that if $f,g\colon V_{\infty}\to V_{\infty}$ are of orders $m$ and $n$, respectively, then $g\circ f$ is of order $m+n$. We also have the following lemma.

Lemma 5.4**.**

Consider two maps $f,g\colon V_{\infty}\to V_{\infty}$, where $f$ is differentiable and $g$ is of order $m$. Suppose that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $f\colon V_{k}\to V_{l}$ is differentiable, and $\|f^{\prime}(v)\|_{\mathcal{B}(V_{k},V_{l})}=O(\|v\|^{n}_{V_{k}})$ as $v\to 0$ in $V_{k}$. Then the map $f^{\prime}\circ g\colon V_{\infty}\to V_{\infty}$, mapping $v\in V_{\infty}$ to $f^{\prime}(v)(g(v))$ is of order $n+m$.

Proof.

Let $l\in\mathbb{N}$. Let $k\in\mathbb{N}$ be such that $f\colon V_{k}\to V_{l}$ is differentiable, and its derivative satisfies the estimate in the lemma. Let $k^{\prime}\in\mathbb{N}$ be such that $\|g(v)\|_{V_{k}}=O(\|v\|^{m}_{V{k^{\prime}}})$ as $v\to 0$ in $V_{k^{\prime}}$. Then $\|(f^{\prime}\circ g)(v)\|_{V_{l}}=O(\|v\|_{V_{k^{\prime}}}^{m+n})$ as $v\to 0$ in $V_{k^{\prime}}$. ∎

An example of a situation where the condition on $f$ in Lemma 5.4 is satisfied is the following.

Lemma 5.5**.**

Let $p\in\operatorname{Pol}^{n}(V_{\infty})$, for $n\geq 2$. Then for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $\|p^{\prime}(v)\|_{\mathcal{B}(V_{k},V_{l})}=O(\|v\|^{n-1}_{V_{k}})$ as $v\to 0$ in $V_{k}$.

Proof.

Let $p\in\operatorname{Pol}^{n}(V_{\infty})$, and let $k,l\in\mathbb{N}$ be such that $p\in\operatorname{Pol}^{n}(V_{k},V_{l})$. As in the proof of Lemma 3.8, $p^{\prime}\in\operatorname{Pol}^{n-1}(V_{k},\mathcal{B}(V_{k},V_{l}))$. So the claim follows from Lemma 3.9. ∎

Remark 5.6**.**

Everything in this subsection generalises directly to polynomial and (compactly) differentiable maps between two different Fréchet spaces that are given as intersections of nested sequences of Banach spaces. We will not need this generalisation, however.

5.2 Comparable sequences of Banach spaces

In the rest of this section, we discuss some relevant properties and examples of comparable sequences of Banach spaces (Definition 2.6). Particularly relevant to Theorem 2.17 are sequences of Banach spaces comparable to sequences of separable Hilbert spaces, which we discuss in Section 5.3. We will see relevant examples in Section 5.4.

Suppose that $\{V_{k}\}_{k=1}^{\infty}$ and $\{W_{k}\}_{k=1}^{\infty}$ are comparable nested sequences of Banach spaces. Then $V_{\infty}=W_{\infty}$ as sets.

Lemma 5.7**.**

The two spaces $V_{\infty}$ and $W_{\infty}$ are equal as Frećhet spaces.

Proof.

Let $(v_{j})_{j=1}^{\infty}$ be a sequence in $V_{\infty}$ such that for every $k\in\mathbb{N}$, $\lim_{j\to\infty}\|v_{j}\|_{V_{k}}=0$. Let $k\in\mathbb{N}$, and choose $l\in\mathbb{N}$ such that we have a bounded inclusion $V_{l}\subset W_{k}$. Then there is a constant $C>0$ such that for every $j$, $\|v_{j}\|_{W_{k}}\leq C\|v_{j}\|_{V_{l}}$, which goes to zero as $j\to\infty$. ∎

The following fact follows directly from the definitions, and the fact that the classes of maps in question are closed under composition with bounded linear maps.

Lemma 5.8**.**

If $f\colon V_{\infty}\to V_{\infty}$ is a (compact) polynomial map or a (compactly) differentiable map, then it also defines a map of the same type on $W_{\infty}$.

This lemma in particular states that $\operatorname{Pol}^{n}(V_{\infty})=\operatorname{Pol}^{n}(W_{\infty})$ as vector spaces. We will use the fact that this equality includes natural topologies on these spaces (Corollary 5.10 below) to prove Corollary 5.13.

Lemma 5.9**.**

For all $l\in N$, there is an $l^{\prime}\in\mathbb{N}$ such that for every $k^{\prime}\in\mathbb{N}$ with $k^{\prime}\geq l^{\prime}$, there is a $k\in\mathbb{N}$ such that we have a bounded inclusion map $\mathcal{B}^{n}(V_{k^{\prime}},V_{l^{\prime}})\subset\mathcal{B}^{n}(W_{k},W_{l})$

Proof.

Let $l\in\mathbb{N}$. Choose $l^{\prime}\in\mathbb{N}$ and $C_{1}>0$ such that for every $v\in V_{\infty}$, $\|v\|_{W_{l}}\leq C_{1}\|v\|_{V_{l^{\prime}}}$. Let $k^{\prime}\geq l^{\prime}$. Choose $k\in\mathbb{N}$ and $C_{2}>0$ such that for every $v\in V_{\infty}$, $\|v\|_{V_{k^{\prime}}}\leq C_{2}\|v\|_{{W_{k}}}$. Then for all $\lambda\in\mathcal{B}^{n}(V_{k^{\prime}},V_{l^{\prime}})$,

[TABLE]

∎

For a sequence $(p_{j})_{j=1}^{\infty}$ in $\operatorname{Pol}^{n}(V_{\infty})$, we define $p_{j}\to 0$ in $\operatorname{Pol}^{n}(V_{\infty})$ to mean that for every $l\in\mathbb{N}$, there is a $k\in\mathbb{N}$ such that $p_{j}\to 0$ in $\operatorname{Pol}^{n}(V_{k},V_{l})$. (This includes the requirement that $p_{j}\in\operatorname{Pol}^{n}(V_{k},V_{l})$ for every $j$.)

Corollary 5.10**.**

We have $\operatorname{Pol}^{n}(V_{\infty})=\operatorname{Pol}^{n}(W_{\infty})$, including topologies.

Proof.

Let $(p_{j})_{j=1}^{\infty}$ be a sequence in $\operatorname{Pol}^{n}(V_{\infty})$ converging to zero. Let $l\in\mathbb{N}$. Choose $l^{\prime}$ as in Lemma 5.9. Choose $k^{\prime}\in\mathbb{N}$ such that $p_{j}\to 0$ in $\operatorname{Pol}^{n}(V_{k^{\prime}},V_{l^{\prime}})$. Choose $k\in\mathbb{N}$ as in Lemma 5.9.

