# A generic framework of adiabatic approximation for nonlinear evolutions II

Jingxuan Zhang (张景宣 )1,2 111Email<EMAIL_ADDRESS>
(1 University of Copenhagen
Department of Mathematical Sciences
Universitetsparken 5
2100 Copenhagen, Denmark
2 University of Toronto
Department of Mathematics
40 St George Street
Toronto, ON, M5S 2E4, Canada
)

###### Abstract

In this paper, we continue the development of a generic adiabatic scheme for
nonlinear evolutions. We consider an abstract gradient flow of some energy
functional, together with a given manifold of static solutions arising from
broken symmetries. First, we list a number of explicit and generic conditions
on the energy functional that ensures the validity of our adiabatic scheme.
Then, we construct some explicit low-energy but no necessarily static
configurations, which form a stable manifold with finite codimensions for the
given gradient flow. Thirdly, we show that the gradient flow is globally well-
posed with initial configuration from the stable manifold. Finally, we show
that any solution to the full gradient flow starting from the stable manifold
is essentially governed by an effective equation on the manifold of static
solutions, up to a uniformly small and dissipating error term.

##  1 Introduction

Consider the (negative) gradient flow

$\partial_{t}u=-E^{\prime}(u).$ (1.1)

Here $u=u(t)\in U,\,t\geq 0$ is a $C^{1}$ path of vectors in some open set
$U\subset X^{\prime}$ in a real Hilbert space $X^{\prime}$, sitting in a
larger Hilbert space $X$. The map $E:U\subset X^{\prime}\to\mathbb{R}$ is some
energy functional which is $C^{2}$ on $U$. The vector $E^{\prime}(u)\in X$ is
the gradient of $E$ at $u$ w.r.t. the inner product on $X$. See Appendix for
relevant definitions.

Throughout this paper, we assume the energy functional $E$ has a non-trivial
symmetry Lie group, and there exists a static solution $u\equiv u_{0}\in X$ to
(1.1) (i.e. an equilibrium state), which simultaneously breaks a symmetry Lie
subgroup $\Sigma$ with $1\leq\dim\Sigma<\infty$. We call this $\Sigma$ the
reduced symmetry (Lie) group for (1.1).

If we shrink $U$ to be a small ball around $u_{0}$, then the standing
assumptions above give rise to a finite-dimensional manifold of static
solutions (i.e. ensemble of equilibrium states)

$\mathcal{M}_{0}:=f(\Sigma)\cap U,$

where $f:\Sigma\to X^{\prime}$ is an immersion of the reduced symmetry Lie
group, given in (2.7) below. See Section 2.1 for details on the set up for
(1.1).

For the main result of this paper, Theorem 2.2, we prove that there exists a
finite-codimensional manifold $\mathcal{M}$ with

$\mathcal{M}_{0}\subset\mathcal{M}\subset U,$

s.th. the full dynamics (1.1) with initial configurations from $\mathcal{M}$
essentially reduces to an explicit effective (i.e. adiabatic) dynamics, namely
(2.26), posed on the finite-dimensional manifold of static solutions
$\mathcal{M}_{0}$.

In a previous paper [EffDyn], under a number of generic sufficient conditions,
we have developed an adiabatic approximation scheme for (1.1). There, we have
assumed global well-posedness for (1.1) with initial configurations in $U$, as
well as the existence of what we call approximate solitons. In this paper, we
present a new set of sufficient conditions on the linearized operator of the
map $u\mapsto E^{\prime}(u)$ at the manifold $\mathcal{M}_{0}$. These
conditions are similar but less restrictive than the ones from [EffDyn]. The
larger manifold $\mathcal{M}$ plays a similar role as the manifold of
approximate solitons in [EffDyn], except for the key difference that the
effective equation is not posed on $\mathcal{M}$, but rather on the smaller
manifold $\mathcal{M}_{0}$. See Section 2.3 for a discussion.

For motivations and backgrounds, we refer the readers to [EffDyn, AdiabMCF].
We single out a paper by W. Schlag [MR2480603] from which we have drawn much
inspiration. An application of the abstract theory developed in this paper is
given in [AdiabMCF], where we study the dynamical stability of the
singularities of the mean curvature flow.

###  1.1 Arrangement

This paper is organized as follows. In Section 2.1, we discuss in details the
set up for the evolution (1.1). In Section 2.2, we present the main
assumptions of this paper, (C1)-(C4). Then we state the main result in Section
2.3.

The main result, Theorem 2.2, is proved in two steps in Sections 3-6. The
upshot is to eliminate the destabilizing effect of the zero-unstable modes of
the linearized operator at the manifold of trivial static solution, see
(2.13).

At the first order, in Section 3,we eliminate the zero modes using modulation
method. Yet this is not enough due to the presence of unstable modes, see
(C2). Such modulation method is customary in the analysis of solitary wave
dynamics in hyperbolic systems, see [MR820338, MR1071238, MR1170476,
MR2094474].

At the next order, we establish a fixed point scheme to eliminate the unstable
modes that have persisted the modulation. In Section 4, we show that Theorem
2.2 is equivalent to a certain fixed point problem (see Proposition 4.2). In
Section 5, we solve the fixed point problem posed in the preceding section.
This gives rise to the set $\mathcal{M}$ mentioned above. In Section 6, we
prove certain smallness estimates on $\mathcal{M}$. These estimate then ensure
that $\mathcal{M}$ form a non-degenerate stable manifold with finite
codimensions.

The second part of the paper can be viewed as constructing a stable manifold
to (1.1), and compared to the established invariant manifold theory for
dissipative systems, see e.g. [MR1000974, MR1160925, MR1445489, MR1675237,
MR2439610]. Here we emphasize that our result gives global-wellposedness as a
by product (in contrast to the cited works above), and produce an explicit and
tractable effective dynamics for (1.1), namely (2.26).

### Notations

Throughout this paper, when no confusion arises, we shall drop the time
dependence $t$ in subscripts. An estimate $A\lesssim B$ means there is some
$C>0$ independent of time and all the parameters in question, s.th. $A\leq
CB$. The expression $A\sim B$ means that $A\lesssim B$ and $B\lesssim A$ hold
simultaneously. The summation convention for repeated indices is always
understood.

##  2 Set up and the main results

In this section, we explain the set up for a generic adiabatic scheme for
(1.1). Then we present a list of sufficient conditions for the validity of
this scheme, and justify their generality. These assumptions consist of some
secondary ones, (f1)-(f2), (h1)-(h2) from Section 2.1, which concern with
generic properties for (1.1) in the presence of symmetry. In Section 2.2, we
present a group of non-trivial assumptions, (C1)-(C4), in terms of certain
linearized operators. Lastly, in Section 2.3, we state the main result,
Theorem 2.2, under these assumptions, and discuss its implications. This
theorem is proved in Section 3-Section 6.

###  2.1 Set up

Let $X$ be a Hilbert space with norm $\lVert\cdot\rVert_{X}$ induced by the
inner product $\left\langle\cdot,\,\cdot\right\rangle_{X}$. Let $X^{\prime}$
be a subspace of $X$, together with a stronger norm
$\lVert\cdot\rVert_{X^{\prime}}$. We understood that $X^{\prime}$ inherits the
inner product $\left\langle\cdot,\,\cdot\right\rangle_{X}$.

A typical example is $X=L^{2}(M,d\mu)$, the Lebesgue space of order $2$ over a
measure space $(M,d\mu)$, and $X^{\prime}:=H^{s}$, the Sobolev space of order
$s\geq 0$. The inner product on both spaces are the $L^{2}$-one, but
$X^{\prime}$ is equipped with the stronger $H^{s}$-norm.

Let $E:X\to\mathbb{R}$ be a $C^{2}$ energy functional in the the sense that
there exists an open set $U\subset X^{\prime}$ (w.r.t. the topology induced by
$\lVert\cdot\rVert_{X^{\prime}}$) s.th. the map $E^{\prime}:U\subset
X^{\prime}\to X$, sending a vector $u$ to the $X$-gradient $E^{\prime}(u)$, is
well-defined and continuous.

Denote by $E^{\prime\prime}(u):X^{\prime}\to X$ the Fréchet derivative of
$E^{\prime}$ at $u\in U$. Then for every $u$, $E^{\prime\prime}(u)$ is a
bounded self-adjoint operator defined on all of $X^{\prime}$. See Appendix for
relevant definitions.

A typical example arising from statistical field theory is as follows. Let
$(M,g)$ be a sufficiently regular Riemannian manifold, corresponding to a
physical domain of interest. Let $X:=L^{2}(M,d\mu;\mathbb{R})$, where $d\mu$
is the canonical measure induced by $g$. This $X$ describes a space of order
parameter. We take $X^{\prime}=H^{1}(M,d\mu;\mathbb{R})$, and define

$E(u)=\int\left\lvert\nabla u\right\rvert^{2}+G(u),$

where $G:\mathbb{R}\to\mathbb{R}$ is an explicit nonlinear function. This $G$
incorporates certain nonlinear phenomen, e.g. due to mean field approximation.
Suppose there is a critical point $u_{0}\in X^{\prime}$ to $E$ with
$\left\lvert E(u_{0})\right\rvert<\infty$, say, describing a pure state, which
amounts to a trivial static solution to (1.1). Suppose $G$ satisfies certain
regularity assumptions depending on this $u_{0}$. Then one can show using
Sobolev inequalities that $E$ is $C^{2}$ in a neighbourhood $U\subset
X^{\prime}$ around $u_{0}$.

We assume evolution (1.1) has a finite-dimensional full symmetry Lie group $G$
with $\dim G\geq 1$. Precisely, we assume each element $g\in G$ acts on the
configuration space $X$ as a linear transformation

$T(g)\in Aut(X),$ (2.1)

satisfying

$E(T(g)u)=T(g)E(u)\quad(u\in X,g\in G).$ (2.2)

In applications, such $T(g)$ can be translation, rotation, gauge transform,
etc.. Denote by

$T:g\mapsto T(g)$ (2.3)

the representation of $G$ via automorphism of $X$.

We assume there exists some $u_{0}\in X^{\prime}$ s.th. $E^{\prime}(u_{0})=0$.
In applications, this $u_{0}$ is usually a well-understood ground state.
Denote by $H\subset G$ the maximal closed normal subgroup of $G$ s.th.
$T(g)u_{0}=u_{0}$ for every $g\in H$, and define the reduced symmetry Lie
group by

$\Sigma:=G/H.$ (2.4)

We assume

$\dim\Sigma\geq 1.$

Physically, this means that $u_{0}$ break parts of the full symmetry group of
the energy functional $E$ (i.e. simultaneous symmetry breaking).

In the remaining of this paper, we will only be concerned with the reduced
symmetry group $\Sigma$, instead of the full symmetry group $G$.

We assume there exists a Riemannian metric $g$ on $\Sigma$, which turns the
latter into a Riemannian manifold. For example, $\Sigma$ is a matrix Lie
subgroup, then one can take $g(A,B)=\operatorname{Tr}A^{*}B$. Denote by

$d(\sigma,\sigma^{\prime}),\quad\sigma,\sigma^{\prime}\in\Sigma$

the distance induced by this metric $g$ on $\Sigma$, and

$\lVert\cdot\rVert_{Y}$

the norm induced by $g$ on the tangent space $T_{\sigma}\Sigma$. For
simplicity, we assume $\lVert\cdot\rVert_{Y}$ does not change as the base
point $\sigma$ varies in $\Sigma$.

Define a map

$\displaystyle\begin{array}[]{ccrcl}{f}&\colon&{\Sigma}&\longrightarrow&{X}\\\
\mbox{}&\mbox{}&{\sigma}&\longmapsto&{T(\sigma)u_{0}}\end{array}$ (2.7)

We assume the map $f$ has the following properties:

  1. (f1)

$f(\mathbf{1}_{\Sigma})=u_{0}$, $f(\Sigma)\subset X^{\prime}$, and $f$ is a
$C^{2}$ immersion as a map from $\Sigma\to X^{\prime}$.

  2. (f2)

There exists $c_{f}>0$ s.th.

