THE DYNAMICS OF PATTERN SELECTION

FOR THE CAHN-HILLIARD EQUATION

by

Christopher Prince Grant

A dissertation submitted to the faculty ofThe University of Utah

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Department of Mathematics

The University of Utah

August 1991

Copyright c© Christopher Prince Grant 1991

All Rights Reserved

THE UNIVERSITY OF UTAH GRADUATE SCHOOL

SUPERVISORY COMMITTEE APPROVAL

of a dissertation submitted by

Christopher Prince Grant

This dissertation has been read by each member of the following supervisory committeeand by majority vote has been found to be satisfactory.

Chair: Paul Fife

James P. Keener

Nicholas Korevaar

Dragan Milicic

Klaus Schmitt

THE UNIVERSITY OF UTAH GRADUATE SCHOOL

FINAL READING APPROVAL

To the Graduate Council of The University of Utah:

I have read the dissertation of Christopher Prince Grant in itsfinal form and have found that (1) its format, citations, and bibliographicstyle are consistent and acceptable; (2) its illustrative materials includingfigures, tables, and charts are in place; and (3) the final manuscript issatisfactory to the Supervisory Committee and is ready for submission tothe Graduate School.

Date Paul FifeChair, Supervisory Committee

Approved for the Major Department

Klaus SchmittChair/Dean

Approved for the Graduate Council

B. Gale DickDean of The Graduate School

ABSTRACT

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equa-

tion which is one of the leading models for the study of phase separation in isother-

mal, isotropic mixtures. The goal of this dissertation is to provide insight into the

qualitative dynamic properties of solutions of the one-dimensional Cahn-Hilliard

equation. The main focus is on the early stages of evolution of solutions whose initial

data is nearly uniform. Linear and numerical analysis has led to the conjecture

of the existence of a large class of such solutions that evolve relatively quickly to

become nearly periodic with large amplitude and small period. Such solutions would

correspond to the experimentally-observed phenomenon of spinodal decomposition,

the fine-grained decomposition of a molten binary alloy after it has been rapidly

quenched. In this dissertation, I present a rigorous mathematical justification for

the process of spinodal decomposition. I believe that this is the first rigorous

treatment of this phenomenon.

As a complement to these precise results, the last part of this dissertation

deals with certain formal and asymptotic methods for studying aspects of the

Cahn-Hilliard equation which have not yet been settled by rigorous techniques.

In particular, models which approximate the Cahn-Hilliard equation with a finite

system of ordinary differential equations are derived in various contexts, and their

behavior is described.

CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

CHAPTERS

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. DERIVATION OF THE EQUATION . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Generalized Chemical Potential . . . . . . . . . . . . . . . 82.2.2 Gradient Flows . . . . . . . . . . . . . . . . . . . . . . . . 9

3. LINEAR ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 The Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . 173.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 The Bistable Equation . . . . . . . . . . . . . . . . . . . . . . . . 20

4. EVOLUTION IN A HILBERT SPACE . . . . . . . . . . . . . . . . . . . . . 22

4.1 The Abstract Setting . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Growth Estimates Near an Equilibrium . . . . . . . . . . . . . . . 234.3 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Flow near the Pseudo-unstable Manifold . . . . . . . . . . . . . . . 31

4.4.1 Evolution in Cones . . . . . . . . . . . . . . . . . . . . . . 314.4.2 Regions Tangent to the Pseudo-Stable Subspace . . . . . . 364.4.3 Regions with Boundaries Satisfying a Power Law . . . . . . 374.4.4 Measure-theoretic Results . . . . . . . . . . . . . . . . . . . 39

5. SPINODAL DECOMPOSITION . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 The Equation in an Abstract Setting . . . . . . . . . . . . . . . . . 495.2 Properties of the ω-limit Points . . . . . . . . . . . . . . . . . . . . 535.3 Orbits Approaching the ω-limit Points . . . . . . . . . . . . . . . . 58

6. THE BIFURCATION DIAGRAM . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1 Self-similarity and Energy . . . . . . . . . . . . . . . . . . . . . . . 616.2 Finite-dimensional Approximations . . . . . . . . . . . . . . . . . 63

6.2.1 Reduced Equations on a Center Manifold . . . . . . . . . . 646.2.2 Galerkin Approximations . . . . . . . . . . . . . . . . . . . 70

7. OPEN PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

vi

LIST OF FIGURES

2.1 Free energy at high temperature. . . . . . . . . . . . . . . . . . . . . . 6

2.2 Free energy at low temperature. . . . . . . . . . . . . . . . . . . . . . . 6

3.1 Dispersion relation for the Cahn-Hilliard equation. . . . . . . . . . . . 16

3.2 Phase portrait for (3.4). . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Scaled phase portrait for (3.4). . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Typical evolution sequence for the Cahn-Hilliard equation. . . . . . . . 19

3.5 Typical evolution sequence for the bistable equation. . . . . . . . . . . 21

6.1 Center manifold bifurcation diagrams. . . . . . . . . . . . . . . . . . . 69

6.2 Two-variable Galerkin bifurcation diagrams. . . . . . . . . . . . . . . 72

6.3 Two-variable Galerkin schematic diagrams; s = stable; us = unstable. 73

ACKNOWLEDGMENTS

I wish to thank Paul Fife for his patient support of my work and his many helpful

suggestions. Others whose advice contributed significantly to this dissertation

include Peter Bates and Hans Othmer. I would also like to express my gratitude

to my family for their encouragement of me in my work. Financial support during

the period of research for this dissertation was provided by a National Science

Foundation Graduate Research Fellowship and by a University of Utah Graduate

Research Fellowship.