For each $j$, write $p_{j}=p_{\lambda_{j}}$, for $\lambda_{j}\in S\mathcal{B}^{n}(V_{k^{\prime}},V_{l^{\prime}})$. By Lemma 5.9, there is a $C>0$ such that for every $j$,

[TABLE]

which goes to zero as $j\to\infty$. Hence $p_{j}\to 0$ in $\operatorname{Pol}^{n}(W_{\infty})$. ∎

5.3 Sequences of Banach spaces comparable to sequences of Hilbert spaces

As before, we suppose that $\{V_{k}\}_{k=1}^{\infty}$ and $\{W_{k}\}_{k=1}^{\infty}$ are comparable nested sequences of Banach spaces. Now we make the additional assumption that the spaces $W_{k}$ are separable Hilbert spaces. Suppose that $\{e_{j}\}_{j=1}^{\infty}$ is a subset of $W_{\infty}$ that is orthogonal in all spaces $W_{k}$, with dense span. Then taking inner products with $e_{j}$ defines bounded functionals, all denoted by $e^{j}$, on all spaces $W_{k}$, and hence in $V_{k}$ for $k$ large enough.

Corollary 5.11**.**

Suppose $f$ is an $n+1$ times differentiable map from $V_{\infty}$ to itself, and that $f$ is $m$ times compactly differentiable for every $m\leq n$. Then there are unique complex numbers $a^{k}_{q}$ such that

[TABLE]

where $\rho\colon V_{\infty}\to V_{\infty}$ is of order $n+1$, and the part of the sum where $|q|=m$ converges as a function of $h$ in $\operatorname{Pol}^{m}(V_{\infty})$, for $m=0,\ldots,n$.

Proof.

Let $l\in\mathbb{N}$. Choose $k,k^{\prime},l^{\prime}\in\mathbb{N}$ be such that we have bounded inclusions $V_{l^{\prime}}\subset W_{l}$ and $W_{k}\subset V_{k^{\prime}}$, and $f\colon V_{k^{\prime}}\to V_{l^{\prime}}$ is $n+1$ times differentiable, and $m$ times compactly differentiable for every $m\leq n$. Then the same is true for $f\colon W_{k}\to W_{l}$. So Corollary 4.14 implies that 5.1 holds, for unique $a_{q}^{j}$, where the sum converges in $\operatorname{Pol}(W_{\infty})$, and hence in $\operatorname{Pol}(V_{\infty})$ by Corollary 5.10, and $\rho$ is of order $n+1$ as a map from $W_{\infty}$ to itself, and hence as a map from $V_{\infty}$ to itself. ∎

Remark 5.12**.**

A useful feature of Corollary 5.11 is that it is not assumed that the spaces $V_{k}$ have the approximation property. The point is that the separable Hilbert spaces $W_{k}$ do have this property.

The following corollary is an important way in which we use comparable sequences of Banach spaces. It is used in the proof of Lemma 7.7.

Corollary 5.13**.**

Let $a_{q}^{j}\in\mathbb{C}$ be given such that

[TABLE]

converges in $\operatorname{Pol}^{n}(V_{\infty})$. Then for all subsets $A\subset\{q\in\mathbb{N}_{0}^{\infty}:|q|=n\}\times\mathbb{N}$, the series

[TABLE]

converges in $\operatorname{Pol}^{n}(V_{\infty})$.

Proof.

By Corollary 5.10, the series 5.2 converges in $\operatorname{Pol}^{n}(W_{\infty})$, and it is enough to show that 5.3 converges in $\operatorname{Pol}^{n}(W_{\infty})$. And that follows from Lemma 4.10. ∎

Remark 5.14**.**

In Corollary 5.13, the two series converge to elements of $\mathcal{K}\operatorname{Pol}^{n}(V_{\infty})$.

5.4 Example: Sobolev spaces and $C^{k}$-spaces

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{d}$ or of an $d$-dimensional Riemannian manifold, and suppose that the boundary of $\Omega$ is $C^{1}$. Set

[TABLE]

Lemma 5.15**.**

The above sequences $\{V_{k}\}_{k=1}^{\infty}$ and $\{W_{k}\}_{k=1}^{\infty}$ of Banach spaces are comparable.

Proof.

Set $r:=\lceil d/2\rceil$. By a Sobolev embedding theorem, we have bounded inclusions

[TABLE]

for all $1\leq p<2<q<\infty$ and every $l\in\mathbb{N}_{0}$. Now for all such $p,q$ and $l$,

[TABLE]

So we have bounded inclusions

[TABLE]

Furthermore, since $\Omega$ has finite volume, we have bounded inclusions $W^{l,p^{\prime}}(\Omega)\subset W^{l,p^{\prime\prime}}(\Omega)$ for every $l$ and all $p^{\prime}\geq p^{\prime\prime}$.

The above arguments imply that we have bounded inclusions

[TABLE]

∎

Remark 5.16**.**

In this example, the spaces $V_{k}$ are separable Hilbert spaces.

For another example of comparable sequences of Banach spaces, fix $p>n$. For $k\in\mathbb{N}$, set

[TABLE]

The space $C^{k}(\overline{\Omega})$ is complete in the norm given by the maximum of the sup-norms of the partial derivatives of functions up to order $k$.

Lemma 5.17**.**

The sequences $\{V_{k}\}_{k=1}^{\infty}$ and $\{W_{k}\}_{k=1}^{\infty}$ of Banach spaces are comparable.

Proof.

We have a bounded inclusion $W_{k}\subset V_{k}$ for every $k$. So it remains to show that for every $k\in\mathbb{N}$, there are $l_{1},l_{2}\in\mathbb{N}$ such that we have bounded inclusions

[TABLE]

By a Sobolev embedding theorem, we have a bounded inclusion

[TABLE]

for all $k,l\in\mathbb{N}$ such that

[TABLE]

For $k\in\mathbb{N}$, set $l_{1}:=k+1+\lceil n/p\rceil$ and $l_{2}:=\max\{k-1-\lfloor n/p\rfloor,1\}$. Then this Sobolev embedding theorem yields the desired inclusions 5.4. ∎

Remark 5.18**.**

If $n=1$, then we may take $p=2$, so that the spaces $V_{k}$ are Hilbert spaces.

Proposition 4.20 and Lemma 5.15 together imply Theorem 2.21. The extension of Proposition 4.20 to coeficients $a_{q}$ depending on a real (time) parameter $t$ in a smooth way, and the extension of Lemma 5.15 to vector-valued functions, are straightforward.

6 A coordinate transform

6.1 A residual

Recall the setting of Section 2.2. In this section and the next, based upon the details of some given dynamical system 2.2 we construct both a coordinate transformation 2.6 and a corresponding ‘normal form’ system 2.7, such that solutions $X$ to 2.7, transformed by 2.6, satisfy the given dynamical system 2.2 up to residuals of a specified order $p$. See Theorem 2.17. We do this inductively, by showing how to construct such a transformed system to satisfy 2.2 with residual of order $p+1$ from a version with residual of order $p$.

In Section 7, we construct a more specific choice of the general coordinate transform constructed in this section, in order to establish exact invariant manifolds, and study their properties, for constructed systems arbitrarily close to the given system 2.2.

Remark 6.1**.**

In Section 2.2, we assumed that the sequence $\{V_{k}\}_{k=1}^{\infty}$ is comparable to a nested sequence $\{W_{k}\}_{k=1}^{\infty}$ of separable Hilbert spaces in which the vectors $e_{j}$ are orthogonal. Lemma 5.8 implies that we may equivalently assume that $\{V_{k}\}_{k=1}^{\infty}$ itself is a nested sequence of separable Hilbert spaces, because all maps from $V_{\infty}$ to itself we use transfer to maps from $W_{\infty}$ to itself of the same type (e.g. compact polynomial and compactly differentiable maps). However, the formulation where $\{V_{k}\}_{k=1}^{\infty}$ is only comparable to a nested sequence of separable Hilbert spaces makes it clearer that we have the flexibility to consider maps between Banach spaces. This is natural for example in the context of Proposition 4.20.