$\lVert f\rVert_{C^{2}}:=\sup_{\sigma\in\Sigma_{0}}\left(\lVert
f(\sigma)\rVert_{X^{\prime}}+\lVert df(\sigma)\rVert_{Y\to X^{\prime}}+\lVert
d^{2}f(\sigma)\rVert_{Y\to L(Y,X^{\prime})}\right)\leq c_{f}.$ (f2)

Condition (f1) allows one to view $f(\Sigma)$ as a manifold in $X^{\prime}$.
Condition (f2) is a technical assumption used to derive various uniform
estimates (see the remark below). These conditions are analogous to the one in
[EffDyn]*Sec 2.2. In applications, (f1)-(f2) mostly follow easily from the
explicit form of symmetry $T(\sigma)$ in (2.1).

######  Remark 1.

If $f$ satisfies condition (f1), then there exists an open connected
submanifold

$\Sigma_{0}\subset\Sigma$ (2.8)

around the identity element $\mathbf{1}_{\Sigma}$, s.th.

$\mathcal{M}_{0}:=f(\Sigma_{0})\subset U$ (2.9)

forms a finite-dimensional immersed manifold, and (f2) holds. This gives rise
to a manifold of static solution to (1.1). In the setting of [EffDyn], this
$f(\Sigma_{0})$ amounts to a manifold of exact solitons.

Shrinking $\Sigma_{0}$ if necessary, there exists a local trivialization

$h:\mathbb{R}^{\dim\Sigma}\times\Sigma_{0}\to\bigcup_{\sigma\in\Sigma_{0}}T_{\sigma}\Sigma.$
(2.10)

This allows us to write a typical element in $T_{\sigma}\Sigma\cong
Lie(\Sigma)$ as

$a^{i}\xi_{i}(\sigma),$

where

$\xi_{i}(\sigma):=h(E_{i},\sigma)\quad(1\leq i\dim\Sigma),$ (2.11)

and $\left\\{E_{i}\right\\}$ is a basis of $\mathbb{R}^{\dim\Sigma}.$

We assume the trivilization $h$ satisfies

  1. (h1)

For every $\sigma\in\Sigma_{0}$, there holds the orthogonality condition

$\left\langle
df(\sigma)\xi_{i}(\sigma),\,df(\sigma)\xi_{j}(\sigma)\right\rangle_{X}=\delta_{ij}.$
(h1)

  2. (h2)

There exists $c_{h}>0$ independent of $i$ s.th.

$\lVert\xi_{i}(\cdot)\rVert_{C^{1}}:=\sup_{\sigma\in\Sigma_{0}}\lVert\xi_{i}(\sigma)\rVert_{Y}+\lVert
d\xi_{i}(\sigma)\rVert_{Y\to Y}\leq c_{h}$ (h2)

Recall that $\lVert\cdot\rVert_{Y}$ denotes the norm on $T_{g}\Sigma$ induced
by the Riemannian metric $g$ on $\Sigma$.

The role of condition (h1) will be clear after Definition 1 below. Condition
(h2) plays a similar role as (f2). For bounded $\Sigma_{0}\subset\Sigma$, this
follows from the regularity assumption (h1).

Now we can define a class of key objects for the subsequent analysis.

######  Definition 1 (symmetry zero modes).

A vector $\phi\in X^{\prime}$ is called a symmetry zero mode of the linearized
operator $L(\sigma)$ if and only if

$\phi=a^{i}df(\sigma)\xi_{i}(\sigma)$ (2.12)

for some $a^{i}\in\mathbb{R},\,1\leq i\leq\dim\Sigma$.

######  Remark 2.

Condition (h1) ensures that these modes are orthonormal in $X$ as $\sigma$
varies in $\Sigma_{0}$.

Denote by

$L(\sigma):=E^{\prime\prime}(f(\sigma))$ (2.13)

the linearized operator of $E^{\prime}$ at $f(\sigma)$. This is well-defined
and continuous in $\Sigma_{0}$, by the assumption (f1) on $f(\sigma)$ and the
$C^{2}$ regularity assumption of $E$ on $U$.

######  Lemma 2.1.

If $\phi$ is a symmetry zero mode of $L(\sigma)$, then $L(\sigma)\phi=0$.

###### Proof.

Write $\phi=a^{i}df(\sigma)\xi_{i}(\sigma)$ as in (2.12). Take a path
$\sigma(s)\in\Sigma,\,\left\lvert s\right\rvert\ll 1$ with
$\sigma(0)=\sigma,\,\partial_{s}\sigma(0)=a^{i}\xi_{i}(\sigma)$ in terms of
the basis (2.11). By (2.2), we have

$E^{\prime}(f(\sigma(s)))\equiv 0.$ (2.14)

Differentiating this w.r.t. $s$ and using the chain rule, we find

$0=\partial_{s}E^{\prime}(f(\sigma(s)))=E^{\prime\prime}(f(\sigma(s)))df(\sigma(s))\partial_{s}\sigma(s).$

By the initial condition for $\sigma(s)$, evaluating this at $s=0$ yields
$L(\sigma)\phi=0$. ∎

######  Remark 3.

We note a small caveat here regarding the Definition 1. It is not necessary
for the symmetry modes to be in $X^{\prime}$, which is usually taken to be the
domain of $L(\sigma)$. For example, the Ginzburg-Landau vortices

$u^{(n)}:\mathbb{R}^{2}\to\mathbb{C},n\in\mathbb{Z}$

are solution to the semilinear elliptic equation

$-\Delta u+(\left\lvert u\right\rvert^{2}-1)u=0,$

with $\deg u^{(n)}=n$. This equation is naturally posed on
$L^{2}=L^{2}(\mathbb{R}^{2},\mathbb{C})$, and has translation symmetry, which
generates the symmetry zero modes

$\partial_{x^{j}}u^{(n)},\quad j=1,2$

as in Definition 1. Yet, for every $n\neq 0$, these zero modes are not in the
natural configuration space $L^{2}$ (though they are in $L^{p}$ for every
$p>2$).

As we show in Section 3, symmetry zero modes can be easily eliminated by a
first order correction, known as the modulation equation. This method was
first introduced in [MR1071238, MR1170476] to study a class of nonlinear
Schödinger equations. It effectively eliminates the influence of $\xi_{i}$
from the dynamics.

###  2.2 Main assumptions

The main assumptions in this paper are concerned with the linearized operator
at

$L(\sigma)=E^{\prime\prime}(T(\sigma)u_{0}):X^{\prime}\subset X\to X,$ (2.15)

defined in (2.8) above.

Main assumptions. Fix $\sigma\in\Sigma_{0}$ with $\Sigma_{0}\subset\Sigma$
given by a neighbourhood around $1_{\Sigma}$, see (2.9).

  * (C1)

$L(\sigma)$ is self-adjoint, and there exists $c_{1}>0$ independent of
$\sigma$ s.th. (2.15) is uniformly bounded as

$\lVert L(\sigma)\rVert_{X^{\prime}\to X}\leq c_{1}.$ (C1)

  * (C2)

$0$ is an embedded eigenvalue of $L(\sigma)$, and the associated eigenvectors
consist solely of the symmetry zero modes (2.12). The discrete spectrum
$\sigma_{\rm d}(L(\sigma))$ consists of isolated eigenvalues

$E_{1}(\sigma)\leq E_{2}(\sigma)\leq\ldots\leq E_{N}(\sigma)\leq E<0.$ (C2)

for some $N\geq 0$ and $E<0$ independent of $\sigma$. The multiplicity of
$E_{p}(\sigma),\,0\leq p\leq N$ is finite and independent of $\sigma$.

We denote by $m_{p}$ the multiplicity of $E_{p}(\sigma)$, so that $1\leq
m_{p}<\infty$. The eigenvectors associated to $E_{j}(\sigma)$ are denoted by

$\phi_{j_{p}}(\sigma),\quad 1\leq j_{p}\leq m_{p}.$ (2.16)

  * (C2)

There holds the propagator estimate

$\lVert e^{-tL(\sigma)}P_{S}(\sigma)\rVert_{X\to X^{\prime}}\leq
c_{2}\left\langle t\right\rangle^{-\alpha}\quad(t\geq 0),$ (C3)

for some $\alpha\geq 1,\,c_{2}>0$ independent of $\sigma,\,w,\,t$. Here
$P_{S}(\sigma)$ denotes the orthogonal projection onto the stable range

$\mathcal{S}(\sigma):=\left\\{w\in X:\left\langle
w,\,\phi_{j_{p}}(\sigma)\right\rangle_{X}=0\text{ for every }1\leq j_{p}\leq
m_{p},\;0\leq p\leq N\right\\}$ (2.17)

  * (C4)

There exists $0<\delta_{0}\ll 1$ s.th. the for every
$\sigma,\sigma^{\prime}\in\Sigma$ with
$d(\sigma,\sigma^{\prime})\leq\delta_{0}$, there holds

$\lVert L(\sigma)-L(\sigma^{\prime})\rVert_{X^{\prime}\to
X}\leq\delta_{0}d(\sigma,\sigma^{\prime}).$ (C4)

A number of remarks are in order to justify these conditions.

######  Remark 4.

Condition (C1) is customary for equation arising from physics. For example, in
applications to classical field theory, $L(\sigma)$ is usually a Schrödinger
operator of the form $L(\sigma)=-\Delta+V(\sigma),$ where $V(\sigma):X\to X$
is a multiplication operator, arising from the linearization process.

######  Remark 5.

Condition (C2) is customary for spectral analysis of Schrödinger operators,
e.g. [MR1639709, MR1639713]. By [MR2006008]*thm(D), if $E$ is $C^{3}$ on $U$,
then (C2) holds for all $\sigma\in\Sigma_{0}$.

######  Remark 6.

Condition (C3) compensates the presence of unstable modes of $L(\sigma)$. For
fixed $\sigma$ and all large $t$, (C3) holds, for example, if $0\in\sigma_{\rm
ess}$ is not a resonance, in the sense that $L(\sigma)P_{S}(\sigma)\geq c>0$.
For a discussion of the latter spectral gap condition, see [EffDyn]*Sect. 2.2.

######  Remark 7.

Condition (C4) can be stated independent of the linearized operator
$L(\sigma)$. Indeed, the point here is to require Fréchet derivatives of $E$
to vanish at a critical point $u_{*}$ at least up to the second order. This
holds, for example, if $E$ is analytic at the critical point $u_{*}$.

######  Remark 8.

Condition (C4) should be understood as an assumption on the nonlinearity.
Indeed, for $\sigma\in\Sigma,\,w\in X^{\prime}$ define the nonlinearity

$N(\sigma,w):=E^{\prime}(f(\sigma)+w)-L(a)w.$ (2.18)

If $E$ is $C^{2}$ on $U$, then $N(\sigma,w)$ is continuous on $\Sigma_{0}$
times a small ball in $X^{\prime}$, and satisfies $\lVert
N(\sigma,w)\rVert_{X}=o(\lVert w\rVert_{X^{\prime}})$. In principle, one would
like to upgrade this remainder estimate to a quadratic one of the form $\lVert
N(\sigma,w)\rVert_{X}\lesssim\lVert w\rVert_{X^{\prime}}^{2}$, where the
implicit constant is independent of $\sigma$. However, in application, this is
sometimes difficult when one uses non-standard norms with strong decay
properties. For example, in [MR3374960], the nonlinearity fails to be
$O(\lVert w\rVert_{X^{\prime}}^{1+\epsilon})$ for any $\epsilon>0$ due to the
use of Gaussian weighted Sobolev norm. See Sect. 4 of that paper.

######  Remark 9.

In the sequel, we will mostly use the following consequence of (C4): There
exists $0<\delta=\delta(c_{f},\delta_{0})\ll 1$ (see (C4) and (f2)) s.th. for
every $\sigma,\,\sigma^{\prime}\in\Sigma_{0}$ and $w,w^{\prime}\in X^{\prime}$
with $\lVert w\rVert_{X^{\prime}}+\lVert w\rVert_{X^{\prime}}\leq\delta$,
there holds

$\lVert
N(\sigma,w)-N(\sigma^{\prime},w^{\prime})\rVert_{X}\leq\delta(d(\sigma,\sigma^{\prime})+\lVert
w-w^{\prime}\rVert_{X^{\prime}}).$ (2.19)

###  2.3 Main result

In the remaining of this paper, we fix some $\sigma_{0}\in\Sigma_{0}$. In
application, this choice amounts to a stationary symmetry frame for (1.1). For
$r>0$ we write

$B_{r}(\eta):=\left\\{\eta^{\prime}\in
X^{\prime}:\lVert\eta-\eta^{\prime}\rVert_{X^{\prime}}<r\right\\},\quad
B_{r}\equiv B_{r}(0).$

The main result of this paper is the following statement about the key map
$\Phi$ defined in (5.28).