CHAPTER 1

INTRODUCTION

When a molten binary alloy with a given concentration of components is rapidly

quenched to a lower temperature, the sample may become inhomogeneous very

quickly, decomposing into a fine-grained mixture of particles in different phases.

This phenomenon is known as spinodal decomposition. Subsequently, there is

usually a coarsening process in which the particles of the fine-grained mixture slowly

grow in size. One of the leading models devised for the study of this phenomenon

is the Cahn-Hilliard equation

∂u

∂t= −∆

(ε2∆u−W ′(u)

)x ∈ Ω (1.1)

∂u

∂ν=∂∆u

∂ν= 0 x ∈ ∂Ω,

where Ω is a bounded open subset of Rn with sufficiently smooth boundary, ν

is the unit outward normal, and W is a C5 function qualitatively similar to the

function u 7→ (u2 − 1)2. To be more precise, it is usually assumed that there exist

real numbers α1 < α2 < α3 < α4 < α5 such that W is strictly decreasing on

(−∞, α1] and on [α3, α5], strictly increasing on [α1, α3] and on [α5,∞), concave up

on (−∞, α2) ∪ (α4,∞), and concave down on (α2, α4). The variable u represents

the concentration of one of the two components of the alloy, so∫Ω udx represents

the total mass of that component. Note that the boundary conditions imply that

mass is conserved, since

d

dt

∫Ωudx =

∫Ω

∂u

∂tdx =

∫Ω

[−∆(ε2∆u−W ′(u))]dx

2

= −∫∂Ω

[∂

∂ν(ε2∆u−W ′(u))

]ds = 0.

In Chapter 2, a reasonable energy functional is defined and justified, and then the

Cahn-Hilliard equation is derived in two different ways from this functional. First, it

is derived in the context of materials science using a generalization of the concept of

chemical potential. Second, it is derived in a more precise mathematical sense as a

generalization of the concept of a gradient flow for an ordinary differential equation.

Chapter 3 presents a linear analysis of the Cahn-Hilliard equation and shows how

the results of this analysis correspond to spinodal decomposition. Comparisons

with a very simple linear ordinary differential equation and with a closely related

partial differential equation are made.

Chapters 4 and 5 contain the main results of this dissertation. These two

chapters demonstrate rigorously that typical solutions of the one-dimensional Cahn-

Hilliard equation that have initial data that are close to a constant M contained

in the interval (α2, α4) where W is concave down exhibit spinodal decomposition.

This result is obtained by considering the Cahn-Hilliard equation as an ordinary

differential equation in an appropriate Hilbert space. Chapter 4 contains results

valid in such an abstract setting based on the theory of analytic semigroups.

This theory yields estimates establishing the existence of an invariant manifold

which governs the evolution of nearby solutions. The sense in which this manifold

dominates other solutions is made precise using the terminology of measure theory.

Although these abstract results are significant in and of themselves, they prove to

be even more powerful when the special structure of the Cahn-Hilliard equation is

taken into account. Results that make use of this special structure are contained

in Chapter 5.

In Chapter 6, the bifurcation diagram for the one-dimensional Cahn-Hilliard

equation is considered. Although the exact form of this diagram is still unknown,

certain important properties can be established. In this chapter finite-dimensional

3

approximations to the Cahn-Hilliard equation are derived via Galerkin approxima-

tion and reduction to a center-unstable manifold. The bifurcation diagrams of these

reduced systems are analyzed in light of the known properties of the bifurcation

diagram of the full partial differential equation.

Finally, Chapter 7 discusses questions regarding the extent to which the results

of the preceding chapters can be generalized. For example, can the justification of

spinodal decomposition be extended to the more natural two-dimensional or three-

dimensional setting? Also, can anything be said about the evolution of solutions in

certain nongeneric situations?

In order to fully appreciate the results contained in this dissertation, it is im-

portant to understand something of the history of the Cahn-Hilliard equation and

the work that has been done on it thus far. The Cahn-Hilliard equation is the

dynamical equation corresponding to a particular free energy functional; details on

this functional and the ways in which the Cahn-Hilliard equation can be derived

from it can be found in Chapter 2. The functional was apparently first proposed by

van der Waals [36] in 1893; it was over 60 years later that Cahn and Hilliard [8] [5]

derived the related evolution equation (1.1). Existence and uniqueness theorems

for solutions of (1.1) in various function spaces have been proven by, among others,

Elliott and Zheng [13], Nicolaenko and Scheurer [29], Temam [35], and Rankin [32].

Because the Cahn-Hilliard equation represents a dissipative system, each of its

solutions approaches a time-independent solution (or possibly a collection of such

solutions) as t becomes large. Thus, the steady-state problem

0 = −∆(ε2∆u−W ′(u)

)x ∈ Ω (1.2)

∂u

∂ν=∂∆u

∂ν= 0 x ∈ ∂Ω

is important in understanding the large-time behavior of the dynamical equation.