Let $p\in\mathbb{N}$, with $p\geq 2$. Let $\xi_{p},F_{p}\colon I\times V_{\infty}\to V_{\infty}$ be such that $\xi_{p}-\operatorname{id}$ and $F_{p}$ are compact polynomial maps of order at most $p-1$ in the $V_{\infty}$ component, and infinitely differentiable in $I$. Suppose, furthermore, that $\xi_{p}$ is a near-identity at zero, and that $F_{p}=\mathcal{O}(2)$.

Recall that our goal is to relate the dynamics of maps $x$ satisfying 2.2 to the dynamics of maps $X\colon I\to V_{\infty}$ satisfying 2.7 when $x$ and $X$ are related by the coordinate transform $\xi_{p}$ as in 2.6.

For maps $f,g\colon I\times V_{\infty}\to V_{\infty}$, with $f$ differentiable, we write $f^{\prime}_{V_{\infty}}\circ g$ for the map from $I\times V_{\infty}$ to $V_{\infty}$ given by

[TABLE]

for all $t\in I$ and $v\in V_{\infty}$. (Note that this is different from $(f\circ g)^{\prime}_{V_{\infty}}(t,v)=f^{\prime}_{V_{\infty}}(t,g(t,v))(g(t,v))$.) If $g$ is a map from $V_{\infty}$ to $V_{\infty}$ to itself, then the composition $f^{\prime}_{V_{\infty}}\circ g$ is defined analogously. Also recall the notation for compositions of maps to and from $I\times V_{\infty}$ and $V_{\infty}$ under Notation and conventions in Section 1.5.

Define the maps $\Phi_{p},R_{p}\colon I\times V_{\infty}\to V_{\infty}$ by

[TABLE]

The map $R_{p}$ is the residual of the transformed ode, in the following sense.

Lemma 6.2**.**

For all smooth maps $X\colon I\to V_{\infty}$ satisfying 2.7, and with $x\colon I\to V_{\infty}$ determined from $X$ by 2.6,

[TABLE]

Proof.

For $X$ and $x$ as in the lemma, the chain rule (Lemma 3.2) and 2.7 imply that for all $t\in I$,

[TABLE]

∎

Lemma 6.3**.**

The maps $\Phi_{p}$ and $R_{p}$ are infinitely compactly differentiable.

Proof.

Because the Banach spaces $V_{k}$ are compactly nested, it is enough to show that $\Phi_{p}$ and $R_{p}$ are infinitely differentiable. And that is true by the chain rule, because $f$ is infinitely differentiable, and so are $F_{p}$ and $\xi_{p}$, by Lemma 3.8. ∎

To recursively construct 2.7 and 2.6, suppose that $R_{p}=\mathcal{O}(p)$. We proceed to show that 2.7 and 2.6 may be refined to make the new residual of $\mathcal{O}(p+1)$. By Lemma 6.3, Corollary 5.11 (where $u=0$ and $h=v$), and Lemmas 3.9 and 3.10, there are unique, infinitely differentiable maps

[TABLE]

for all multi-indices $q\in\mathbb{N}_{0}^{\infty}$ with $|q|=p$, and a map

[TABLE]

such that for all $t\in I$, and $v\in V_{\infty}$,

[TABLE]

where the sum converges in $\operatorname{Pol}^{p}(V_{\infty})$, differentiably in $t$, and $\rho_{p}=\mathcal{O}(p+1)$. The monomial $v^{q}$ is defined as in 3.10, with respect to the set of functionals $e^{j}=(e_{j},\relbar)_{V_{1}}$, for $j\in\mathbb{N}$.

In Section 6.2, we construct maps $\xi_{p+1},F_{p+1}$ such that the order $p$ term $R_{p}(t,X(t))$ in 6.1 may be replaced by an order $p+1$ term $R_{p+1}(t,X(t))$, if 2.7 and 2.6 hold with $p$ replaced by $p+1$.

6.2 Construction of the coordinate transform

For each $j\in\mathbb{N}$, let $\alpha_{j}$ be the eigenvalue of $A$ corresponding to $e_{j}$. For all $q\in\mathbb{N}_{0}^{\infty}$ with $|q|=p$, define $\mu^{q}\in\mathbb{C}$ by

[TABLE]

(This sum has at most $p$ nonzero terms.) For such $q\in\mathbb{N}_{0}^{\infty}$, let $a^{q}$ be as in 6.2. Let $\hat{\xi}^{q},\hat{F}^{q}\colon I\to V_{\infty}$ be smooth maps such that

[TABLE]

Suppose that the sums

[TABLE]

converge in $\operatorname{Pol}^{p}(V_{\infty})$, differentiably in $t$.

Define a new coordinate transform map $\xi_{p+1}\colon I\times V_{\infty}\to V_{\infty}$ and corresponding map $F_{p+1}\colon I\times V_{\infty}\to V_{\infty}$ that replaces $F_{p}$ in 2.7, by

[TABLE]

The following result is the main step in the construction of the coordinate transform we are looking for.

Proposition 6.4**.**

If $X$ and $x$ are as in 2.7 and 2.6, with $p$ replaced by $p+1$, then 6.1 holds, with the residual $R_{p}$ replaced by a residual $R_{p+1}$ satisfying

[TABLE]

The maps $\xi_{p+1}-\operatorname{id}$ and $F_{p+1}$ are compact polynomials in $\mathcal{K}\operatorname{Pol}(V_{\infty})$ of order at most $p$, and $\xi_{p+1}$ is a near-identity.

Remark 6.5**.**

The maps $\hat{\xi}^{q}$ and $\hat{F}^{q}$ can be found explicitly if we decompose 6.4 with respect to the basis $\{e_{j}\}_{j=1}^{\infty}$. This will be done in Section 7. One solution to 6.4 is $\hat{F}^{q}=a^{q}$ and $\hat{\xi}^{q}=0$. However, for our purposes, we need the function $F^{q}$ to be of a specific form. The main purpose of this work is to find $\hat{F}^{q}$ such that the $e_{j}$-component of $F^{q}$ is zero for certain combinations of $q$ and $j$, in such a way that an exact separation of stable, centre and unstable modes is maintained. See Proposition 7.1.

6.3 Proof of Proposition 6.4

Define the map $\hat{\Phi}\colon I\times V_{\infty}\to V_{\infty}$ by

[TABLE]

Lemma 6.6**.**

We have

[TABLE]

Proof.