######  Theorem 2.2.

Suppose (f1)-(f2), (h1)-(h2), and (C1)-(C4) hold. Then there exist
$0<\delta\ll 1$, $c_{0}>0$ independent of $\delta$, together with a map

$\Phi:\mathcal{S}(\sigma_{0})\cap B_{c_{0}\delta}\to
X^{\prime}\quad\text{defined in \eqref{Phi} below, }$ (2.20)

s.th. for every $\eta,\,\eta^{\prime}$ in the domain of $\Phi$,

$\displaystyle\lVert\Phi(\eta)\rVert_{X^{\prime}}=$ $\displaystyle
o(\lVert\eta\rVert_{X^{\prime}})\quad(\delta\to 0),$ (2.21)
$\displaystyle\lVert\Phi(\eta)-\Phi(\eta^{\prime})\rVert_{X^{\prime}}=$
$\displaystyle
O(\delta\lVert\eta-\eta^{\prime}\rVert_{X^{\prime}})\quad(\delta\to 0),$
(2.22)

and the following holds:

  1. 1.

(Global existence) For every $\eta\in\mathcal{S}(\sigma_{0})$ with
$\lVert\eta_{0}\rVert_{X^{\prime}}<c_{0}\delta$, there exists a unique global
solution

$u\in C^{1}(\mathbb{R}_{\geq 0},\Sigma\times X)\cap Lip(\mathbb{R}_{\geq
0},\Sigma\times X^{\prime})$

to the gradient flow (1.1), with initial configuration

$u(0)=f(\sigma_{0})+\eta+\Phi(\eta).$ (2.23)

  2. 2.

(Effective dynamics) For all $t\geq 0$, this solution $u$ is of the form

$u=f(\sigma)+w,$ (2.24)

where $f$ is defined in (2.7), the path $w=w(t),\,t\geq 0$ satisfies

$\lVert w(t)\rVert_{X^{\prime}}\leq\delta\left\langle
t\right\rangle^{-\alpha},$ (2.25)

and $\sigma(t)\in\Sigma,\,t\geq 0$ evolves according to the effective dynamics

$\dot{\sigma}=h(a(\sigma,w),\sigma),\quad\sigma(0)=\sigma_{0}.$ (2.26)

Here, $h$ is the trivilization given in (2.10), the vector
$a=a(\sigma,w)\in\mathbb{R}^{\dim\Sigma}$ is the unique solution to the
equation $G(a,\sigma,w)=0$, with $G$ defined in (3.13). Moreover, for all
$t\geq 0$, there holds

$\lVert\dot{\sigma}(t)\rVert_{Y}=O(\delta\left\langle
t\right\rangle)\quad(\delta\to 0).$ (2.27)

######  Definition 2 (stable manifold).

With notations as in Theorem 2.2 above, the set

$\mathcal{M}=\mathcal{M}(\sigma_{0},\delta):=\left\\{f(\sigma_{0})+\eta+\Phi(\eta):\eta\in\mathcal{S}(\sigma_{0}),\,\lVert\eta\rVert_{X}^{\prime}<c_{0}\delta\right\\}$
(2.28)

is called the stable manifold for (1.1),

This set $\mathcal{M}$ forms a manifold. Indeed, by Rademacher’s theorem
[MR1373829], Lipschitz estimate (2.22) ensures that $\Phi$ is differentiable
a.e. with respect to a suitable Radon measure on $X^{\prime}$, and the Fréchet
derivative $d\Phi(\eta):X^{\prime}\to X^{\prime}$ satisfies $\lVert
d\Phi(\eta)\rVert_{X^{\prime}\to X^{\prime}}\lesssim\delta$ for fixed small
$\delta$. This, together with definition (2.28) above, implies that
$\mathcal{M}$ is an a.e. immersed manifold in $X^{\prime}$.

######  Remark 10.

By (2.21), the set $\mathcal{M}$ is a small graph over the finite-
codimensional affine space $f(\sigma_{0})+\mathcal{S}(\sigma_{0})$. By the
preceding remark, this graph can be viewed as a finite-codimensional manifold
inside the small ball $B_{\delta}(f(\sigma_{0}))\subset U$. This justifies the
generality of Theorem 2.2, as Items 1-2 above only have to do with initial
configuration from $\mathcal{M}$.

We understand Theorem 2.2 in two ways. First, in physical terms, this can be
viewed as an adiabatic theorem for (1.1) as follows. By (2.21) and the
assumption $0<\delta\ll 1$ (i.e. the adiabatic condition), $\mathcal{M}$
consists of low energy solution in the configuration space for (1.1) (recall
that $E^{\prime}(f(\sigma_{0}))=0$ by (2.2)). Then assertions (2.24)-(2.26)
show that the full dynamics (1.1) for $u$, which a priori is an infinite-
dimensional dynamical system on $X$, reduces to an effective dynamics for
$\sigma$, which is a system of nonlinear ODEs posed on
$\mathbb{R}^{\dim\Sigma}$.

By construction, the manifold $\mathcal{M}_{0}\equiv f(\Sigma_{0})\subset U$
of static solution is an ensemble of equilibrium states (see (2.9)), and
therefore (2.26) is the adiabatic approximation for (1.1). The validity of
this adiabatic dynamics is guaranteed, since (2.25) ensures that a generic
initial perturbation (see Remark 10) dissipates to $0$ as $t\to\infty$.

Secondly, in the theory of invariant manifold, the set $\mathcal{M}$ can be
viewed as a stable manifold for (1.1). Indeed, the dissipative estimate (2.25)
implies that the trivial solutions $u_{0}$ (or equivalently, $f(\sigma_{0})$
for any fixed $\sigma_{0}\in\Sigma_{0}$) is asymptotically stable with initial
perturbations of the form (2.23).

The remaining sections are devoted to the proof of Theorem 2.2.

##  3 First order correction

We would like to reduce the full dynamics (1.1) to an effective one on the
manifold of static solution $f(\Sigma_{0}$). The main obstacle is the presence
of the zero modes (2.12), and the unstable modes (2.16). In this section, we
use the method of modulation equation to eliminate the symmetry zero modes.
The unstable modes are handled in Sections 4-5. At least heuristically, once
we remove all the zero-unstable modes, the desired dissipative estimate (2.25)
would follow from the propagator estimate (C2).

The main result of this section is the following:

######  Theorem 3.1.

Let $u(t)\in U,\,t\geq 0$ be a solution to (1.1). Suppose $u=u(t)$ is of the
form

$u=f(\sigma)+w,$ (3.1)

where $\sigma=\sigma(t)$ is a path in $\Sigma_{0}$, the vector $f(\sigma)\in
U$ (see (2.7)-(2.8)), and $w:=u-f(\sigma)$ satisfies $\lVert
w(t)\rVert_{X^{\prime}}\ll 1$.

Suppose $w(0)$ is orthogonal to all symmetry zero modes of $L(\sigma(0))$,
i.e.

$\left\langle w(0),\,df(\sigma(0))\xi_{i}\right\rangle=0\quad(1\leq
i\leq\dim\Sigma).$ (3.2)

Then $w(t),t>0$ satisfies

$\left\langle w(t),\,df(\sigma(t))\xi_{i}(\sigma(t))\right\rangle=0\quad(1\leq
i\leq\dim\Sigma)$ (3.3)

if and only if

$\dot{\sigma}(t)=h(a(t),\sigma(t))=a^{j}(t)\xi_{j}(\sigma(t)),$

were $h$ is the trivialization given in(2.10), and each $a^{i}=a^{i}(t)$
solves the algebraic equation

$\displaystyle a^{i}=$
$\displaystyle\rho_{ij}(\sigma,w)a^{j}+\tau_{i}(\sigma,w)$ (3.4)
$\displaystyle\rho_{ij}(\sigma,w):=$ $\displaystyle\left\langle
w,\,(d^{2}f(\sigma)\xi_{j}(\sigma))\xi_{i}(\sigma)\right\rangle+\left\langle
w,\,(d\xi_{i}(\sigma))\xi_{j}(\sigma)\right\rangle$ (3.5)
$\displaystyle\tau_{i}(\sigma,w):=$ $\displaystyle-\left\langle
N(\sigma,w),\,df(\sigma)\xi_{i}(\sigma)\right\rangle.$ (3.6)

###### Proof.

(3.3) holds if and only if it holds at $t=0$, i.e. (3.2) is satisfied, and

$\left\langle\dot{w},\,df(\sigma)\xi_{i}(\sigma)\right\rangle=-\left\langle
w,\,\partial_{t}(df(\sigma)\xi_{i}(\sigma))\right\rangle.$ (3.7)

We now calculate both sides of (3.7) explicitly, and show that it is
equivalent to (3.7).

Differentiating (3.1), we find

$\dot{u}=df(\sigma)\dot{\sigma}+\dot{w}=a^{i}df(\sigma)\xi_{j}(\sigma)+\dot{w},$
(3.8)

where we write the velocity $\dot{\sigma}=h(a,\sigma)=a^{j}\xi_{j}(\sigma)$ in
terms of the basis (2.11). Plugging this back to the l.h.s. of (3.7), we find

$\left\langle\dot{w},\,df(\sigma)\xi_{i}(\sigma)\right\rangle=-\left\langle
E^{\prime}(f(\sigma)+w),\,df(\sigma)\xi_{i}(\sigma)\right\rangle-a^{j}\left\langle
df(\sigma)\xi_{j}(\sigma),\,df(\sigma)\xi_{i}(\sigma)\right\rangle$ (3.9)

Since $f(\sigma)\in U$ and $\lVert w\rVert_{X^{\prime}}\ll 1$, the following
expansion is valid:

$E^{\prime}(f(\sigma)+w)=E^{\prime}(f(\sigma))+L(\sigma)w+N(\sigma,w).$

The first term in the r.h.s. vanishes by (2.2) and the definition of $f$, see
(2.14). By Lemma 2.1 and the self-adjointness of $L(\sigma)$ from condition
(C1), the inner product

$\left\langle L(\sigma)w,\,df(\sigma)\xi_{i}\right\rangle=\left\langle
w,\,L(\sigma)df(\sigma)\xi_{i}\right\rangle=0.$

Hence, the first term in the r.h.s. of (3.9) simplifies to $-\left\langle
N(\sigma,w),\,df(\sigma)\xi_{i}(\sigma)\right\rangle$. This, together with
(h1), implies that

$\left\langle\dot{w},\,df(\sigma)\xi_{i}(\sigma)\right\rangle=-a_{i}-\left\langle
N(\sigma,w),\,df(\sigma)\xi_{i}(\sigma)\right\rangle.$ (3.10)

For the r.h.s. of (3.7), we calculate

$\displaystyle\partial_{t}(df(\sigma)\xi(\sigma))$ (3.11) $\displaystyle=$
$\displaystyle\partial_{t}(df(\sigma))\xi_{i}(\sigma)+df(\sigma)\partial_{t}\xi_{i}(\sigma)$
$\displaystyle=$ $\displaystyle
a^{j}((d^{2}f(\sigma)\xi_{j})\xi_{i}+d\xi_{i}(\sigma)\xi_{j})\quad(\sigma=a^{j}\xi_{j}).$

Here $d^{2}f(\sigma):T_{\sigma}\Sigma\to L(T_{\sigma}\Sigma,X^{\prime})$ and
$d\xi_{i}(\sigma):T_{\sigma}\Sigma,X^{\prime}$ are respectively the Fréchet
derivatives of $df$ and $\xi_{i}=h(\cdot,E_{i})$ evaluated at $\sigma$, see
(2.7) and (2.11). Plugging (3.10)-(3.11) back to (3.7) gives (3.4).