Carr, Gurtin, and Slemrod [9] showed that when Ω is one-dimensional, all locally

4

stable solutions of (1.2) are monotone. Modica [26] proved an analogous result for

multidimensions: A limit of locally stable solutions to (1.2) as ε → 0 has a range

which is confined to the local minimizers of W almost everywhere, and the interface

between these two phases has minimal area. When W is assumed to be a quartic

polynomial and Ω is one-dimensional, Novick-Cohen and Segel [30] have derived a

representation of solutions of (1.2) in terms of Jacobi elliptic functions. Under the

same assumptions, Zheng [37] has calculated the number of the solutions of (1.2)

with a given mass in certain special cases and has claimed that, in general, the

number of such solutions is finite.

Some miscellaneous mathematical results of interest include the following. Pego

[31] performed a formal asymptotic expansion of the Cahn-Hilliard equation on an

infinite domain and showed that typical solutions have fronts that move with a

velocity corresponding to their mean curvature. Eilbeck, Furter, and Grinfeld [11]

used numerical techniques to examine the bifurcation diagram of (1.2) with respect

to ε, the way energy varies along branches of the bifurcation diagram, and the

manner in which the diagram depends on the mean value of the solutions. Bates

and Fife [3] derived a relationship between the spectrum of the linearization of the

right-hand side of (1.1) about a solution of (1.2) and the spectrum of other linearized

partial differential operators. Alikakos, Bates, and Fusco [1] proved the existence of

extremely slowly evolving solutions of the one-dimensional Cahn-Hilliard equation

that connect a steady-state solution with two interfaces to a monotone steady-state

solution.

CHAPTER 2

DERIVATION OF THE EQUATION

The Cahn-Hilliard equation was derived by John W. Cahn and John E. Hilliard

[8] [5] over 30 years ago. Cahn and Hilliard were involved in the field of materials

science, the study of the relationship between the structure and physical properties

of materials such as ceramics, metals, polymers, and composites, and it is in this

field that there has been the most interest and activity involving the Cahn-Hilliard

equation. Nevertheless, the ideas involved in the derivation are sufficiently general

that the equation is applicable to many other areas of science. For example, J.

D. Murray derives it in the context of biological cell formation and aggregation

[28]. The materials science approach to deriving the Cahn-Hilliard equation will be

presented here.

2.1 Statics

Consider a molten binary alloy, i.e., an alloy consisting of two different com-

ponents, and suppose this alloy is confined to a bounded region Ω. Let u be the

concentration of one of the components, and let W (u) be the associated bulk free

energy of the alloy, i.e., the free energy density that the alloy would have if each

component had a uniform spatial distribution. If the temperature of the alloy is

high, then typically W is a convex function of u (see Figure 2.1). If the temperature

6

u

W

Figure 2.1. Free energy at high temperature.

u

W

Figure 2.2. Free energy at low temperature.

of the alloy is lowered, then W may become double-welled. (see Figure 2.2). This

phenomenon is predicted by the equation

∂W

∂T= −S,

where T represents temperature and S represents entropy. This fundamental law

of thermodynamics says that as the temperature of a material is lowered, the bulk

free energy increases, and the increase is most pronounced where the entropy is

greatest. Since entropy can be thought of as a statistical measure of disorder, the

7

entropy should be low for extremely low or extremely high values of u, and the

entropy should be higher for values of u in between. Therefore, the central part

of the graph of W may rise enough faster than the rest of the graph to produce a

local maximum with local minima on either side.

Suppose now that the alloy is rapidly quenched to a temperature at which W

is as in Figure 2.2. The region where W ′′ is negative is called the spinodal region.

It is anticipated that the total free energy of the alloy will consist not only of

bulk free energy but interfacial energy as well. Suppose that the alloy is isotropic,

i.e., that its relevant physical properties are independent of orientation. Then if

this interfacial energy, which can be thought of as surface tension, is a function

of ∇u, then symmetry considerations would imply that its Taylor series expansion

will have only even powers of ∇u, since odd powers would indicate a preferred

direction. Taking the simplest possible nontrivial interfacial term, the total energy

of the alloy is

E(u) =∫

Ω

[W (u(x)) +

ε2

2|∇u(x)|2

]dx. (2.1)

If the interfacial energy is assumed to be small then ε can be taken to be a small

parameter.

2.2 Dynamics

Note that although it is not explicitly expressed in (2.1), u is a function of

time as well. There are two related, but different, paths that lead from the free

energy functional E to the Cahn-Hilliard equation for the time evolution of the

system. The first that will be described uses the concept of the generalized chemical

potential and can be found, for example, in [8] and [28]. The second generalizes

the idea of gradient flows from the theory of ordinary differential equations, and

the development to be presented below is based largely on the work of Fife in [14].

8

2.2.1 Generalized Chemical Potential

Whenever a system has a functional which assigns an energy to each possible

state of the system, it is natural to consider the way the energy changes as the state

changes. If the total energy is equal to the bulk energy W (u) alone, then the rate of

change of the energy W with respect to the state u is the chemical potential W ′(u).