The left-hand side of 6.7 equals

[TABLE]

By Lemma 5.5, the derivative $\hat{\xi}^{\prime}_{V_{\infty}}$ satisfies the condition of Lemma 5.4, with $n=p-1$. Since $F_{p}=\mathcal{O}(2)$, Lemma 5.4 implies that $\hat{\xi}^{\prime}_{V_{\infty}}\circ F_{p}=\mathcal{O}(p+1)$. Similarly, $\hat{\xi}^{\prime}_{V_{\infty}}\circ\hat{F}=\mathcal{O}(2p-1)$. Now $\xi_{p}$ is a polynomial map, and a near-identity. By Lemma 5.5, this implies that $(\xi_{p})^{\prime}_{V_{\infty}}-\operatorname{id}$ satisfies the condition of Lemma 5.4, with $n=1$. So Lemma 5.4 implies that $((\xi_{p})^{\prime}_{V_{\infty}}-\operatorname{id})\circ\hat{F}=\mathcal{O}(p+1)$. ∎

Lemma 6.7**.**

For $q\in\mathbb{N}_{0}^{\infty}$ such that $|q|=p$, let $a^{q}$ be as in 6.2. Then for all $t\in I$ and $v\in V_{\infty}$,

[TABLE]

Proof.

First of all, $\hat{\xi}^{\prime}_{I}(t,v)=\sum_{q\in\mathbb{N}_{0}^{\infty};\,|q|=p}(\hat{\xi}^{q})^{\prime}(t)v^{q}.$ By Lemma 3.12,

[TABLE]

Now, because the vectors $\{e_{j}\}_{j=1}^{\infty}$ are orthogonal with respect to $(\relbar,\relbar)_{V_{1}}$, we have $(e_{j},Av)_{V}=\alpha_{j}(e_{j},v)_{V_{1}}$ for every $j$. So

[TABLE]

We find that the left-hand side of 6.8 equals

[TABLE]

with $\mu^{q}$ as in 6.3. So the claim follows from 6.4. ∎

Define the maps $\Phi_{p+1},R_{p+1}\colon I\times V_{\infty}\to V_{\infty}$ by

[TABLE]

Lemma 6.8**.**

The residual $R_{p+1}$ satisfies $R_{p+1}=\mathcal{O}(p+1).$

Proof.

By Lemmas 6.6 and 6.7,

[TABLE]

where, for $t\in I$ and $v\in V_{\infty}$,

[TABLE]

By 6.2, the last expression in 6.9 equals

[TABLE]

Let $l\in\mathbb{N}$ be given, and choose $k,k^{\prime}\in\mathbb{N}$ such that $f\colon I\times V_{k^{\prime}}\to V_{l}$ is differentiable, and $\xi_{p},\xi_{p+1}\in\operatorname{Pol}(V_{k},V_{k^{\prime}})$. Using Theorem 3.4, we write

[TABLE]

for $t\in I$ and $v\in V_{k^{\prime}}$. Lemma 3.9 implies that $\|\hat{\xi}(t,v)\|_{V_{k^{\prime}}}=O(\|v\|_{V_{k}})$ uniformly in $t$ in compact sets. So

[TABLE]

uniformly in $t$ in compact sets. And then, as in the proof of Lemma 5.4, the assumption 2.1 and Lemma 5.4 imply that $\|f^{\prime}(t,\xi_{p}(t,v))\hat{\xi}(t,v)\|_{V_{l}}=O(\|v\|_{k^{\prime}}^{p+1})$, uniformly in $t$ in compact sets. ∎

Proof of Proposition 6.4.

The correction terms $\hat{\xi}$ and $\hat{F}$ lie in $\mathcal{K}\operatorname{Pol}^{p}(V_{\infty})$. Hence $\xi_{p+1}-\operatorname{id}$ and $F_{p+1}$ are compact polynomials, because $\xi_{p}-\operatorname{id}$ and $F_{p}$ are. By Lemma 3.9, this also implies that $\xi_{p+1}$ is a near-identity because $\xi_{p}$ is. The desired property of $R_{p+1}$ is Lemma 6.8. ∎

7 Centre, stable and unstable coordinates

There is considerable flexibility in choosing the maps $\hat{\xi}^{q}$ and $\hat{F}^{q}$ in Section 6.2. In this section, we discuss how to make specific choices, in terms of the eigenvalues of $A$, so that the normal form 2.7 is useful for detecting invariant manifolds.

7.1 Centre, stable and unstable components of $\hat{F}^{q}$

We use the notation from Section 2.5. In particular, let $\alpha$, $\beta$, $\gamma$ and $\tilde{\mu}$ be spectral gap parameters defined there. Recall the definition of polynomial growth in Definition 2.16.

Proposition 7.1**.**

Suppose that $\beta-(p+1)\alpha>\tilde{\mu}$ and $\gamma-(p+1)\alpha>\tilde{\mu}$. Suppose that $R_{p}$ has polynomial growth. The maps $\hat{\xi}^{q}$ and $\hat{F}^{q}$ in Section 6.2 can be chosen such that

• •

if either $q^{s}=0$ and $q^{u}\not=0$, or $q^{u}=0$ and $q^{s}\not=0$, then $\hat{F}^{q}_{c}=0$;

• •

if $q^{s}=0$, then $\hat{F}^{q}_{s}=0$; and

• •

if $q^{u}=0$, then $\hat{F}^{q}_{u}=0$.

Proposition 7.1 is proved in Section 7.2, after some preparation done in this subsection.

Definition 7.2**.**

Let $\mu\in\mathbb{C}$, such that $|\Re(\mu)|>\tilde{\mu}$. Set $a:=\inf I$ and $b:=\sup I$. Let $u$ be a continuous function on $\mathbb{R}$ such that $u(t)=O(e^{\tilde{\mu}|t|})$ as $t\to\infty$ if $b=\infty$ and as $t\to-\infty$ if $a=-\infty$. Then we define the function $e^{\mu(\cdot)}\star u$ on $I$ by

[TABLE]

The integrals occurring in this definition are $\tilde{\mu}$-regular, in the sense defined in Section 2.5.

Lemma 7.3**.**

In the setting of Definition 7.2,

[TABLE]

Proof.

This is a straightforward computation. ∎

Lemma 7.4**.**

Let $u\colon I\to\mathbb{C}$ be a smooth function, and suppose that $u$ and all its derivatives grow at most polynomially. Then, for every $\mu$ as in Definition 7.2, $e^{\mu(\cdot)}\star u$ and all its derivatives grow at most polynomially.

Proof.

First of all, because $(e^{\mu(\cdot)}\star u)^{(k)}=e^{\mu(\cdot)}\star(u^{(k)})$, it is enough to consider the function $u$ itself rather than all its derivatives.

Let $C>0$ and $n\in\mathbb{N}_{0}$ be such that for all $t\in I$, $|u(t)|\leq C(1+|t|^{n})$. We prove by induction on $n$ that there is a constant $C^{\prime}>0$ such that for all $t\in I$, $|(e^{\mu(\cdot)}\star u)(t)|\leq C^{\prime}(1+|t|^{n})$. We consider the case where $\Re(\mu)<-\tilde{\mu}$; the case where $\Re(\mu)>\tilde{\mu}$ is similar.

If $n=0$, then for all $t\in I$,

[TABLE]

Suppose that the claim holds for $n$, and suppose that $|u(t)|\leq C(1+|t|^{n+1})$ for a constant $C$. Using integration by parts, we find that

[TABLE]

which implies the claim by the induction hypothesis. ∎

Let $q\in\mathbb{N}_{0}^{\infty}$, with $|q|=p$. Consider the differentiable maps

[TABLE]

where the sum converges in $V_{1}$, uniformly and differentiably in $t$ in compact sets in $I$.