∎

######  Corollary 3.2.

Suppose $u=f(\sigma)+w$ is a solution to (1.1) as in Theorem 3.1, and
satisfies (3.2)-(3.3). Then $\dot{\sigma}=h(a,\sigma)$ with $\left\lvert
a\right\rvert=o(\lVert w\rVert_{X^{\prime}})$.

###### Proof.

By Theorem 3.1, the path $\dot{\sigma}$ satisfies

$\dot{\sigma}=h(\sigma,a),\quad G(a,\sigma,w)=0,$ (3.12)

where

$\begin{array}[]{ccrcl}{G}&\colon&{\mathbb{R}^{\dim\Sigma}\times\Sigma\times
X^{\prime}}&\longrightarrow&{\mathbb{R}^{\dim\Sigma}}\\\
\mbox{}&\mbox{}&{(a,\sigma,w)}&\longmapsto&{a^{i}-\rho_{ij}(\sigma,w)a^{j}-\tau_{i}(\sigma,w)}\end{array},$
(3.13)

c.f. (3.4)-(3.6). We use (3.12)-(3.13) to establish the estimate for $a$.

Fix $1\leq i,j\leq\dim\Sigma$. We first bound $\rho_{ij}$ from (3.5). The
first term in $\rho_{ij}$ can be bounded by

$\left\lvert\left\langle
w,\,(d^{2}f(\sigma)\xi_{j}(\sigma))\xi_{i}(\sigma)\right\rangle\right\rvert\leq
c_{f}c_{h}\dim\Sigma\lVert\xi\rVert_{Y}\lVert w\rVert_{X},$ (3.14)

and the second term by

$\left\lvert\left\langle
w,\,(d\xi_{i}(\sigma))\xi_{j}(\sigma)\right\rangle\right\rvert\leq
c_{h}\dim\Sigma\lVert w\rVert_{X}.$ (3.15)

These follow from Cauchy-Schwartz and the assumptions (f2), (h2). Combining
these we find $\left\lvert\rho_{ij}(\sigma,w)\right\rvert\lesssim\lVert
w\rVert_{X}$.

Similarly, we find that $\tau_{i}$ is of sub-leading order, as

$\left\lvert\left\langle
N(\sigma,w),\,df(\sigma)\xi_{i}(\sigma)\right\rangle\right\rvert\leq
c_{f}c_{h}\lVert N(\sigma,w)\rVert_{X}=o(\lVert w\rVert_{X^{\prime}}).$ (3.16)

The last bound follow from the regularity of $E$ on $U$.

Plugging (3.14)-(3.16) into (3.12), and using the definition of $G$ in (3.13),
we find that

$\left\lvert a^{i}\right\rvert\leq c\lVert w\rVert_{X}\left\lvert
a\right\rvert+o(\lVert w\rVert_{X^{\prime}})$ (3.17)

for some absolute constant $c>0$. If $\lVert
w\rVert_{X^{\prime}}\leq(2c\dim\Sigma)^{-1}$, then (3.16) implies
$\tfrac{1}{2}\left\lvert a\right\rvert\leq o(\lVert w\rVert_{X^{\prime}})$.

∎

######  Proposition 3.3.

For every $(\sigma,w)\in\Sigma_{0}\times X^{\prime}$ with $\lVert
w\rVert_{X^{\prime}}\ll 1$, there exists a unique solution
$a=a(\sigma,w)\in\mathbb{R}^{\dim\Sigma}$ to the equation $G(a,\sigma,w)=0$,
where $G$ is as in (3.13)

Moreover, there exist constants $c,\,c^{\prime}>0$ independent of
$(\sigma,w),(\sigma^{\prime},w^{\prime})$ s.th.

$\displaystyle\left\lvert a(\sigma,w)\right\rvert\leq$ $\displaystyle c\lVert
w\rVert_{X^{\prime}},$ (3.18) $\displaystyle\left\lvert
a(\sigma,w)-a(\sigma^{\prime},w^{\prime})\right\rvert\leq$ $\displaystyle
c^{\prime}(d(\sigma,\sigma^{\prime})+\lVert w-w^{\prime}\rVert_{X^{\prime}}).$
(3.19)

###### Proof.

By formula (3.5)-(3.6), we find that equation (3.13) has trivial solution
$(a,\sigma,w)=(0,0,0)$. As a part of the standing assumptions, $f$ is $C^{2}$
and $\xi_{i}$ is $C^{1}$ on $\Sigma_{0}$, and $N(\sigma,w)$ is continuous from
$X^{\prime}\to X$ for $\left\lvert w\right\rvert_{X^{\prime}}\ll 1$. Hence the
map $G(a,\sigma,w)$ is $C^{1}$ in $a$ for fixed $(\sigma,w)\in\Sigma_{0}\times
X^{\prime}$ with $\lVert w\rVert_{X^{\prime}}\ll 1$. The Jacobian matrix of
$G$ at the trivial solution is given by
$\partial_{a^{j}}G_{i}|_{(0,0,0)}=\delta_{ij}$, since $\rho_{ij}(\sigma,w)\to
0$ for $w\to 0$, as we have shown in the proof of Corollary 3.2. Combining
these facts, the existence claim follows from Inverse Function Theorem.

Estimate (3.18) follows from Corollary 3.2. The Lipschitz estimate (3.19)
follows from the uniform estimate assumptions (f2), (h2), and the Lipschitz
estimate assumption on the linearity, (2.19). ∎

##  4 Fixed point scheme

In this section, we present a fixed point scheme to remove the destabilizing
effect of the unstable modes (2.16) of the linearized operator. This amounts
to a second order correction in the sense of (6.2)-(6.3) below. As we show in
Proposition 4.2, this fixed point scheme, together with the modulation from
Section 3, yields global existence as well as the dissipative estimate in the
main result, Theorem 2.2.

Let

$\mathcal{X}:=C^{1}(\mathbb{R}_{\geq 0},\Sigma\times X)\cap
Lip(\mathbb{R}_{\geq 0},\Sigma\times X^{\prime}).$ (4.1)

Fix $0<\delta\ll 1$. Let

$\mathcal{A}_{\delta}=\left\\{(\sigma,w)\in\mathcal{X}:\lVert
w(t)\rVert_{X^{\prime}}\leq\delta\left\langle
t\right\rangle^{-\alpha},\lVert\dot{\sigma}(t)\rVert_{Y}\leq
c^{\prime}(c_{h}+1)\delta\left\langle t\right\rangle^{-\alpha}\right\\}.$
(4.2)

Here $\alpha$ is the exponent in the propagator estimate (C2), $c^{\prime}$ is
the prefactor in (3.19), and $c_{h}$ is from (h2).

The space $\mathcal{A}_{\delta}$ is non-empty, since the path
$\sigma\equiv\sigma_{0},\,w\equiv 0$ lies in it. Notice that this path
corresponds to the trivial static solution to (1.1) given by $f(\sigma_{0})$,
see (2.7). Now we show that $\mathcal{A}_{\delta}$ can be turned into a Banach
space.

######  Lemma 4.1.

For every $\alpha\geq 1$ and $c=c^{\prime}(c_{h}+1)>0$, the function

$\lVert(\sigma,w)\rVert=\sup_{t\geq 0}(\left\langle
t\right\rangle^{\alpha}\lVert
w(t)\rVert_{X^{\prime}}+c^{-1}t^{-1}\int_{0}^{t}\left\langle
t^{\prime}\right\rangle^{\alpha}\lVert\dot{\sigma}(t^{\prime})\rVert_{Y}\,dt^{\prime})$
(4.3)

defines a norm on $\mathcal{X}$. Moreover,
$(\mathcal{A}_{\delta},\lVert\cdot\rVert)$ is complete.

###### Proof.

For $c>0$, (4.3) defines a faithful sublinear form on $\mathcal{X}$. For
$\alpha\geq 1$, this (4.3) is stronger than the uniform norm on $\mathcal{X}$.
Hence $(\mathcal{X},\lVert\cdot\rVert)$ is complete. Direct calculation shows
that $\mathcal{A}_{\delta}\subset\mathcal{X}$ is a closed subset of the unit
ball of size $\delta$ w.r.t. the topology induced by (4.3), and it follows
that $(\mathcal{A}_{\delta},\lVert\cdot\rVert)$ is also complete. ∎

Fix a path $(\sigma^{(0)},w^{(0)})\in\mathcal{X}$. Consider the following
linear evolution equation for a path $\sigma(t)\in\Sigma,\,t\geq 0$:

$\dot{\sigma}=a^{j}(\sigma^{(0)},w^{(0)})\xi_{j}(\sigma^{(0)}),$ (4.4)

where $a^{j}=a^{j}(\sigma^{(0)},w^{(0)})$ is the unique solution to the
equation

$G(a,\sigma^{(0)},w^{(0)})=0,$

c.f. (3.13). This is well defined for sufficiently small $\delta$ by
Proposition 3.3.

Couple (4.4) to the following linear evolution equation for $w=w(t)\in
X^{\prime},\,t\geq 0$:

$\dot{w}=-L(\sigma^{(0)})w-N(\sigma^{(0)},w^{(0)})-a^{j}(\sigma^{(0)},w^{(0)})df(\sigma^{(0)})\xi_{i}(\sigma^{(0)}).$
(4.5)

Since, by assumption, $L(\sigma^{(0)})$ is self-adjoint, (4.5) is globally
well-posed for $X^{\prime}$-small initial data by standard semiflow theory.

We associate the following initial configurations to the system (4.4)-(4.5):

$\displaystyle\sigma(0)=$ $\displaystyle\sigma_{0}\in\Sigma_{0},$ (4.6)
$\displaystyle w(0)=$
$\displaystyle\eta+\beta^{j_{1}}\phi_{j_{1}}(\sigma_{0})+\ldots+\beta^{j_{N}}\phi_{j_{N}}(\sigma_{0}),$
(4.7)

where $\eta\in\mathcal{S}(\sigma_{0})$, $1\leq j_{p}\leq m_{p}$ with $m_{p}$
denoting the multiplicity of the eigenvalue $E_{p}<0$ from (C2), various
$\phi$’s are the unstable modes from (2.16), and various $\beta$’s are some
real numbers.

Fix $\sigma_{0}\in\Sigma_{0}$. For every $\eta\in\mathcal{S}(\sigma_{0})$, and
$\beta^{j_{p}}=\beta^{j_{p}}(\eta)$ to be determined in Section 5 (see
(5.27)), define a key solution map

$\begin{array}[]{ccrcl}{\Psi(\eta)}&\colon&{\mathcal{A}_{\delta}\subset\mathcal{X}}&\longrightarrow&{\mathcal{X}}\\\
\mbox{}&\mbox{}&{(\sigma^{(0)},w^{(0)})}&\longmapsto&{\text{ the unique
solution to }\eqref{4.2}-\eqref{4.5}.}\end{array}$ (4.8)

This $\Psi$ is well-defined by the preceding discussion regarding the global
well-posedness of (4.4)-(4.5).

######  Proposition 4.2.

Fix $\sigma_{0}\in\Sigma_{0}$. Suppose
$\Psi(\eta),\,\eta\in\mathcal{S}(\eta_{0})$ has a fixed point in
$\mathcal{A}_{\delta}$, denoted by $(\sigma,w)$. Then $u=f(\sigma)+w$ is a
global dissipating solution to (1.1), in the sense that $u(t)$ converges to
the static solution $f(\sigma_{0})$ as $t\to\infty$.

###### Proof.

If the expansion (3.8) holds, then $u=f(\sigma)+w$ solves (1.1) if and only if
$w$ solves the under-determined equation

$\dot{w}=-E(f(\sigma)+w)-a^{j}df(\sigma)\xi_{j}(\sigma).$ (4.9)

If $w$ comes from a path in $\mathcal{A}_{\delta}$, then $\lVert
w\rVert_{X^{\prime}}\leq\delta\ll 1$ for all time, and therefore (3.8) is
valid. If, moreover, $w$ comes from a fixed point of $\Psi(\eta)$, then (4.9)
is satisfied, see the definition of (4.8) and (4.5). This proves that
$u=T(\sigma)+w$ is indeed a global solution to (1.1). As for the dissipation,
we use the fact that the definition of $\mathcal{A}_{\delta}$ implies $\lVert
w\rVert_{X^{\prime}}\leq\delta\left\langle t\right\rangle^{-\alpha}\to 0$ as
$t\to 0$. ∎

By Proposition 4.2 above, Items 1-2 in Theorem 2.2 are proved once we
establish the contraction property of $\Psi(\eta)$. This is the goal of the
next section.