If, however, the total energy E has other terms, then the analogous quantity is the

generalized chemical potential δE/δu, the variational derivative of E with respect to

u. Assuming that u satisfies the natural boundary condition ∂u∂ν

∣∣∣∂Ω

= 0, where ν is

the outward normal on the boundary, the divergence theorem implies that

E(u+ h)− E(u) =∫

Ω

[W (u+ h)−W (u) +

ε2

2(2(∇u · ∇h) + |∇h|2)

]dx

=∫

Ω[W ′(u)h− ε2(∆u)h+ · · ·]dx,

where the omitted terms are higher-order in h. Hence, δE/δu = W ′(u)− ε2∆u.

Now corresponding to this generalized chemical potential, there is a flux

Jdef= −D∇

(δEδu

)= −D∇(W ′(u)− ε2∆u),

where D is some proportionality parameter which will here be assumed to be a

scalar constant. As in the derivation of the classical heat equation, it is reasonable

to take ∂u∂t

proportional to the divergence of the flux (with negative proportionality

constant). In this case this implies that, after rescaling,

∂u

∂t= −∆(ε2∆u−W ′(u)),

which is the Cahn-Hilliard equation. In addition to the Neumann boundary condi-

tion, a no-flux boundary condition

∂

∂ν(ε2∆u−W ′(u))

∣∣∣∣∣∂Ω

= 0

is typically assumed. Notice that the combination of this new boundary condi-

tion with the Neumann boundary condition is equivalent to the pair of boundary

conditions given in (1.1).

9

2.2.2 Gradient Flows

Let V ⊂ Rn be a linear manifold (i.e., a translate of a linear subspace V0),

and let F be a real-valued function whose domain includes V . Then a constrained

gradient flow of F on V is an ordinary differential equation

u′(t) = −K∇V F (u(t)) (2.2)

where K > 0 is a constant and ∇V is the constrained gradient, i.e., the projection

of the regular gradient onto V0. The importance of (2.2) can be seen by noting that

along a solution u(t),

d

dtF (u(t))

∣∣∣∣∣t=t0

= ∇V F (u(t0)) · u′(t0) = −K|∇V F (u(t0))|2 ≤ 0,

so F is nonincreasing along trajectories of its gradient flow. Moreover, (2.2) says

that at any instant, u is moving in the direction of the fastest decrease in F ; thus,

u(t) can be thought of as the position of a particle sliding down the surface (x, F (x))

in V ×R while its kinetic energy is constantly kept negligible by friction or some

other means.

In order to generalize this useful concept to a linear manifold in an infinite-

dimensional Hilbert space H , it is necessary to determine what is meant by∇V F (u).

The obvious choice is to define∇V F (u) to be the projection of the Frechet derivative

onto the closure V0 of V0. This definition is, however, too restrictive, since in

applications F is typically not Frechet differentiable with respect to the inner

product in H , even if F is restricted to V . Instead, for u0 ∈ V define ∇V F (u0) to

be an element of V0 such that for any smooth path γ(t) in V with γ(t0) = u0,

d

dtF (γ(t))

∣∣∣∣∣t=t0

= 〈∇V F (u0), γ′(t0)〉H ,

if such an element exists. In particular, this definition requires that F be Gateaux

differentiable with respect to any direction v ∈ V0 and that the Gateaux derivative

of F with respect to v be given by 〈∇VF (u0), v〉H .

10

Returning to the energy functional E defined in (2.1), the goal is to find a partial

differential equation (with boundary conditions) that is a constrained gradient flow

of E on an appropriate linear manifold V in an appropriate Hilbert space H . In

particular, V should be restricted to functions that satisfy the natural Neumann

boundary conditions, and all the functions on V should have the same mass |Ω|M

so that mass will be conserved. Also, all functions in V should be smooth enough

to be in the domain of E .

Let γ(t) be a path in V satisfying γ(t0) = u0 and γ′(t0) = v0 ∈ V0. Then

d

dtE(γ(t))

∣∣∣∣∣t=t0

=d

dt

∫Ω

[W (γ(t)) +

ε2

2|∇γ(t)|2

]dx

∣∣∣∣∣t=t0

=∫

Ω

[W ′(u0)v0 + ε2(∇u0 · ∇v0)

]dx. (2.3)

Applying the divergence theorem to (2.3) gives

d

dtE(γ(t))

∣∣∣∣∣t=t0

=∫

Ω[W ′(u0)− ε2∆u0]v0dx. (2.4)

Now

∫Ω

[W ′(u0)− ε2∆u0]dx =∫

ΩW ′(u0)dx− ε2

∫∂Ω

∂u0

∂νds =

∫ΩW ′(u0)dx,

so (2.4) can be rewritten as

d

dtE(γ(t))

∣∣∣∣∣t=t0

=∫

Ω

[W ′(u0)− ε2∆u0 −

∫ΩW ′(u0(y))dy

]v0dx.

Thus, if H = L2(Ω), W ′(u0) − ε2∆u0 −∫Ω W

′(u0(y))dy will be ∇V E(u0) since it

has average value 0. This means that in order to get a gradient flow for E in

L2(Ω) that conserves mass, it is necessary to have a nonlocal term in the partial

differential equation satisfied by u. Generally, such nonlocal terms are considered

to be physically unrealistic, so this choice of H is, in some sense, undesirable. It

should also be noted that similar problems arise when H is chosen to be a Sobolev

space Hk(Ω) for any k > 0.