For $q\in\mathbb{N}_{0}^{\infty}$ with $|q|=p$, let $J^{q}\subset\mathbb{N}$ be the set of $j\in\mathbb{N}$ such that either

• •

$j\in J_{c}$ and either $q^{s}=0$ and $q^{u}\not=0$, or $q^{u}=0$ and $q^{s}\not=0$;

• •

$j\in J_{s}$ and $q^{s}=0$; or

• •

$j\in J_{u}$ and $q^{u}=0$.

For $q\in\mathbb{N}_{0}^{\infty}$ with $|q|=p$, and $j\in\mathbb{N}$, write $\mu^{q}_{j}:=\mu^{q}-\alpha_{j}$, with $\mu^{q}$ as in 6.3.

Lemma 7.5**.**

If $\beta-(p+1)\alpha>\tilde{\mu}$ and $\gamma-(p+1)\alpha>\tilde{\mu}$, then for every $j\in J^{q}$, $|\Re(\mu^{q}_{j})|>\tilde{\mu}$.

Proof.

If $q^{s}=0$ and $q^{u}\not=0$, and $j\in J_{c}$, then

[TABLE]

If $q^{u}=0$ and $q^{s}\not=0$, and $j\in J_{c}$, then

[TABLE]

If $q^{s}=0$, and $j\in J_{s}$, then

[TABLE]

And if $q^{u}=0$, and $j\in J_{u}$, then

[TABLE]

∎

7.2 Update terms for $\xi_{p}$ and $F_{p}$

Suppose that $R_{p}$ has polynomial growth. Then the functions $a^{q}_{j}$ and their derivatives grow at most polynomially, uniformly in $q$ and $j$.

For every $q\in\mathbb{N}_{0}^{\infty}$ with $|q|=p$ and $j\in\mathbb{N}$, consider the ode for $\hat{\xi}^{q}_{j}$ and $\hat{F}^{q}_{j}$

[TABLE]

Define the maps $\hat{\xi}^{q}_{j},\hat{F}^{q}_{j}\colon I\to\mathbb{C}$ as follows. If $j\in J^{q}$, then

[TABLE]

This definition makes sense because of Lemma 7.5 and the growth behaviour of the functions $a^{q}_{j}$. If $j\not\in J^{q}$, then we set

[TABLE]

Lemma 7.6**.**

With the above definitions, the ode 7.1 is satisfied for all $q$ and $j$.

Proof.

If $j\not\in J^{q}$, the statement is immediate from the definitions. If $j\in J^{q}$, it follows from Lemma 7.3. ∎

Lemma 7.7**.**

*Suppose that $\beta-(p+1)\alpha>\tilde{\mu}$ and $\gamma-(p+1)\alpha>\tilde{\mu}$. Then the sums *

[TABLE]

converge in $\operatorname{Pol}^{p}(V_{\infty})$, differentiably in $t$.

Proof.

The first sum in 7.4 equals

[TABLE]

Since $\{V_{k}\}_{k=1}^{\infty}$ is comparable to a nested sequence of separable Hilbert spaces in which the set $\{e_{j}\}_{j=1}^{\infty}$ is orthogonal, Corollary 5.13 implies that this series converges in $\operatorname{Pol}^{p}(V_{\infty})$.

Write $J^{q}_{\pm}:=\{j\in J^{q}:\pm\Re(\mu^{q}_{j})>\tilde{\mu}\}.$ Lemma 7.5 states that $J^{q}=J^{q}_{+}\cup J^{q}_{-}$. So the second sum in 7.4 equals

[TABLE]

Tonelli’s theorem implies that convergence of the first of these sums is equivalent to convergence of

[TABLE]

The sum inside the brackets converges in $\operatorname{Pol}^{p}(V_{\infty})$, uniformly in $\tau$, by convergence of 6.2 and Corollary 5.13. Since the functions $a^{q}_{j}$ grow at most polynomially, uniformly in $q$ and $j$, the value of that sum grows at most polynomially as well, when viewed as a convergent series in $\operatorname{Pol}^{p}(V_{k},V_{l})$. So the integral over $\tau$ converges in $\operatorname{Pol}^{p}(V_{\infty})$, by completeness of the spaces $\operatorname{Pol}^{p}(V_{k},V_{l})$. By continuity of 7.7 in $t$, the convergence is uniform in $t$ on compact subsets of $I$. The derivatives of 7.7 with respect to $t$ are linear combinations of 7.7 and

[TABLE]

and therefore converge as well.

By an analogous argument, the second sum in 7.6 converges as well, differentiably in $t$. ∎

Proposition 7.1 follows from Lemmas 7.6 and 7.7.

7.3 Proof of Theorem 2.17

Lemma 7.8**.**

If $f,\xi_{p},F_{p},\Phi_{p}$ and $R_{p}$ have polynomial growth, then so do $\xi_{p+1}$, $F_{p+1}$, $\Phi_{p+1}$ and $R_{p+1}$.

Proof.

If $R_{p}$ has polynomial growth, then the functions $a^{q}_{j}$ and all their derivatives grow at most polynomially, uniformly in $q$ and $j$. Hence, by 7.2 and 7.3, the map $\hat{F}$ has polynomial growth. By Lemma 7.4, the same is true for $\hat{\xi}$. So $F_{p+1}$ and $\xi_{p+1}$ have polynomial growth.

Polynomial growth is preserved under composition and derivatives in the $I$ and $V_{\infty}$ directions. Hence the map $\hat{\Phi}$ as in 6.6 has polynomial growth, and therefore so do $\Phi_{p+1}$ and $R_{p+1}$. ∎

Combining Lemma 7.8 with Propositions 6.4 and 7.1, we prove the following slightly more explicit version of Theorem 2.17.

Theorem 7.9**.**

*Let $p\in\mathbb{N}$ be such that $p\geq 2$, $\beta-(p+1)\alpha>\tilde{\mu}$ and $\gamma-(p+1)\alpha>\tilde{\mu}$. Suppose that $f$ has polynomial growth. Then there are infinitely differentiable maps *

[TABLE]

where $R_{p}=\mathcal{O}(p)$, $F_{p}$ is a polynomial map that separates invariant subspaces, $\xi_{p}$ is a near-identity and $\xi_{p}-\operatorname{id}$ and $F_{p}$ are compact polynomials of orders at most $p-1$, such that if $X$ and $x$ are as in 2.7 and 2.6, then 6.1 holds. Finally, there is a construction of the map $\xi_{p}$ in which all integrals over $I$ that occur are $\tilde{\mu}$-regular.

Proof.

We use induction on $p$ to prove that the claim holds for every $p$, including the auxiliary statement that $\xi_{p},F_{p},\Phi_{p}$ and $R_{p}$ have polynomial growth.

If $p=2$, then we may take $F_{p}(t,v)=0$ and $\xi_{p}(t,v)=v$ for all $t\in I$ and $v\in V_{0}$. Then $R_{2}=f$, so $\xi_{p},F_{p},\Phi_{p}$ and $R_{p}$ have polynomial growth because $f$ does.

The induction step follows from Lemmas 7.8, 6.4 and 7.1. ∎

8 Dynamics of the normal form equation

It remains to prove Lemma 2.8 and Proposition 2.9, which we use to justify Definition 2.10 based on Theorem 2.17. Throughout this section, we suppose that $F\colon I\times V_{\infty}\to V_{\infty}$ is a smooth map that separates invariant subspaces.