##  5 Second order correction

In this section, we define the coefficients $\beta^{j_{p}}(\eta)$ from (4.7).
If we view the removal of zero-unstable modes from
$\eta\in\mathcal{S}(\sigma_{0})$ (see (2.17)) as a first order correction in
the initial perturbation (4.7), then the result in this section amounts to a
second order correction. Whereas the modulation method from Section 3
eliminates the zero modes from the evolution for all subsequent time $t>0$,
the second order correction from this section effectively does the same with
unstable modes (2.16).

###  5.1 Initial choice of initial perturbations

Throughout this subsection, fix a symmetry $\sigma_{0}\in\Sigma_{0}$, together
with a vector $\eta\in\mathcal{S}(\sigma_{0})$. From now on till the end of
this section, dependence of various functions on $\sigma_{0},\eta$ are not
displayed but always understood, and all (explicit or implicit) constants are
independent of $\sigma_{0},\eta$ unless otherwise stated.

######  Theorem 5.1.

Let $\sigma_{0}\in\Sigma_{0},\,0<\delta\ll 1$. Suppose
$\eta\in\mathcal{S}(\sigma_{0})$ with $\lVert\eta\rVert_{X}\leq
c_{0}\delta,\,c_{0}:=(4c_{2})^{-1}$, where $c_{0}>0$ is as in Theorem 2.2,
$c_{2}>0$ as in (C2). Let $(\sigma^{(0)},w^{(0)})\in\mathcal{A}_{\delta}$ be a
path satisfying initial condition (4.6)-(4.7).

Then there exist unique constants

$\beta^{j_{p}}(\sigma^{(0)},w^{(0)})\quad(1\leq j_{p}\leq m_{p},0\leq p\leq
N),$ (5.1)

in place of (4.7) s. th. $\Psi(\sigma^{(0)},w^{(0)})\in\mathcal{A}_{\delta}$,
where $\Psi=\Psi(\eta)$ is the map from (4.8)

###### Proof.

1\. Write

$(\sigma^{(1)},w^{(1)}):=\Psi(\sigma^{(0)},w^{(0)}).$

By Corollary 3.2, together with (4.4), we have
$\lVert\partial_{t}\sigma^{(1)}(t)\rVert_{Y}=o\left(\lVert
w^{(0)}(t)\rVert_{X^{\prime}}\right)$. For sufficiently small $\delta$ and
$\lVert w^{(0)}(t)\rVert_{X^{\prime}}\leq\delta\left\langle
t\right\rangle^{-\alpha}$, this gives
$\lVert\sigma^{(1)}(t)\rVert_{Y}\leq\delta\left\langle
t\right\rangle^{-\alpha}$. Hence, it suffices to show that $\lVert
w^{(1)}(t)\rVert_{X^{\prime}}\leq\delta\left\langle t\right\rangle^{-\alpha}$.

2\. Let $P_{j_{p}}(\sigma^{(0)}(t)):X\to X,\,t\geq 0$ be the orthogonal
projection onto the span of unstable mode $\phi_{j_{p}}$. Let
$P_{S}(\sigma^{(0)}(t))=1-\sum_{p}\sum_{j_{p}}P_{j_{p}}(\sigma^{(0)}(t))$ be
the complement projection, c.f. (2.17).

Denote by $w_{\\#}=P_{\\#}w,\\#=j_{p},S$ the projection of $w$ into various
eigenspaces, and define the conjugated linearized operator

$L_{S}(\sigma^{(0)})=P_{S}(\sigma^{(0)})L(\sigma^{(0)})P_{S}(\sigma^{(0)}).$

Since $L(\sigma)$ is self-adjoint, the various eigenvectors are mutually
orthogonal. By this fact, we find that (4.5) is equivalent to the following
system:

$\displaystyle\dot{w}_{S}=$ $\displaystyle-
L_{S}(\sigma^{(0)})w_{S}-P_{S}N(\sigma^{(0)},w^{(0)}),$ (5.2)
$\displaystyle\dot{\beta}^{j_{p}}=$ $\displaystyle-
E_{p}\beta^{j_{p}}-\left\langle
N(\sigma^{(0)},w^{(0)}),\,\phi_{j_{p}}\right\rangle.$ (5.3)

Notice that zero modes are already eliminated by the modulation equation
(4.4), see Theorem 3.1.

We associate to (5.2)-(5.3) system the following initial configurations:

$\displaystyle w_{S}(0)=$ $\displaystyle\eta$ (5.4)
$\displaystyle\beta^{j_{p}}(0)=$
$\displaystyle\beta^{j_{p}}(\sigma^{(0)},w^{(0)}),$ (5.5)

where r.h.s. of (5.5) is to be determined. The system (5.2)-(5.3) is
decoupled, so we solve the equations above separately.

3\. First, for (5.2), we use Duhamel’s principle to rewrite

$w_{S}(t)=e^{-tL_{S}(\sigma^{(0)})}\eta+\int_{0}^{t}e^{-(t-t^{\prime})}P_{S}N(\sigma^{(0)}(t^{\prime}),w^{(0)}(t^{\prime}))\,dt^{\prime}.$
(5.6)

By the stability condition (C2), see (2.2), we have

$\lVert w_{S}(t)\rVert_{X^{\prime}}\leq c_{2}\left(\left\langle
t\right\rangle^{-\alpha}\lVert\eta\rVert_{X}+\int_{0}^{t}\left\langle
t^{\prime}\right\rangle^{-\alpha}\lVert
N(\sigma^{(0)}(t^{\prime}),w^{(0)}(t^{\prime}))\rVert_{X}\right)\,dt^{\prime}.$
(5.7)

By the assumption on the nonlinearity, and the choice of $w^{(0)}$ (see the
definition of $\mathcal{A}_{\delta}$ in(4.2)), for every fixed $t\geq 0$ there
holds

$\lVert N(\sigma^{(0)}(t),w^{(0)}(t))\rVert_{X}=o(\lVert
w^{(0)}(t)\rVert_{X^{\prime}})=o(\delta\left\langle
t\right\rangle^{-\alpha})\quad(\delta\to 0).$ (5.8)

Since $\alpha\geq 1$, (5.8) implies that for fixed $t$,

$\int_{0}^{t}\left\langle t^{\prime}\right\rangle^{-\alpha}\lVert
N(\sigma^{(0)}(t^{\prime}),w^{(0)}(t^{\prime}))\rVert_{X}\,dt^{\prime}=o(\left\langle
t\right\rangle^{-2\alpha+1})=o(\left\langle t\right\rangle^{-\alpha}).$ (5.9)

Plugging this back to (5.7), and using the assumption
$\lVert\eta\rVert_{X}\leq(4c_{2})^{-1}\delta$, we find that for every
$0<\delta\ll 1$ there holds

$\lVert w_{S}(t)\rVert_{X^{\prime}}\leq\frac{1}{2}\delta\left\langle
t\right\rangle^{-\alpha}.$ (5.10)

4\. Next, consider the equation (5.3). The variation of parameter formula
gives solution

$\beta^{j_{p}}(t)=e^{-\int_{0}^{t}E_{p}}\left(\beta(0)-\int_{0}^{t}\left\langle
N(\sigma^{(0)}(t^{\prime}),w^{(0)}(t^{\prime})),\,\phi_{j_{p}}(t^{\prime})\right\rangle
e^{\int_{0}^{t^{\prime}}E_{p}}\,dt^{\prime}\right),$ (5.11)

where $E_{p}=E_{p}(t),\,0\leq p\leq N$ denotes the $p$-th negative eigenvalue
of $-L(\sigma^{(0)}(t))$ with eigenvectors $\phi_{j_{p}}(t),1\leq j_{p}\leq
m_{p}$.

Now, for the initial condition (5.5), we define

$\beta^{j_{p}}(\sigma^{(0)},w^{(0)}):=\int_{0}^{\infty}\left\langle
N(\sigma^{(0)}(t^{\prime}),w^{(0)}(t^{\prime})),\,\phi_{j_{p}}(t^{\prime})\right\rangle
e^{\int_{0}^{t^{\prime}}E_{p}}\,dt^{\prime}.$ (5.12)

Since $E_{p}(t)<0$ for all $t$, the integral on the r.h.s. converges by the
remainder estimate (5.8) for every $\alpha\geq 1$. For later reference, here
we record that (5.12) also depends on $\sigma_{0},\,\eta$ through the initial
condition on $(\sigma^{(0)},w^{(0)})$, see (4.6)-(4.7).

We claim now that solution (5.11) satisfies

$\beta^{j_{p}}(t)=o(\delta\left\langle
t\right\rangle^{-\alpha})\quad(\delta\to 0)$ (5.13)

for every fixed $t\geq 0$, if and only if the initial condition is given by
(5.12).

Suppose $\beta^{j_{p}}(t)\to 0$. Multiplying both sides of (5.11) by
$e^{\int_{0}^{t}E}$, and then taking $t\to\infty$, we find that
$\beta^{j_{p}}(t)e^{\int_{0}^{t}E_{p}}\to 0$ as $E_{p}(t)<0$, and therefore
$\beta^{j_{p}}(0)$ is given by (5.12).

Conversely, if (5.12) holds, then the formula (5.11) simplifies to

$\beta^{j_{p}}(t)=\int_{t}^{\infty}\left\langle
N(\sigma^{(0)}(t^{\prime}),w^{(0)}(t^{\prime})),\,\phi_{j_{p}}(t^{\prime})\right\rangle
e^{\int_{t}^{t^{\prime}}E_{p}}\,dt^{\prime}.$ (5.14)

Let

$E_{p}:=\sup_{t\geq 0}E_{p}(t).$

Then $E_{p}<0$ by (C2). Hence, by (5.8), we conclude from formula (5.14) that
for each fixed $t\geq 0$,

$\left\lvert\beta^{j_{p}}(t)\right\rvert=o(\delta)\int_{t}^{\infty}{t^{\prime}}^{-\alpha}e^{Et^{\prime}}\,dt^{\prime}\quad(\delta\to
0).$ (5.15)

Notice that in the integrand we have $t$ rather than $\left\langle
t\right\rangle$, due to the smoothing property of the exponential at $t=0$.

For fixed $\alpha\geq 1$ and all $t\gg 0$, r.h.s. above is bounded by

$k(t):=\int_{t}^{\infty}{t^{\prime}}^{-\alpha}e^{Et}.$

At $t=0$, this reduces to the well known exponential integral, see
[MR0167642]*Chapt. 5. In particular, for each fixed $\delta\ll 1$ we have

$\left\lvert\beta^{j_{p}}(t)\right\rvert\leq
k(t)=o(t^{-\alpha})\quad(t\to\infty).$ (5.16)

This, together with (5.15), implies (5.13).

5\. By (5.10), (5.15)-(5.16), we conclude that for every fixed $0<\delta\ll 1$
and all $t\geq 0$,

$\lVert w^{(1)}(t)\rVert_{X^{\prime}}\leq\lVert
w_{S}(t)\rVert_{X^{\prime}}+\sum_{p}\sum_{j_{p}}\beta^{j_{p}(t)}\leq\delta\left\langle
t\right\rangle^{-\alpha}.$ (5.17)

This, together with Step 1, proves the theorem. ∎

For simplicity, below we write elements in $\mathcal{X}$ as

$U=(w,\sigma),\quad U^{(n)}=(w^{(n)},\sigma^{(n)})\quad(n=0,1,\ldots).$

######  Proposition 5.2.

Fix $1\leq j_{p}\leq m_{p},\,0\leq p\leq N$. Let
$\beta\equiv\beta^{j_{p}}:\mathcal{X}\to\mathbb{R}$ be the functional defined
in (5.11). Then for every $U^{(0)},U^{(1)}\in\mathcal{A}_{\delta}$, there hold

$\displaystyle\beta(U^{(0)})=$ $\displaystyle o(\lVert U^{(0)}\rVert),$ (5.18)
$\displaystyle\left\lvert\beta(U^{(0)})-\beta(U^{(1)})\right\rvert=$
$\displaystyle O(\delta\lVert U^{(0)}-U^{(0)}\rVert).$ (5.19)

###### Proof.

1\. Since the index $j_{p}$ is fixed, in this proof the dependence on $j_{p}$
is not displayed. Various implicit constants below do not depend on $j_{p}$.