11

Now the Fredholm alternative, in combination with the existence theory for the

Neumann Laplacian, says that for reasonable Ω and for any v ∈ L2(Ω) there is a

unique solution z = N(v) to the problem

∆z = v − 1

|Ω|

∫Ωvdx

∂z

∂ν

∣∣∣∣∣∂Ω

= 0∫Ωzdx = 0. (2.5)

Since the function v0 defined above has mean value 0, if z0 = N(v0) then ∆z0 = v0,

so applying the divergence theorem to (2.3) twice yields

d

dtE(γ(t))

∣∣∣∣∣t=t0

=∫

Ω[W ′(u0)∆z0 + ε2(∇u0 · ∇v0)]dx

=∫

Ω[W ′(u0)∆z0 − ε2(∆u0∆z0)]dx

=∫

Ω∇(ε2∆u0 −W ′(u0)) · ∇z0dx. (2.6)

Examining the last line of (2.6) leads one to consider the space H = H−1(Ω),

the continuous dual space of the Sobolev space H1(Ω). There is a dense subspace

of H−1(Ω) which can be identified with L2(Ω) in such a way that

‖v‖H−1 = sup‖w‖H1=1

〈v, w〉L2

for v in this subspace, and H−1 is the completion of L2 with respect to this norm.

This norm is somewhat difficult to use directly, so consider the following inner

product and its induced norm. Given v1, v2 ∈ L2, let

(v1|v2) = |Ω|v1v2 +∫

Ω∇z1 · ∇z2dx,

where vj = |Ω|−1∫Ω vjdx and zj = N(vj), as defined by (2.5). This is clearly an

inner product on L2. If v ∈ L2, z = N(v), and ‖w‖H1 = 1, then

|〈v, w〉L2 | = |〈∆z + v, w〉L2 | =∣∣∣∣∫

Ωw∆zdx+ v

∫Ωwdx

∣∣∣∣

12

≤∣∣∣∣∫

Ω∇w · ∇zdx

∣∣∣∣+ |v| ∣∣∣∣∫Ω

1 · wdx∣∣∣∣

≤(∫

Ω∇w · ∇wdx

)1/2(∫Ω∇z · ∇zdx

)1/2

+ |v|(∫

Ω1dx

)1/2(∫Ωw2dx

)1/2

≤√

(v|v)− |Ω|v2 +√|Ω|v2

≤ 2√

(v|v). (2.7)

Taking the supremum over all such w ∈ H1 in (2.7) gives

‖v‖H−1 ≤ 2√

(v|v). (2.8)

Estimating in the other direction,

∣∣∣(v|v)− |Ω|v2∣∣∣ =

∣∣∣∣∫Ω∇z · ∇zdx

∣∣∣∣=

∣∣∣∣∫Ωz∆zdx

∣∣∣∣=

∣∣∣∣∫Ωz(v − v)dx

∣∣∣∣=

∣∣∣∣∫Ωzvdx

∣∣∣∣≤ ‖v‖H−1‖z‖H1 . (2.9)

Since∫Ω zdx = 0, one of the Poincare inequalities implies that

‖z‖2H1 =

∫Ωz2dx+

∫Ω∇z · ∇zdx ≤ C2

∫Ω∇z · ∇zdx (2.10)

for some C > 0 depending only on Ω. Substituting (2.10) into (2.9) gives

∣∣∣(v|v)− |Ω|v2∣∣∣ ≤ C‖v‖H−1

(∫Ω∇z · ∇zdx

)1/2

= C‖v‖H−1

∣∣∣(v|v)− |Ω|v2∣∣∣1/2,

so ∣∣∣(v|v)− |Ω|v2∣∣∣1/2 ≤ C‖v‖H−1. (2.11)

Setting w = |Ω|−1/2 in the definition of ‖v‖H−1 gives ‖v‖H−1 ≥ |Ω|1/2v. Using this

and (2.11),

√(v|v) ≤

√C2‖v‖2

H−1 + |Ω|v2

13

≤√C2‖v‖2

H−1 + ‖v‖2H−1

≤ (C + 1)‖v‖H−1 . (2.12)

The estimates (2.8) and(2.12) together imply that (·|·) induces a norm equivalent

to the original H−1 norm on L2 and, therefore, makes H−1 a Hilbert space.

Now suppose that the functions u0 on the manifold V are all required not only

to have the same mass and satisfy ∂u0

∂ν

∣∣∣∂Ω

= 0, but also to be smooth and satisfy

∂∆u0

∂ν

∣∣∣∂Ω

= 0 . Then ∂∂ν

(ε2∆u0 −W ′(u0))∣∣∣∂Ω

= 0, so the divergence theorem implies

that ∆(ε2∆u0 −W ′(u0)) has mass 0. Using this fact and restating (2.6) in terms

of this inner product gives, for some constant C,

d

dtE(γ(t))

∣∣∣∣∣t=t0

=(∆(ε2∆u0 −W ′(u0) + C)|∆z0

)=

(∆(ε2∆u0 −W ′(u0))|v0

)=

(∆(ε2∆u0 −W ′(u0))|γ′(t0)

).