Proof of Lemma 2.8..

First, suppose that $a=c$. For all $v\in V_{c}$ and all $q\in\mathbb{N}_{0}^{\infty}$ with $|q|\leq p$ and $q^{s}\not=0$ or $q^{u}\not=0$, we have $v^{q}=0$. So the properties 2.3 of the map $F$ imply that $F(I\times V_{c})\subset V_{c}$. This, in turn, implies that for all maps $X\colon I\to V_{\infty}$ satisfying 2.7, if $X(t)\in V_{c}$ for a given $t$ then $\dot{X}(t)\in V_{c}$. So $X(t)\in V_{c}$ for all $t\in I$.

Next, suppose that $a=s$. If $v\in V_{s}$ and $q\in\mathbb{N}_{0}^{\infty}$, then $v^{q}=0$ if $q^{u}\not=0$. So $F_{u}(t,v)=0$ for all $t\in I$. And the components of $F_{c}(t,v)$ with $q^{u}\not=0$ are zero for the same reason, while its components with $q^{s}=0$ are zero since $v\in V_{s}$. Hence $F_{c}(t,v)=0$. We conclude that $F(I\times V_{s})\subset V_{s}$. As in the case $a=c$, this implies the claim for $a=s$.

The argument for $a=u$ is entirely analogous to the case $a=s$. ∎

To prove Proposition 2.9, we start with a general comparison estimate for solutions of odes in Hilbert spaces.

Lemma 8.1**.**

Let $V$ be a Hilbert space, $W\subset V$ a subspace, $I$ an open interval containing [math], and $g$ a map from $I\times V$ into the space of linear operators from $W$ to $V$. Let $X\colon I\to W$ be a differentiable map (as a map into $V$), such that for all $t\in I$,

[TABLE]

If $\zeta\in\mathbb{R}$ is such that $g(t,w)+\zeta$ is negative semidefinite for all $t\in I$ and $w\in W$, then for all $t\in I$ with $t\geq 0$,

[TABLE]

Proof.

First, suppose that $\zeta=0$. Then for all $t\in I$,

[TABLE]

So $\|X\|_{V}^{2}$ is a nonnegative, non-increasing function on $I$, and the claim for $\zeta=0$ follows.

Next, let $\zeta\in\mathbb{R}$ be arbitrary. Then

[TABLE]

Applying the claim for $\zeta=0$, with $X(t)$ replaced by $X(t)e^{\zeta t}$ and $g(t,w)$ by $g(t,w)+\zeta$, now yields the claim for $\zeta$. ∎

For any homogeneous polynomial map $p=p_{\lambda}$ between normed vector spaces $V$ and $W$, where $\lambda\in S\mathcal{B}^{n}(V,W)$, define the map $\tilde{p}\colon V\to\mathcal{B}(V,W)$ by

[TABLE]

Here $n-1$ copies of $v_{1}$ are inserted into $\lambda$ on the right hand side.

For all $t\in I$, the map $F(t,\relbar)$ lies in $\operatorname{Pol}(V_{k},V_{1})$ for some $k$. The operator $A$ lies in $\mathcal{B}(V_{l},V_{1})$ for some $l$. By replacing the smaller of $k$ or $l$ by the larger of these two numbers, we henceforth assume $k=l$. Applying the construction 8.1 to each homogeneous term of $F(t,\relbar)$ and adding the resulting maps, we obtain a map $\tilde{F}\colon V_{k}\to\mathcal{B}(V_{k},V_{1})$, such that for all $v\in V_{k}$,

[TABLE]

For $a\in\{c,s,u\}$, we write $\tilde{F}_{a}$ for $\tilde{F}$ composed with orthogonal projection onto $V_{a}$.

For $j\in\mathbb{N}$, let $q^{(j)}\in\mathbb{N}_{0}^{\infty}$ be defined by $q^{(j)}_{m}=1$ if $m=j$, and $q^{(j)}_{m}=0$ otherwise.

Lemma 8.2**.**

Let $v\in V_{k}$. Write $v=v_{c}+v_{s}+v_{u}$, where $v_{a}\in V_{a}$ for $a\in\{c,s,u\}$. Then for all $t\in I$, the components of $F(t,v)$ in $V_{s}$, $V_{u}$ and $V_{c}$ satisfy

[TABLE]

Proof.

Let $t\in I$ and $v\in V_{k}$. To prove 8.2, we use the fact that by 2.3b,

[TABLE]

So

[TABLE]

where $\{e^{j}\}_{j\in\mathbb{N}}$ is the basis of $V_{1}^{*}$ dual to $\{e_{j}\}_{j\in\mathbb{N}}$. Hence

[TABLE]

which implies 8.2. The equality 8.3 can be proved anaogously.

To prove 8.4, we note that by 2.3a,

[TABLE]

where

[TABLE]

The right hand side of 8.5 only depends on $v_{c}$, and the right hand side of 8.6 is zero if $v_{s}=0$ or $v_{u}=0$. So, under that condition, $F_{c}(t,v)=F_{c}(t,v_{c})=\tilde{F}(t,v_{c})v_{c}.$ ∎

Proof of Proposition 2.9..

Set

[TABLE]

Because $F$ is a sum of polynomials of degrees at least two, we have $\tilde{F}(t,0)=0$ for all $t$. So $D_{\tilde{\mu}}$ contains $I\times\{0\}$. It is open by continuity of $\tilde{F}$.

Let $X\colon I\to V_{\infty}$ be a solution of the constructed system 2.7. As in the proof of Lemma 2.8,

[TABLE]

where we used the first equality in Lemma 8.2 and the fact that $A$ preserves $V_{s}$. For all $(t,v)\in D_{\tilde{\mu}}$, the operator

[TABLE]

is negative semidefinite. Hence the claim about $X_{s}$ follows from the second part of Lemma 8.1. The claim about $X_{u}$ can be proved similarly, via a version of Lemma 8.1 for positive-definite operators.

Next, suppose that $X_{s}(0)=0$ or $X_{u}(0)=0$. By Lemma 2.8, either $X_{s}(t)=0$ for all $t\in I$ or $X_{u}(t)=0$ for all $t\in I$. Similarly to 8.8, the third equality in Lemma 8.2 implies that

[TABLE]

for all $t\in I$. And for all $(t,v)\in D_{\tilde{\mu}}$, the operator

[TABLE]

is negative semidefinite. So by Lemma 8.1,

[TABLE]

for all $t\geq 0$ in $I$. It similarly follows that for all $t\leq 0$ in $I$,

[TABLE]

∎

9 Example: a non-autonomous version of Burgers’ equation

Let $r\in\mathbb{R}$, and consider the non-autononomous, nonlinear pde

[TABLE]

with $2\pi$-periodic boundary conditions in $\theta$. Then Theorem 2.21 applies, where $\Omega$ is the circle.

Using Theorem 2.17, we compute the centre manifold of the normal form system approximating 9.1 up to residuals of order three, in Section 9.1. Via a direct approach, we compute all invariant manifolds for residuals of orders three and four, in Section 9.2. We find that the order three centre manifolds computed in the two ways agree. These computations illustrate Remark 2.20, that the construction from Theorem 2.17 is guaranteed to give a result, while a direct computation may be more efficient in concrete situations.