2\. By (5.12), together with the spectral condition (C2), we have

$\beta(U^{(0)})=\int_{0}^{\infty}e^{E^{(0)}t^{\prime}}o(\sup_{0\leq
t^{\prime\prime}\leq t^{\prime}}\lVert
w^{(0)}(t^{\prime\prime})\rVert_{X^{\prime}})\,dt^{\prime},$ (5.20)

where $E^{(0)}:=\sup_{t\geq 0}E^{(0)}_{p}$ is strictly negative. The integrand
in (5.20) can be bounded as

$e^{E^{(0)}t^{\prime}}o(\sup_{0\leq t^{\prime\prime}\leq t^{\prime}}\lVert
w^{(0)}(t^{\prime\prime})\rVert_{X^{\prime}})=\lVert U^{(0)}\rVert
o(\left\langle t^{\prime}\right\rangle^{-\alpha}e^{Et^{\prime}}).$

The integral of the r.h.s. over $t\geq 0$, which is convergent since $E<0$ and
$\alpha\geq 1$, is of the order $o(\lVert U^{(0)}\rVert)$. This implies
(5.18).

3\. For $n=0,1$, define the linear functional

$\begin{array}[]{ccrcl}{l^{(n)}(t)}&\colon&{X}&\longrightarrow&{\mathbb{R}}\\\
\mbox{}&\mbox{}&{u}&\longmapsto&{\left\langle
u,\,\phi^{j_{p}}(\sigma^{(n)}(t))\right\rangle}\end{array}.$ (5.21)

Then we have by (5.12) that

$\displaystyle\left\lvert\beta(U^{(0)})-\beta(U^{(1)})\right\rvert\lesssim$
$\displaystyle\int_{0}^{\infty}\left\lvert(l^{(1)}(t^{\prime})-l^{(0)}(t^{\prime}))N(U^{(0)}(t^{\prime}))\right\rvert\,dt^{\prime}$
(5.22) $\displaystyle+\int_{0}^{\infty}\left\lvert
l^{(0)}(t^{\prime})(N(U^{(1)}(t^{\prime}))-N(U^{(0)}(t^{\prime})))\right\rvert.$

The integrand in the first term on the r.h.s. can be bounded as

$\left\lvert(l^{(1)}(t^{\prime})-l^{(0)}(t^{\prime}))N(U^{(0)}(t^{\prime}))\right\rvert=o(\delta\left\langle
t^{\prime}\right\rangle^{-\alpha})\sup_{0\leq t^{\prime\prime}\leq
t^{\prime}}d(\sigma^{(1)}(t^{\prime\prime}),\sigma^{(0)}(t^{\prime\prime}),$
(5.23)

by estimate (5.8). The second term in the r.h.s. of (5.22) is bounded as

$\left\lvert
l^{(0)}(t^{\prime})(N(U^{(1)}(t^{\prime}))-N(U^{(0)}(t^{\prime})))\right\rvert=O(\delta\sup_{0\leq
t^{\prime\prime}\leq t^{\prime}}\left\langle
t^{\prime\prime}\right\rangle^{-\alpha}\lVert
w^{(1)}(t^{\prime\prime})-w^{(0)}(t^{\prime\prime})\rVert),$ (5.24)

where we use estimate (2.19) on the nonlinearity, and the uniform bound

$\sup_{t\geq 0}\lVert l^{(0)}(t)\rVert_{X\to\mathbb{R}}\lesssim 1.$ (5.25)

To see (5.25), we first use the definition of $\mathcal{A}_{\delta}$ and
$\alpha\geq 1$ to get $d(\sigma^{(0)}(t),\sigma^{(0)}(0))\lesssim 1$. Then
(5.25) follows from this, together with condition (C4).

Plugging (5.23)-(5.24) back to (5.22), and using the definition (4.3) as in
Step 1, we conclude (5.19) after integration. ∎

###  5.2 Adaptive choice of initial perturbations

Fix a symmetry $\sigma_{0}\in\Sigma_{0}$ and $\eta\in\mathcal{S}(\sigma_{0})$.
For $n=0$, we solve the Cauchy problem (4.4)-(4.7) with initial condition
(4.7) given by Theorem 5.1. Denote the solution by
$U^{(1)}(\eta)\equiv(\sigma^{(1)}(\eta),w^{(1)}(\eta)):=\Psi(\eta)(\sigma^{(0)},w^{(0)}).$
By Theorem 5.1, we have $U^{(1)}(\eta)\in\mathcal{A}_{\delta}$. Hence, we can
repeat this process for $n=1,2,\ldots$ and a sequence
$U^{(n)}(\eta)\in\mathcal{A}_{\delta}$.

######  Lemma 5.3.

If $\Psi(\eta)$ is a contraction (see (4.8)), then the following limits exist:

$\displaystyle U_{\infty}(\eta):=$
$\displaystyle\lim_{n\to\infty}U^{(n)}(\eta)\in\mathcal{A}_{\delta},$ (5.26)
$\displaystyle\beta^{j_{p}}(\eta):=$
$\displaystyle\lim_{n\to\infty}\beta^{j_{p}}(\eta,U^{(n)})\quad(1\leq
j_{p}\leq m_{p},0\leq p\leq N).$ (5.27)

The path $U_{\infty}$ is the unique fixed point of $\Psi(\eta)$ in
$\mathcal{A}_{\delta}$.

######  Remark 11.

Notice that (5.27) depend on $\eta$ only, in contrast to the initial choice
(5.12).

###### Proof.

If the limit (5.26) exists, then it is the unique fixed point of
$\Psi(\eta)\in\mathcal{A}_{\delta}$ by construction. If $\Psi(\eta)$ is a
contraction, then the sequence $U^{(n)}(\eta)$ is Cauchy in
$\mathcal{A}_{\delta}$ by Theorem 5.1. By Lemma 4.1,
$(\mathcal{A}_{\delta},\lVert\cdot\rVert)$ forms a Banach space. By the
Lipschitz estimate (5.19), for each $j_{p}$ and $\beta^{j_{p}}(\eta,U)$ as in
Proposition 5.2, the number sequence
$\beta^{j_{p}}(\eta,U^{(0)}),\beta^{j_{p}}(\eta,U^{(1)}),\ldots$ is also
Cauchy, and therefore converges in $\mathbb{R}$. ∎

Fix $\sigma_{0}\in\Sigma_{0}$ and a number $0<\delta\ll 1$. Define the key map

$\begin{array}[]{ccrcl}{\Phi}&\colon&{\mathcal{S}(\sigma_{0})\cap
B_{\delta}}&\longrightarrow&{X^{\prime}}\\\
\mbox{}&\mbox{}&{\eta}&\longmapsto&{\beta^{j_{1}}(\eta)\phi_{j_{1}}(\sigma_{0})+\ldots+\beta^{j_{N}}(\eta)\phi_{j_{N}}(\sigma_{0})}\end{array}.$
(5.28)

This map $\Phi$ depends on the fixed symmetry frame $\sigma_{0}$, as well as
the number $\delta$, which goes into the definition of $\Psi$ through the
space $\mathcal{A}_{\delta}$. To be consistent in notation, we do not display
the dependence on the fixed small number $\delta$.

If $\eta=0$, then $\Phi(\eta)=0$, since in this case we have the trivial
solution $u\equiv f(\sigma_{0})$ is a global solution to (1.1) satisfying
(4.6)-(4.7). This, together with Definition 2, implies
$\mathcal{M}\supset\mathcal{M}_{\delta}=f(\Sigma_{0}).$

Suppose the initial condition (4.7) is given by $\eta+\Phi(\eta)$. Then by
construction, the limit $U_{\infty}(\eta)$ is the unique fixed point of
$\Psi(\eta)$. By (4.2), this implies Items 1-2 in the main result, Theorem
2.2.

It remaining of this subsection, we prove $\Psi(\eta)$ is a contraction. As in
Section 5.1, since $\sigma_{0},\eta$ is fixed, dependence of various functions
on $\sigma_{0},\eta$ are not displayed but always understood, and all
(explicit or implicit) constants are independent of $\sigma_{0},\eta$.

######  Theorem 5.4.

Let $\sigma_{0},\,\eta$ be as in Theorem 5.1. Then $\Psi=\Psi(\eta)$ is a
contraction on $\mathcal{A}_{\delta}$.

###### Proof.

1\. Take two paths $U^{n}\in\mathcal{A}_{\delta},\,n=0,1$. Let
$U^{n+2}:=\Psi(U^{n})$. Then $U^{n+2}\in\mathcal{A}_{\delta}$ by thm5.1. By
the definition of $\Psi$ in (4.8), the difference $U^{3}-U^{2}$ solves the
linear system

$\displaystyle\begin{split}\partial_{t}(w^{3}-w^{2})=&-L(\sigma^{0})(w^{3}-w^{2})+V(\sigma^{0},\sigma^{1})w^{3}\\\
&-df(\sigma^{0})(\dot{\sigma}^{3}-\dot{\sigma}^{2})-(df(\sigma^{1})-df(\sigma^{0}))\dot{\sigma}^{3}\\\
&-N(U^{1})+N(U^{0}),\end{split}$ (5.29) $\displaystyle
V(\sigma^{0},\sigma^{1}):=$ $\displaystyle
L(\sigma^{1})-L(\sigma^{0}):X^{\prime}\to X,$ (5.30)
$\displaystyle(w^{3}-w^{2})|_{t=0}=$
$\displaystyle(\beta^{j_{p}}(U^{1})-\beta^{j_{p}}(U^{0}))\phi_{j_{p}}(\sigma_{0}).$
(5.31)

2\. We study (5.29)-(5.31) as in the proof of Theorem 5.1. Let

$\displaystyle P_{S}(t):=\text{stable projection onto
}\mathcal{S}(\sigma^{0}(t))\subset X,$ (5.32) $\displaystyle
L_{S}=L_{S}(t):=P_{S}(t)L(\sigma^{0}(t))P_{S}(t),$ (5.33) $\displaystyle
l_{j_{p}}(t):u\mapsto\left\langle
u,\,\phi^{j_{p}}(\sigma^{0}(t))\right\rangle\text{as in \eqref{5.15}}.$ (5.34)

Projecting (5.29)-(5.31) onto the eigenspaces of $L(\sigma^{0})$, we find that
this system is equivalent to

$\displaystyle\begin{split}\partial_{t}P_{S}(w^{3}-w^{2})=&-L_{S}P_{S}(w^{3}-w^{2})\\\
&+P_{S}(V(\sigma^{0},\sigma^{1})w^{3}-(df(\sigma^{1})-df(\sigma^{0}))\dot{\sigma}^{3}-N(U^{1})+N(U^{0})),\end{split}$
(5.35) $\displaystyle\dot{\gamma}^{j_{p}}=$ $\displaystyle-
E_{p}\gamma^{j_{p}}-f^{j_{p}}(U^{0},U^{1},U^{3}),$ (5.36) $\displaystyle
f^{j_{p}}(U^{0},U^{1},U^{3}):=$ $\displaystyle
l^{j_{p}}(V(\sigma^{0},\sigma^{1})w^{3}-(df(\sigma^{1})-df(\sigma^{0}))\dot{\sigma}^{3}-N(U^{1})+N(U^{0})),$
(5.37) $\displaystyle P_{S}(w^{3}-w^{2})|_{t=0}=$ $\displaystyle 0,$ (5.38)
$\displaystyle\gamma^{j_{p}}(0)=$
$\displaystyle\beta^{j_{p}}(U^{1})-\beta^{j_{p}}(U^{0}).$ (5.39)

Here $\gamma^{j_{p}}:=l^{j_{p}}(w^{3}-w^{2})$. To get (5.35)-(5.39), we use
the orthogonality among various eigenvectors of the self-adjoint operator
$L(\sigma^{0})$.