Moreover, ∆(ε2∆u0 − W ′(u0)) lies in V0 since it has mass 0 and the boundary

conditions imposed on V and V0 do not restrict V0; hence,

∇V E(u0) = ∆(ε2∆u0 −W ′(u0)).

A gradient flow of E on V (with respect to (·|·)) is, therefore,

∂u

∂t= −∆(ε2∆u−W ′(u)),

which is, again, the Cahn-Hilliard equation (1.1).

It needs to be mentioned that although this derivation is precise about some

points, it skirts over such things as the loss of smoothness via differentiation,

etc. To deal with such issues rigorously would essentially constitute an existence

proof, and that has not been the goal of this section. The aim here has been to

argue convincingly that the Cahn-Hilliard equation is a natural sort of equation to

14

consider in the reasonably natural space H−1, and that framing the evolution of E

in terms of another function space leads to some restrictions that are undesirable

from a physical standpoint, if mass is to be conserved.

CHAPTER 3

LINEAR ANALYSIS

3.1 The Dispersion Relation

Consider the Cahn-Hilliard equation when Ω = [0, 1]:

ut = −(ε2uxx −W ′(u))xx x ∈ (0, 1) (3.1)

ux = uxxx = 0 x ∈ 0, 1.

If the mass M is in the spinodal region then β2 def= −W ′′(M) > 0. Linearizing (3.1)

about u ≡M gives

ut = −ε2uxxxx − β2uxx x ∈ (0, 1) (3.2)

ux = uxxx = 0 x ∈ 0, 1.

The eigenfunctions of the linear operator on the right-hand side of the first line of

(3.2), subject to the boundary conditions in the second line, are

cosnπx : n = 0, 1, 2, . . ..

Substituting u = an cosnπx into (3.2), where an is a function of t, gives

a′n cosnπx = −ε2(nπ)4an cosnπx + β2(nπ)2an cosnπx,

so

a′n = (nπ)2[β2 − ε2(nπ)2]an.

Hence,

an(t) = an(0) exp(λnt),

16

where

λn = (nπ)2[β2 − ε2(nπ)2]. (3.3)

In Figure 3.1, the dispersion relation, the relationship between the wavelength of

a mode and its corresponding growth rate, imposed by (3.3) is plotted. Although

a continuous curve is shown, because of the boundary conditions only the integer

values of n (represented as dots on the curve) correspond to eigenfunctions. For 0 <

n < β/(επ), λn > 0, so the corresponding Fourier modes grow as time progresses in

the linearized equation (3.2), while the other modes remain steady or shrink. Thus,

for ε < β/π spatially homogeneous equilibria in the spinodal region are linearly

unstable. Elementary calculus implies that the maximum of the continuous curve

in Figure 3.1 occurs when n = β/(επ√

2), so the one or two integer values of n

whose squares are nearest to β2/(2ε2π2) correspond to the modes which grow the

most rapidly under (3.2). From Figure 3.1 it is clear that the largest eigenvalue

of the linearized operator is simple except when λn = λn+1 for some n ∈ N. This

holds when

(nπ)2[β2 − ε2(nπ)2] = ((n+ 1)π)2[β2 − ε2((n + 1)π)2]

⇔ ε2[((n+ 1)π)4 − (nπ)4] = β2[((n + 1)π)2 − (nπ)2]

n

n

λ

Figure 3.1. Dispersion relation for the Cahn-Hilliard equation.

17

⇔ ε2[((n + 1)π)2 + (nπ)2] = β2

⇔ ε =β

π√

2n2 + 2n+ 1.

In particular, there is generically a unique fastest-growing mode. Also, the wave-

length of this mode is O(ε), so if ε is small, as was assumed in Chapter 2, the

wavelength of the fastest-growing mode is small, also.

3.2 Ordinary Differential Equations

To understand the implications of the linearized analysis done in the preceding

section, consider the behavior of the linear system of ordinary differential equations

d

dt

(u1

u2

)=

(λ+ 00 λ−

)(u1

u2

), (3.4)

where λ+ > λ− > 0. The phase portrait of this system is sketched in Figure 3.2.

The fastest-growing solutions of this linear system lie completely on the u1-axis.

Each of the other orbits is tangent to the u2-axis and lies on a curve of the form

u1 = Cuλ+/λ−2 ,

for some constant C. Consider the evolution of the forward semiorbits beginning

in the shaded region. In Figure 3.3, the scale of the phase portrait in Figure 3.2

has been changed so that a larger portion of the plane is visible. This figure makes

it clear that the large-time behavior of solutions starting in the shaded region is

similar to that of the fastest-growing solution. Note also that if a small circular

neighborhood of the origin is chosen, a large fraction of that neighborhood lies in

the shaded region. Furthermore, as the radius of that neighborhood approaches

0, the fraction of the neighborhood that is shaded approaches 1. Thus, in some

sense, most orbits starting near the origin behave like the fastest-growing solution

for large time.

Now if the solutions to the Cahn-Hilliard equation behave in an analogous fashion

then the fastest-growing mode should dominate the behavior of most solutions with

18

2

1

u

u

Figure 3.2. Phase portrait for (3.4).

2

1

u

u

Figure 3.3. Scaled phase portrait for (3.4).