9.1 Centre manifold via Theorem 2.17

In this setting,

[TABLE]

where a prime denotes the derivative in the $\theta$-direction. The eigenfunctions of $A$ are $e_{j}$, for $j\in\mathbb{Z}$, given by $e_{j}(\theta):=e^{ij\theta}$. The eigenvalue corresponding to $e_{j}$ is $\alpha_{j}=r-j^{2}$ (which has multiplicity two when $j\neq 0$). Choose $\alpha,\beta,\gamma$ and $\tilde{\mu}$ such that $0\leq\alpha<\tilde{\mu}<\beta=\gamma<1$, and $\alpha<\frac{1}{2}$. Suppose that $r$ lies within $\alpha$ of an integer of the form $n^{2}$, for a nonzero $n\in\mathbb{Z}$. Then the eigenvalue $\alpha_{n}$ is central up to precision $\alpha$.

We determine a corresponding centre manifold for a system that approximates 9.1 up to a third-order residual. This involves the coordinate transform $\xi_{3}$. To compute this centre manifold, we only need to apply $\xi_{3}$ to elements of $V_{c}=\operatorname{span}\{e_{n},e_{-n}\}$. In other words, we only need to compute $\xi_{3}(t,X_{n}e_{n}+X_{-n}e_{-n})$, for $t\in\mathbb{R}$ and $X_{n},X_{-n}\in\mathbb{C}$. (We do not determine the domain $D_{\tilde{\mu}}$ here.)

For $p=2$, the map $\xi_{2}$ is the identity map. So

[TABLE]

where

[TABLE]

For $j\in\mathbb{Z}$, let $q^{(j)}\in\mathbb{Z}^{\infty}$ be defined by $q^{(j)}_{m}=1$ if $m=j$, and $q^{(j)}_{m}=0$ otherwise. Then, for $q\in\mathbb{Z}^{\infty}$ with $|q|=2$,

[TABLE]

So

[TABLE]

The map $\hat{\xi}^{2q^{(n)}}_{j}$ is expressed in terms of the map $a^{2q^{(n)}}_{j}$ in

[TABLE]

(The order three term in 6.2 now equals zero.) See 7.2 and 7.3. If $u=\sum_{l\in\mathbb{Z}}x_{l}e_{l}$, then

[TABLE]

The equality $x_{k}x_{j-k}=u^{2q^{(n)}}=x_{n}^{2}$ holds precisely if $k=n$ and $j=2n$. Hence

[TABLE]

and $a^{2q^{(n)}}_{j}=0$ if $j\not=2n$. An analogous argument shows that

[TABLE]

and $a^{2q^{(-n)}}_{j}=0$ if $j\not=-2n$. The equality $x_{k}x_{j-k}=u^{q^{(n)}+q^{(-n)}}=x_{n}x_{-n}$ holds precisely if $j=0$ and either $k=n$ or $k=-n$. Hence

[TABLE]

and $a^{q^{(n)}+q^{(-n)}}_{j}=0$ if $j\not=0$.

The relevant numbers $\mu^{q}_{j}$ as in Section 7.1 equal

[TABLE]

(note that $\alpha_{j}=\alpha_{-j}$ for every $j$). Because $n^{2}\geq 1$ and $\alpha<\frac{1}{2}$, the real parts of $\mu^{2q^{(n)}}_{2n}$ and $\mu^{2q^{(-n)}}_{2n}$ are greater than $\alpha$, whereas the real part of $\mu^{q^{(n)}+q^{(-n)}}_{0}$ is smaller than $-\alpha$.

And with $J^{q}$ as in Section 7.1, we have $2n\in J^{2q^{(n)}}$. Indeed,

[TABLE]

so $2n\in J_{s}$ and $(2q^{(n)})^{s}=0$. Similarly, $2n\in J^{2q^{(-n)}}$. And $0\in J_{u}$ and $(q^{(n)}+q^{(-n)})^{u}=0$, so $0\in J^{q^{(n)}+q^{(-n)}}$.

Hence, by 7.2,

[TABLE]

The integral converges since $\Re(r+2n^{2})>0$, and is $\tilde{\mu}$-regular. Similarly,

[TABLE]

And because $\Re(\mu^{q^{(n)}+q^{(-n)}}_{0})<-\alpha$,

[TABLE]

We conclude that for all $t\in\mathbb{R}$ and $X_{n},X_{-n}\in\mathbb{C}$,

[TABLE]

(The last term is a scalar multiple of the constant function $e_{0}$.) If $r=n^{2}$, this simplifies to

[TABLE]

9.2 Invariant manifolds via direct computations

For order of residual $p=2$, the map $\xi_{2}$ is the identity map, $x_{j}=X_{j}$ .

Proceeding to order of residual $p=3$ we construct quadratic corrections to the identity $\xi_{2}$ to form $\xi_{3}$. In the eigenvector basis the field $u(t,\theta)=\sum_{j}x_{j}(t)e^{ij\theta}$ (all sums in this section are over $\mathbb{Z}$), and the pde 9.1 becomes

[TABLE]

Writing

[TABLE]

and solving{journal}333The computer algebra code used for the computations in this section is available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt.

for $g^{kl}_{j}$ such that $x_{j}$ satisfies 9.3 up to terms of order three if $\dot{X}_{k}=\alpha_{k}X_{k}$, we find that $x(t)=\xi_{3}(t,X(t))$ is given by

[TABLE]

For $r\approx n^{2}$ and odd $n$, the denominators in $d_{jk}$ are not small. Then this map, combined with the linear $\dot{X}_{j}=\alpha_{j}X_{j}$ , matches the pde 9.1 to third-order errors.

However, for $r\approx n^{2}$ and even $n>0$, some denominators are small, becoming zero when $r=n^{2}$. Then the divisor being zero becomes $k(k-j)=n^{2}/2$ and hence has zeros for every pair of integer factors of $n^{2}/2$ (including negative pairs). Consequently these terms are excluded from the sum 9.4, and instead lead to nonlinearly modifying the evolution for some $j$ via

[TABLE]

Often the centre manifold is of most interest, so in $\xi_{3}$ setting all $X_{j}=0$ except $X_{\pm n}$, gives the quadratic approximate centre manifold to be $x_{j}=X_{j}$ for all $j$ except

[TABLE]

This is the same result as 9.2.

Proceeding to order of residual $p=4$ we may construct cubic corrections to $\xi_{2}$ to form $\xi_{4}$. For simplicity, restrict attention to the cases of $n$ odd. It is straightforward but tedious to construct that for $\xi_{4}$

[TABLE]

The terms excluded from $(k,l)$ in the sum 9.5 must cause cubic terms in the evolution. For example, when $r=n^{2}=1$ then444The apparent pattern in these odes becomes more complicated—at $\dot{X}_{\pm 6}$ for example.

[TABLE]

By construction, in this case of $r=1$, the coordinate transform 9.5 together with the odes 9.6 creates a dynamical system in $u(t,\theta)=\sum_{j}x_{j}e^{ij\theta}$ which is the same as the pde 9.1 to a residual of order four. In the combined system 9.5 and 9.6, by definition Definition 2.10 three invariant manifolds are: the 1D unstable manifold parametrised by $X_{0}$ with all other $X_{j}=0$; the 2D centre manifold parametrised by $X_{\pm 1}$ with all other $X_{j}=0$; and the stable manifold with $X_{0}=X_{\pm 1}=0$ .