3\. First, consider evolution of the stable mode, (5.35). By Duhamel’s
principle and initial condition (5.39), we have

$\begin{split}&P_{S}(w^{3}-w^{2})\\\
=&\int_{0}^{t}e^{-(t-t^{\prime})L_{S}(t^{\prime})}\underbrace{P_{S}(V(\sigma^{0},\sigma^{1})w^{3}-(df(\sigma^{1})-df(\sigma^{0}))\dot{\sigma}^{3}-N(U^{1})+N(U^{0}))}_{\text{all
depend on }t^{\prime}}\,dt^{\prime}.\end{split}$ (5.40)

For any element $w\in X^{\prime}$ with $\lVert w\rVert_{X^{\prime}}\ll 1$, we
write

$V(\sigma^{0},\sigma^{1})w=(E^{\prime}(f(\sigma^{1})+w)-E^{\prime}(f(\sigma^{0})+w))-(N(\sigma^{1},w)-N(\sigma^{0},w)).$
(5.41)

Since $E$ is $C^{2}$ on $U$, $E^{\prime}$ is $C^{1}$ from $X^{\prime}\to X$.
Hence, the first term on the r.h.s. can be bounded as

$\lVert E^{\prime}(f(\sigma^{1})+w)-E^{\prime}(f(\sigma^{0})+w)\rVert_{X}\leq
c_{1}\lVert f(\sigma^{1})-f(\sigma^{1})\rVert_{X^{\prime}}\leq
c_{1}d_{f}d(\sigma^{0},\sigma^{1}).$

For the first inequality, we use the uniform bound assumption (C1) and the
mean value theorem for Fréchet differentiable maps. For the second inequality,
we use the assumption (f2) on the parametrization $f:\Sigma\to X^{\prime}$.
Similarly, by assumption (2.19), the second term in the r.h.s. of (5.41) can
be bounded by $d(\sigma^{0},\sigma^{1})$ as well. Hence, since
$U^{3}\in\mathcal{A}_{\delta}$, we find

$\lVert V(\sigma^{0}(t),\sigma^{1}(t))w^{3}(t)\rVert_{X}\leq
c_{1}c_{f}\delta\left\langle
t\right\rangle^{-\alpha}d(\sigma^{0},\sigma^{1}).$ (5.42)

Next, since $f$ is $C^{2}$ on $\Sigma_{0}$, we have $\lVert
df(\sigma^{1})-df(\sigma^{0})\rVert_{Y^{\prime}\to X^{\prime}}\lesssim
d(\sigma^{0},\sigma^{1})$. This, together with the fact that
$U^{3}\in\mathcal{A}_{\delta}$, implies

$\lVert(df(\sigma^{1})-df(\sigma^{0}))\dot{\sigma}^{3}\rVert_{X}\leq
c_{f}\delta\left\langle
t\right\rangle^{-\alpha}d(\sigma^{0}(t),\sigma^{1}(t)).$ (5.43)

Using (5.42)-(5.43) and another application of (5.41), we conclude the
Lipschitz estimate

$\displaystyle\lVert
V(\sigma^{0}(t),\sigma^{1}(t))w^{3}(t)-(df(\sigma^{1}(t))-df(\sigma^{0}(t)))\dot{\sigma}^{3}(t))-N(U^{1}(t))+N(U^{0}(t))\rVert_{X}$
(5.44) $\displaystyle\leq$ $\displaystyle\delta(c_{f}(c_{1}+1)\left\langle
t\right\rangle^{-\alpha}+1)(d(\sigma^{0}(t),\sigma^{1}(t))+\lVert
w^{1}(t)-w^{0}(t)\rVert_{X^{\prime}})$ $\displaystyle\leq$
$\displaystyle\delta\lVert U^{1}-U^{0}\rVert\left\langle
t\right\rangle^{-\alpha}(c_{f}(c_{1}+1)\left\langle
t\right\rangle^{-\alpha}+1)$

where in the last step we use the definition (4.3). Plugging this back to
(5.40), and then using the propagator estimate (C3), we conclude

$\lVert P_{S}(w^{3}-w^{2})\rVert_{X}\leq\delta\lVert U^{1}-U^{2}\rVert
c_{\alpha}(t),$ (5.45)

where, with $c_{2}$ denoting the constant from (C3),

$c_{\alpha}(t):=c_{2}\int_{0}^{t}\left\langle
t\right\rangle^{-2\alpha}(c_{f}(c_{1}+1)\left\langle
t\right\rangle^{-\alpha}+1)\,dt.$

This integral is positive and of the order $O(\left\langle
t\right\rangle^{-\alpha})$ for $\alpha\geq 1$.

4\. Next, fix $j_{p}$ and consider the ODE (5.36). For simplicity of notation,
in this step we drop the dependence on $p$ and $j_{p}$. All implicit constants
are independent of $p,\,j_{p}$.

Since $w^{3}(t),w^{2}(t)$ both tends to $0$ in $X^{\prime}$ by the initial
choice of $\beta$ in Theorem 5.1, the function $\gamma(t)=l(\sigma^{0}(t))\to
0$ as $t\to\infty$, Hence, as in Step 4 of Theorem 5.1, we find that

$\left\lvert\gamma(t)\right\rvert\leq\int_{t}^{\infty}f(U^{0}(t^{\prime}),U^{1}(t^{\prime}),U^{3}(t^{\prime}))e^{Et^{\prime}}\,dt^{\prime},$
(5.46)

where $E:=\sup_{t\geq 0}E(t)<0$ by (C2).

Using estimate (5.44), together with the definition (5.39), we find that

$\left\lvert\gamma(t)\right\rvert\leq\delta\lVert U^{1}-U^{0}\rVert
c_{\alpha}^{\prime}(t),$ (5.47)

where

$c_{\alpha}^{\prime}(t):=\int_{t}^{\infty}\left\langle
t\right\rangle^{-\alpha}(c_{f}(c_{1}+1)\left\langle
t\right\rangle^{-\alpha}+1)e^{Et}\,dt.$

This integral is finite, and $O(\left\langle t\right\rangle^{-\alpha})$ for
$\alpha\geq 1$ and $E<0$.

5\. Combining (5.47) and (5.45) gives the estimate

$\lVert w^{3}-w^{2}\rVert_{X^{\prime}}\leq C_{\alpha}\delta\lVert
U^{1}-U^{0}\rVert\left((\sum_{p=1}^{N}m_{p})+1\right)\left\langle
t\right\rangle^{-\alpha},$ (5.48)

where $C_{\alpha}>0$ depends on the decay property of
$c_{\alpha}(t)+c_{\alpha}^{\prime}(t)$ as $t\to\infty$ (this limit exists and
is finite by the large time asymptotics on
$c_{\alpha}(t),c_{\alpha}^{\prime}(t)$).

6\. Next, consider the equation for $\dot{\sigma}^{3}-\dot{\sigma}^{2}$, which
follow from the definition of $\Psi$ in (4.8) and (4.4),(4.9):

$\displaystyle\partial_{t}(\sigma^{3}-\sigma^{2})=$ $\displaystyle
a^{j}(U^{1})\xi_{j}(\sigma^{1})-a^{j}(U^{0})\xi_{j}(\sigma^{0}),$ (5.49)
$\displaystyle(\sigma^{3}-\sigma^{2})|_{t=0}=$ $\displaystyle 0.$ (5.50)

Write

$a^{j}(U^{1})\xi_{j}(\sigma^{1})-a^{j}(U^{0})\xi_{j}(\sigma^{0})=a^{j}(U^{1})(\xi_{j}(U^{1})-\xi_{j}(U^{0}))+(a^{j}(U^{1})-a^{j}(U^{0}))\xi_{j}(U^{0}).$
(5.51)

By Corollary 3.2, the choice $U^{1}\in\mathcal{A}_{\delta}$, and the
assumption (h2), for fixed $\delta\ll 1$, the first term in the r.h.s. of
(5.51) is bounded as

$\lVert a^{j}(U^{1}(t))(\xi_{j}(U^{1}(t))-\xi_{j}(U^{0}(t)))\rVert_{Y}\leq
c_{h}\delta\left\langle t\right\rangle^{-\alpha}\lVert
U^{1}-U^{0}\rVert\quad(t\to\infty).$

By the Lipschitz estimate (3.19), similarly, the second term in the r.h.s. of
of (5.51) is bounded as

$\lVert(a^{j}(U^{1}(t))-a^{j}(U^{0}(t)))\xi_{j}(U^{0}(t))\rVert_{Y}\leq
c^{\prime}c_{h}\delta\left\langle t\right\rangle^{-\alpha}\lVert
U^{1}-U^{2}\rVert.$

Plugging the two preceding estimates back to (5.49), we find

$\lVert\partial_{t}(\sigma^{3}-\sigma^{2})\rVert_{Y}\leq(c^{\prime}(c_{h}+1))\delta\left\langle
t\right\rangle^{-\alpha}.$ (5.52)

7\. Combining (5.48) and (5.52), we conclude that if we choose

$\delta_{0}:=\frac{1}{2}\min\left(\frac{1}{C_{\alpha}\left((\sum_{p=1}^{N}m_{p})+1\right)},\frac{1}{c^{\prime}(c_{h}+1)}\right)$

then for every $0<\delta\leq\delta_{0}$, there holds

$\lVert U^{3}-U^{2}\rVert\leq\frac{1}{2}\lVert U^{1}-U^{0}\rVert.$

This shows that $\Psi$ is a contraction so long as $\delta\ll 1$. ∎

##  6 Stable manifold

Recall that in Definition 2, we have defined the set

$\mathcal{M}=\mathcal{M}(\sigma_{0},\delta):=\left\\{f(\sigma_{0})+\eta+\Phi(\eta):\eta\in\mathcal{S}(\sigma_{0}),\,\lVert\eta\rVert_{X}^{\prime}<\delta,\,\Phi\text{
as in \eqref{Phi}}\right\\}.$ (6.1)

By Proposition 4.2 and Theorem 5.4, each vector $u_{0}\in\mathcal{M}$
generates a global dissipating solution $u\in\mathcal{X}$ (see (4.1)) to (1.1)
with $u(0)=u_{0}$. Moreover, the evolution of this $u$ is essentially governed
by an equation in the manifold of static solutions $f(\Sigma)$, namely (2.26).

In Sections 4-5, for the most parts we have been considering the property of
the $\Psi(\eta)$ map, given in (4.8), with fixed $\eta$. In order to justify
the manifold structure on $\mathcal{M}$, we need to establish suitable
smallness estimates for the map $\Phi$ from (5.28) in terms of $\eta$. This is
the main goal of this section, corresponding to the claimed inequalities
(2.21)-(2.22) in the main theorem.

######  Theorem 6.1.

For every $\eta_{0},\,\eta_{1}\in\mathcal{S}(\sigma_{0})$ with
$\lVert\eta_{0}\rVert+\lVert\eta_{1}\rVert\leq\delta$, there hold

$\displaystyle\lVert\Phi(\eta_{0})\rVert_{X^{\prime}}=$ $\displaystyle
o(\lVert\eta_{0}\rVert_{X^{\prime}})\quad(\delta\to 0),$ (6.2)
$\displaystyle\lVert\Phi(\eta_{0})-\Phi(\eta_{1})\rVert_{X^{\prime}}=$
$\displaystyle
O(\delta\lVert\eta_{0}-\eta_{1}\rVert_{X^{\prime}})\quad(\delta\to 0),$ (6.3)

c.f. (2.21)-(2.22).

######  Remark 12.

Notice the difference between these and Proposition 5.2. Here $\eta$ is
allowed to vary in a small ball in $\Sigma(\sigma_{0})$.

###### Proof of Theorem 6.1.

1\. We first prove (6.2). Take $\eta_{0}\in\mathcal{S}(\sigma_{0})$ with
$\lVert\eta_{0}\rVert_{X^{\prime}}\leq\delta\ll 1$.Fix some $0<\epsilon\ll 1$
to be determined. As in the beginning of Section 5.2, we iterate
$\Psi=\Psi(\eta_{0})$ for $m$ times on a fixed
$U^{(0)}\in\mathcal{A}_{\delta}$ to get a sequence of paths
$U^{(m)}\in\mathcal{A}_{\delta}$. For each fixed $j_{p}$, we also geta
sequence of numbers
$\beta^{(m)}:=\beta^{j_{p}}(\eta_{0},U^{(m-1)})\to\beta:=\beta^{j_{p}}(\eta_{0})$,
where for the transparency of notation the superindex on $j_{p}$ is dropped.