19

initial values that are small perturbations from a constant. Thus, the evolution

of a typical small perturbation from the homogeneous steady-state might be as

depicted in the three cross-sections in Figure 3.4. Somewhat independently of the

initial data, the characteristic periodic structure of the fastest-growing mode would

emerge. Of course, the Cahn-Hilliard equation (1.1) is much more complicated than

(3.4). First, (3.4) is linear while (1.1) is nonlinear. Second, (3.4) is two-dimensional

while (1.1) is infinite-dimensional. Thus, at this point the qualitative description of

solutions to the Cahn-Hilliard equation must be considered to be only a conjecture.

In Chapters 4 and 5, the extent to which this conjectured behavior actually occurs

will be discussed.

3.3 Empirical Results

In Chapter 2, the Cahn-Hilliard equation was derived in the context of phase

transitions in a binary alloy. The conjectures in the present chapter regarding

the behavior of solutions of the Cahn-Hilliard equation have, of course, physical

implications, assuming that the model is a good one. The development of large-

amplitude small-wavelength spatial oscillations from small random perturbations

of a homogeneous solution corresponds to a fine-grained decomposition of the alloy.

u

uu

Figure 3.4. Typical evolution sequence for the Cahn-Hilliard equation.

20

Such spinodal decomposition has been observed many times by materials scientists.

For example, micrographs in [6] and [7] clearly show this fine-grained structure,

with a characteristic length scale, appearing in a variety of materials.

3.4 The Bistable Equation

At this point, it would be instructive to make a brief comparison between the

Cahn-Hilliard equation and the bistable equation:

∂u

∂t= ε2∆u−W ′(u) x ∈ Ω (3.5)

∂u

∂ν= 0 x ∈ ∂Ω,

where W is as in Figure 2.2. Assume, in particular, that W ′ has exactly three zeros

z−, 0, z+ with z− < 0 < z+ and W ′′(z−) > 0, W ′′(0) < 0, and W ′′(z+) > 0. From

(2.4), it is clear that (3.5) will be a gradient flow in L2(Ω) for the total energy E

when the mass conservation constraint is removed.

If ε is set equal to 0, then (3.5) becomes a simple ordinary differential equation.

Note that in this case, 0 is an unstable rest point, and z− and z+ are stable rest

points. Thus, points greater than 0 are attracted to z+, while points less than

0 are attracted to z−. This analysis suggests the following heuristic behavior for

solutions to the full partial differential equation when ε is small. Each point on

the curve of initial data evolves according to the ordinary differential equation just

discussed until steep interfaces develop, at which time the ε2∆u term is no longer

insignificant. Figure 3.5 depicts a typical sequence of cross-sections from such an

evolution process.

For the bistable equation, the only unstable homogeneous steady-state is u ≡ 0.

Unlike the Cahn-Hilliard equation, the bistable equation does not exhibit the

development of fine-grained oscillations from small perturbations from u ≡ 0.

This difference can be explained by the fact that the fastest growing mode of its

21

u

uu

Figure 3.5. Typical evolution sequence for the bistable equation.

linearization is spatially homogeneous. Since the bistable equation is second-order,

a maximum principle exists, so the method of subsolutions and supersolutions can

be used to obtain detailed information about solutions; consequently, the separa-

tion process is fairly well understood. (See, e.g., [10], [15].) The Cahn-Hilliard

equation, on the other hand, is fourth-order and has no maximum principle. The

separation process is, therefore, harder to determine. In the next two chapters, a

mathematically rigorous treatment of spinodal decomposition for the Cahn-Hilliard

equation will be presented.

CHAPTER 4

EVOLUTION IN A HILBERT SPACE

4.1 The Abstract Setting

Consider an evolution equation which can be represented in the form

d

dtu(t) = Au(t) + f(u(t)) t > 0 (4.1)

u(0) = u0,

where u(t) is an element of a Hilbert space X. Suppose the following hold.

1. The operator −A is a sectorial operator on X, and S(t) is the analytic semi-

group generated by A.

2. For some α ∈ [0, 1), f maps Xα into X, where Xα ≡ D((−A)α) is a Hilbert

space with an inner product equivalent to the graph norm. Both the inner

product and the induced norm will be identified with a subscript α.

3. The map f : Xα → X is C1 and satisfies f(0) = 0 and Df(0) = 0.

4. The operator A induces a decomposition of X, X = X− ⊕X+, such that

(a) X− and X+ are invariant under A;

(b) X− and X+ are orthogonal with respect to the inner product on Xα;

(c) X+ is finite-dimensional and X+ ⊂ D(A);

(d) and Re(σ(A−)) < a < b < Re(σ(A+)), where A− : X− ∩D(A)→ X− and

A+ : X+ → X+ are the restrictions of A to X− and X+, respectively, and

σ(L) represents the spectrum of the linear operator L.

23

Extrapolation from the known behavior of ordinary differential equations sug-

gests that there is an invariant manifold for (4.1) which is tangent to X+ at 0 and

which, in some sense, attracts the forward-time flow of nearby solutions. More

precisely, the speed at which nearby solutions move in a direction parallel to the

manifold should be much greater than the speed at which they are repelled by the

manifold. Similarly, there should be an invariant manifold tangent to X− at 0. In

the special case when a < 0 < b, these invariant manifolds are the unstable and

stable manifolds whose existence is proved by Henry [22]. In the special case when

α = 0, i.e., when f maps all of X into X, Bates and Jones [4] have an elegant proof

for the existence of these pseudo-unstable and pseudo-stable manifolds for arbitrary

a and b. This special case is also the subject of some powerful results recently

obtained by Kening Lu [25].