{ArXiv}

See Appendix B for the computer algebra code used the for the computations in this subsection. It is also available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt.

Acknowledgement

Part of this research was supported by the Australian Research Council grant DP150102385.

{ArXiv}

Appendix A Compact and finite-rank operators into Banach spaces

Let $V$ and $W$ be Banach spaces, and suppose that $V^{*}$ has the approximation property (this is true for example if $V$ is a Hilbert space). Let $\{e^{j}\}_{j\in\mathbb{N}}\subset V^{*}$ and $\{f_{k}\}_{k\in\mathbb{N}}\subset W$ be countable subsets with dense spans. (So $V^{*}$ and $W$ are separable.)

In the main text, we use the following, which is standard in the case where $V$ and $W$ are Hilbert spaces.

Proposition A.1**.**

The space $\operatorname{span}\{e^{j}\otimes f_{k}:j,k\in\mathbb{N}\}$ is dense in $\mathcal{K}(V,W)$.

Let $\mathcal{F}(V,W)$ be the space of finite-rank linear operators from $V$ to $W$; that is, operators whose images are finite-dimensional.

Lemma A.2**.**

The space $\operatorname{span}\{e^{j}\otimes f_{k}:j,k\in\mathbb{N}\}$ is dense in $\mathcal{F}(V,W)$.

Proof.

Let $T\in\mathcal{F}(V,W)$. Since the image of $T$ is finite-dimensional, there are $v^{1},\ldots,v^{n}\in V^{*}$ and $w_{1},\ldots,w_{n}\in W$ such that $T=\sum_{l=1}^{n}v^{l}\otimes w_{l}\,.$

Let $\varepsilon>0$. For every $l$, choose $r\in\mathbb{N}$ and $a_{l}^{1},\ldots,a_{l}^{r}\in\mathbb{C}$ and $b_{l}^{1},\ldots,b_{l}^{r}\in\mathbb{C}$ such that

[TABLE]

Using the triangle and Cauchy–Schwartz inequalities, one finds that for all $v\in V$,

[TABLE]

∎

Proof of Proposition A.1..

Since $V^{*}$ has the approximation property, $\mathcal{F}(V,W)$ is dense in $\mathcal{K}(V,W)$. See for example Proposition 4.12(b) in the book by Ryan [31]. So the claim follows from Lemma A.2. ∎

{ArXiv}

Appendix B Computer algebra code for Burgers example computation

1Comment periodic Burgers-like, infinite-D coord transform.

2Written in Reduce, see https://reduce-algebra.sourceforge.io/

3AJR, 3 Jun 2019;

4on div; off allfac; on revpri;

5clear sum;

6

7operator xx; depend xx,t;

8let df(xx(~i),t)=>sub(j=i,dxxjdt);

9operator a; write "a(j) for eigenvalues alpha_j";

10operator b; write "b(j,k)=k*(j-k)";

11let b(j,j-~k)=>b(j,k);

12factor o;

13

14operator sum; linear sum;

15let df(sum(~a,~b),~t)=>sum(df(a,t),b);

16

17operator d;

18write "d(j,k)=1/(-a(j)+a(k)+a(j-k))=1/(n^2+2jk-2k^2)";

19write "d(j,k,l)=1/(-a(j)+a(k)+a(l-k)+a(j-l))

20        =1/2/(n^2+jl+kl-k^2-l^2)";

21let { d(j,k)a(k) => 1-d(j,k)(-a(j)+a(j-k))

22    , d(j,k,l)a(k) => 1-d(j,k)(-a(j)+a(l-k)+a(j-l))

23    };

24

25write "Linear approx evolution";

26xj:=xx(j);

27dxxjdt:=a(j)*xx(j);

28

29write "Quadratic solution shape";

30let o^2=>0;% order of error

31xj:=xj+o/2sum(b(j,k)(d(j,k)*t-d(j,k)^2)*xx(k)*xx(j-k),k);

32resj:=-df(xj,t)+a(j)*xj

33    +ot/2sum(b(j,l)*sub(j=l,xj)*sub(j=j-l,xj),l);

34resj:=sub(l=k,resj);

35

36write "However, zero divisors occur in d(j,k)

37=1/(-a(j)+a(k)+a(j-k)) =1/(n^2+2jk-2k^2).  But not for odd

38n.  For even n>0, find that zero divisors come from all the

39factors of n^2/2 since divisor=0 rewrites as k(k-j)=n^2/2.

40Both the positive and negative factors contribute.  For

41example, n=2 gives n^2/2=2 with factors 2,1 and -2,-1 and

421,2 and -1,-2.  So zero divisor for k=2,j=1 and k=-2,j=-1

43and k=1,j=-1 and k=-1,j=1.";

44

45write "Cubic manifold shape---pv. zero divisors are avoided

46in the quadratic terms by perhaps requiring n be odd.";

47let o^3=>0;% order of error

48resj:=-df(xj,t)+a(j)*xj

49    +ot/2sum(b(j,l)*sub(j=l,xj)*sub(j=j-l,xj),l)$

50depend k,kl; depend l,kl;

51resj:=(resj where sum(~~asum(~b,~l),~k)=>sum(ab,kl))$

52resj:=(resj where sum(~b,l)=>sum(sub(l=k,b),k))$

53resj:=(resj where sum(~a,kl)=>sum(sub(l=j-l,a),kl)

54    when df(a,xx(l)) neq 0)$

55write "Extract the coefficient of X(j-l)X(l-k)X(k)";

56rhsjkl:=(resj where sum(~a,kl) => coeffn( coeffn( coeffn(

57    a,xx(k),1) ,xx(j-l),1) ,xx(l-k),1));

58write "Iterative soln of ODE for coeff---zero divisors excluded";

59cjkl:=0;

60for it:=1:9 do begin

61  write resc:=d(j,k,l)*(rhsjkl-df(cjkl,t))-cjkl;

62  cjkl:=cjkl+resc;

63  if resc=0 then write "success ",it:=10000+it;

64end;

65cjkl:=cjkl;

66write "Update the xj";

67xj:=xj+sum(cjkl*xx(j-l)*xx(l-k)*xx(k),kl);

68

69write "But some RHS resonante and so must be in the

70evolution instead. Need definite r so set ",n:=1,

71"Explore coefficients to a max wavenumber of ",maxj:=10,

72"The negative j cases are the same with k, l, j-l and l-k

73all also changed sign.  It looks like that to get dxx(j)

74correct then we need maxj=j+3, but could be different for

75n>1, guess perhaps maxj=j+3*n?";

76r:=n^2;

77o:=1; % remove order symbol

78factor xx;

79% to get substitution have to change variables!

80rhsjkl:=(rhsjkl where { b(l,k)=>kk*(ll-kk)

81    , b(j,l)=>ll*(jj-ll)

82    , d(l,k)=>1/(r+2llkk-2*kk^2) })$

83% Now sum over all contributions to dXj/dt

84array dxx(maxj);

85for j:=0:maxj do write dxx(j):= (r-j^2)*xx(j)+

86    for k:=-maxj:maxj sum for l:=-maxj:maxj sum

87    if n^2+jl+kl-k^2-l^2=0

88    then sub({jj=j,kk=k,ll=l},rhsjkl)*xx(j-l)*xx(l-k)*xx(k)

89    else 0;

90

91end;