By Lemma 5.3 and Theorem 5.4, for every $\epsilon>0$ there exists
$m=m(\epsilon)\gg 1$ s.th. s.th.

$\left\lvert\beta^{(m)}-\beta\right\rvert<\epsilon.$ (6.4)

Consider the path $w^{(m+1)}$ with $m=m(\epsilon)\gg 1$. This solves the
Cauchy problem

$\displaystyle\dot{w}^{(m+1)}=$
$\displaystyle-L(\sigma^{(m)})w^{(m+1)}-N(\sigma^{(m)},\xi^{(m)})-a^{j}(U^{(m)})df(\sigma^{(m)})\xi_{j}(\sigma^{(m)}),$
(6.5) $\displaystyle w^{(m+1)}(0)=$
$\displaystyle\eta_{0}+\beta_{j_{1}}^{(m)}\phi_{j_{1}}(\sigma_{0})+\ldots+\beta_{j_{N}}^{(m)}\phi_{j_{N}}(\sigma_{0})$
(6.6)

Using (5.7)-(5.9), together (5.13), we find that for every fixed $t$, there
holds

$\lVert w^{(m+1)}(t)\rVert_{X^{\prime}}\leq
2c_{2}\lVert\eta_{0}\rVert_{X^{\prime}}+o(\delta\left\langle
t\right\rangle^{-\alpha})\quad(\delta\to 0).$ (6.7)

Here $c_{2}$ is the constant in the propagator estimate (C3).

We claim now

$\beta^{(m+1)}=o(\lVert\eta_{0}\rVert_{X^{\prime}})\quad(\delta\to 0).$ (6.8)

Recall that by assumption, $\lVert\eta_{0}\rVert\lesssim\delta$. If
$\lVert\eta_{0}\rVert\sim\delta$, then (6.8) follows from (5.18) and estimate
(6.7). If $\lVert\eta_{0}\rVert\sim\delta_{0}\ll 1$, then we can repeat the
argument in Section 5 with $\delta_{0}$ in place of $\delta$ everywhere, to
obtain (6.10) with $\delta_{0}$ in place of $\delta$. This proves the claim.

Estimate (6.8), together with (6.4), gives

$\beta\leq\epsilon+3c_{2}\lVert\eta_{0}\rVert_{X^{\prime}}+\epsilon.$

Since $\epsilon$ is arbitrary, this implies (6.2).

2\. Next, we prove (6.3). Recall the map
$\Psi(\eta),\,\eta\in\mathcal{S}(\sigma_{0})$ is defined in (4.8). For every
fixed $U\in\mathcal{A}_{\delta}$, we claim the following Lipschitz estimates
where the implicit constants are independent of $U$:

$\displaystyle\lVert\Phi(\eta_{0})-\Phi(\eta_{1})\rVert_{X^{\prime}}$
$\displaystyle\lesssim\delta\lim_{m\to\infty}\lVert\Psi^{(m)}(U,\eta_{0})-\Psi^{(m)}(U,\eta_{1})\rVert$
(6.9) $\displaystyle\lVert\Psi(U,\eta_{0})-\Psi(U,\eta_{1})\rVert$
$\displaystyle\lesssim\lVert\eta_{0}-\eta_{1}\rVert_{X^{\prime}}.$ (6.10)

Here the notation $\Psi^{(m)}$ means iterating the map $m$ times.Notice that
the limit in (6.9) exists, since by Theorem 5.4, the sequence
$\Psi^{(m)}(U,\eta_{n}),\,n=0,1$ converges as $M\to\infty$ to the fixed point
of $\Psi(\eta_{n})$ in the space $\mathcal{A}_{\delta}$.

3\. We first prove (6.9), assuming (6.10) holds. We do this by adapting the
construction from Step 1 for $n=0,1$.

As in Step 1, for $n=0,1$ and every fixed $j_{p}$, put

$\displaystyle U^{(0)}\in\mathcal{A}_{\delta},\quad
U^{(m)}_{n}=\Psi(\eta_{n})(U^{(m-1)}),\quad m=1,2,\ldots,$ (6.11)
$\displaystyle\beta^{(m)}_{n}:=\beta^{j_{p}}(\eta_{n},U^{(m-1)}),\quad\beta_{n}:=\lim_{m\to\infty}\beta^{(m)}_{n}.$
(6.12)

The function $\beta^{j_{p}}$ in (6.12) is defined in (5.12).

For every $0<\epsilon\ll 1$, we choose some $m=m(\epsilon)\gg 1$ s.th.

$\left\lvert\beta_{i}(\eta_{1})-\beta_{i}(\eta_{0})\right\rvert\leq\epsilon+\int_{0}^{\infty}\left\lvert
f_{1}(U^{(M)}(t)-f_{0}(t)\right\rvert,$ (6.13)

where, for $n=0,1,$, $m=1,2,\ldots$, and $t\geq 0$, we put

$\begin{array}[]{ccrcl}{f_{n}^{(m)}}&\colon&{\mathbb{R}_{\geq
0}}&\longrightarrow&{\mathbb{R}}\\\
\mbox{}&\mbox{}&{t}&\longmapsto&{\left\langle
N(U_{n}^{(m)}(t)),\,\phi^{j_{p}}(\sigma^{(m)}(t))\right\rangle}\end{array}.$
(6.14)

Here $f_{n}^{(m)}$ depends on $\eta_{n}$ through definition of $U^{(m)}_{n}$,
see(6.11).

The claim now is that there exists $C>0$ independent of $m,t$ s.th.

$\left\lvert f_{1}^{(m)}(t)-f_{0}^{(m)}(t)\right\rvert\leq C\delta\left\langle
t\right\rangle^{-\alpha}\lVert U^{(M)}_{n}-U^{(M)}_{n}\rVert.$ (6.15)

If (6.15) holds, then plugging it into (6.13) and taking $\epsilon\to
0,\,m(\epsilon)\to\infty$ gives (6.9).

In view of the definition of $f_{n}^{(m)}$ from (6.14) and the Lipschitz
estimate (2.19), to get (6.15), it suffices to show that the paths
$U^{(m)}_{0},\,U^{(m)}_{1}$ remains uniformly close in $\mathcal{A}_{\delta}$
for all large $m$. The latter follows from the definition (6.11), and the
uniform bound (6.10), provided $\eta_{0}$ and $\eta_{1}$ are close (which is
indeed the case, since $\lVert\eta_{0}\rVert+\lVert\eta_{1}\rVert\ll\delta$).
This proves (6.9).

4\. It remains to prove (6.10). Fix a path
$U=(\sigma,w)\in\mathcal{A}_{\delta}$, and let

$(\sigma^{n},w^{n}):=\Psi(U,\eta_{n}),\quad n=0,1.$

By the definition of $\Psi$ from (4.8), we find that the difference
$w^{1}-w^{0}$ satisfies the Cauchy problem

$\displaystyle\partial_{t}(w^{1}-w^{0})-L(a)(w^{1}-w^{0})$ $\displaystyle=0,$
(6.16)
$\displaystyle\begin{split}(w^{1}-w^{0})|_{\tau=0}&=\eta_{1}-\eta_{0}\\\
&+\sum_{p}(\beta^{j_{p}}(\eta_{1},U)\phi_{j_{p}}(\sigma)-\beta^{j_{p}}(\eta_{0},U)\phi_{0_{p}}(\sigma)\end{split}$
(6.17) $\displaystyle\partial_{t}(\sigma^{1}-\sigma^{0})$ $\displaystyle=0,$
(6.18) $\displaystyle(\sigma^{1}-\sigma^{0})|_{\tau=0}$ $\displaystyle=0.$
(6.19)

(6.18)-(6.19) implies $\sigma^{1}(t)-\sigma^{0}(t)\equiv 0$. Hence it suffices
to study (6.16)-(6.17) only. As in the proof of Theorem 5.1. Indeed, if we
decompose $w^{1}(t)-w^{0}(t)$ in terms of the eigenfunctions of
$L(\sigma(t))$, then the projection of this difference along the stable modes
of $L(\sigma(t))$ satisfies an decay estimate of the form (5.7), and the
projections along the unstable modes satisfy an remainder estimate of the form
(5.13). The latter is due to the choice of initial condition (6.17), as we
have shown in the proof of Theorem 5.4. We conclude estimate

$\lVert w^{1}(t)-w^{0}(t)\rVert_{X^{\prime}}\lesssim\left\langle
t\right\rangle^{-\alpha}\lVert\eta_{1}-\eta_{0}\rVert_{X^{\prime}}.$

This proves (6.10). ∎

## Acknowledgment

The Author is supported by Danish National Research Foundation grant CPH-
GEOTOP-DNRF151 and The Niels Bohr Grant from the Royal Danish Academy of
Sciences and Letters. The Author thanks IM Sigal for support and hospitality
in the completion of this work during a visit to the University of Toronto.

## Declarations

  * •

Conflict of interest: The Author has no conflicts of interest to declare that
are relevant to the content of this article.

  * •

Data availability: Data sharing is not applicable to this article as no
datasets were generated or analysed during the current study.

##  Appendix A Basic variational calculus

Here we recall some basic elements of variational calculus that have been used
repeatedly. For details, see for instance [MR2431434]*Appendix C,
[MR1336591]*Chapt. 1.

###  A.1 Fréchet Derivative

Let $X,\,Y$ be two Banach spaces. Let $U$ be an open set in $X$. For a map
$g:U\subset X\to Y$ and a vector $u\in U$, the Fréchet derivative $dg(u)$ is a
linear map from $X\to Y$ s.th. $g(u+v)-g(u)-dg(u)v=o(\lVert v\rVert_{X})$ for
every $v\in X$ with $\lVert v\rVert_{X}\ll 1$ . If $dg(u)$ exists at $u$, then
it is unique. If $dg(u)$ exists for every $u\in U$, and the map $u\mapsto
dg(u)$ is continuous from $U$ to the space of linear operators $L(X,Y)$, then
we we say $g$ is $C^{1}$ on $U$. In this case, $dg(u)$ is uniquely given by

$v\mapsto{\frac{dg(u+tv)}{dt}}|_{t=0}\quad(v\in X).$

Iteratively, we can define higher order derivatives this way.

###  A.2 Gradient and Hessian

If $X$ is a Hilbert space over a scalar field $Y$, then by Riesz
representation, we can identify $dg(u)$ as an element in $X$, denoted by
$\Sigma(u)$. The vector $\Sigma(u)$ is called the $X$-gradient of $g$.
Similarly, we denote $\Sigma^{\prime}(u)$ the second-order Fréchet derivative
$d^{2}g(u)$. If $g$ is $C^{2}$, then $\Sigma^{\prime}$ can be identified as a
symmetric linear operator uniquely determined by the relation

$\left\langle\Sigma^{\prime}(u)v,\,w\right\rangle={\tfrac{\partial{{}^{2}}g(u+tv+sw)}{\partial{t}\partial{s}}}|_{s=t=0}\quad(v,w\in
X).$

###  A.3 Remainder and Composition

Let $X$ be a Hilbert space over a scalar field $Y$. Suppose $g$ is $C^{2}$ on
$U\subset X$. Define a scalar function $\phi(t):=g(v+tw)$ for vectors $v,w$
s.th. $v+tw\in U$ for every $0\leq t\leq 1$. Then the elementary Taylor
expansion at $\phi(1)$ gives

$g(v+w)=g(v)+\left\langle\Sigma(v),\,w\right\rangle+\frac{1}{2}\left\langle\Sigma^{\prime}(v)w,\,w\right\rangle+o(\lVert
w\rVert_{X}^{2}).$

Here we have used the definition of $\Sigma$ and $\Sigma^{\prime}$ from the
last subsection.

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth boundary.
Fix $r>d/2,\,f\in C^{r+1}(\mathbb{R}^{n})$. For $u:\Omega\to\mathbb{R}^{n}$,
define a map $g:u\mapsto f\circ u$. Then $g:H^{r}(\Omega)\to H^{r}(\Omega)$ is
$C^{1}$, and the Fréchet derivative is given by $v\mapsto\nabla f\cdot v$.

## References