Here the existence of these invariant manifolds is proved in a slightly more general

context. This is done by constructing estimates similar to those used by Bates

and Jones in [4] and then applying the theory of Hirsch, Pugh, and Shub [23] for

invariant manifolds for maps. These same estimates imply, in a well-defined sense,

that for arbitrarily long finite time, “most” solutions beginning near 0 are drawn

along close to the pseudo-unstable manifold.

4.2 Growth Estimates Near an Equilibrium

It will be useful to consider the problem obtained when the nonlinearity f in

(4.1) is modified so that it is very well-behaved outside of a small neighborhood of

0. From the stated assumptions the following is true.

Lemma 4.1 For any ε > 0, there exists f : Xα → X such that f is C1, is globally

Lipschitz continuous with a Lipschitz constant no greater than ε, and agrees with f

in some neighborhood of 0. If, in addition, ‖Df(u)‖L(Xα,X) = O(‖u‖α) then f can

24

be made to agree with f on a ball of radius ρ centered at 0, where ρ−1 = O(ε−1) as

ε→ 0.

Proof. Let ψ : R→ [0, 1] be a C∞ function which is 1 on [−1, 1], which is 0 on

(−∞,−4] ∪ [4,∞), and which satisfies |ψ′| ≤ 1 on R. Choose ρ > 0 so small that

‖Df(u)‖L(Xα,X) ≤ ε/9 if ‖u‖α ≤ ρ. The mean value theorem then says that f has

a Lipschitz constant less than or equal to ε/9 on the ball of radius ρ centered at 0.

If f is defined by

f(u) = ψ

(4‖u‖2

α

ρ2

)f(u),

then f(u) = f(u) for ‖u‖α ≤ ρ/2 and f(u) = 0 for ‖u‖α > ρ. For h ∈ Xα,

Df(u)h =8

ρ2ψ′(

4‖u‖2α

ρ2

)〈u, h〉αf(u) + ψ

(4‖u‖2

α

ρ2

)Df(u)h,

so

‖Df(u)‖L(Xα,X) ≤8

ρ2

∣∣∣∣∣ψ′(

4‖u‖2α

ρ2

)∣∣∣∣∣ ‖u‖α‖f(u)‖

+

∣∣∣∣∣ψ(

4‖u‖2α

ρ2

)∣∣∣∣∣ ‖Df(u)‖L(Xα,X) ≤ ε

if ‖u‖α ≤ ρ. By the mean value theorem, this proves that f has a global Lipschitz

constant no greater than ε.

It is easy to see that if ‖Df(u)‖L(Xα,X) = O(‖u‖α), then ρ can be chosen so that

ρ−1 = O(ε−1) as ε→ 0. This completes the proof of the lemma.

The equation that results from modifying the nonlinearity is

d

dtu(t) = Au(t) + f(u(t)) t > 0 (4.2)

u(0) = u0.

Unless it is explicitly stated otherwise, when solutions, semiorbits, etc. are men-

tioned, reference is being made to (4.2). The letters v and w will represent elements

of X− and X+, respectively. Also, where the particular topology used is not

specified, implicit reference is being made to the Xα topology.

25

Let S−(t) and S+(t) be the analytic semigroups generated by A− and A+,

respectively. Henry [22] gives the following estimates on these semigroups:

‖S+(t)w‖α ≤ C1ebt‖w‖α, w ∈ X+, t ≤ 0, (4.3)

‖S+(t)w‖α ≤ C2ebt‖w‖, w ∈ X+, t ≤ 0, (4.4)

‖S−(t)v‖α ≤ C3eat‖v‖α, v ∈ X− ∩Xα, t ≥ 0, (4.5)

‖S−(t)v‖α ≤ C4eatt−α‖v‖, v ∈ X−, t ≥ 0. (4.6)

Solutions of (4.2) satisfy the variation-of-constants formula (see, e.g., [18])

u(t) = S(t)u0 +∫ t

0S(t− s)f(u(s))ds (4.7)

for t ≥ 0. Write u = v + w with v ∈ X− and w ∈ X+, and then project (4.7) onto

these two subspaces to get the pair of integral equations

v(t) = S−(t)v(0) +∫ t

0S−(t− s)f−(v(s), w(s))ds (4.8)

w(t) = S+(t)w(0) +∫ t

0S+(t− s)f+(v(s), w(s))ds (4.9)

for t ≥ 0, where f− and f+ are the projections of f onto X− and X+, respectively.

Note that S+(τ) makes sense for negative τ since X+ is finite-dimensional, so a

change of variable in (4.9) can be made to get

w(t+ τ) = S+(τ)w(t) +∫ τ

0S+(τ − s)f+(v(t+ s), w(t+ s))ds (4.10)

for τ ≥ −t. Taking norms in (4.8) and (4.10) and using the estimates in (4.3)

through (4.6) yields

‖v(t)‖α ≤ ‖v(0)‖αC3eat + C4ε

∫ t

0(‖v(s)‖α + ‖w(s)‖α)(t− s)−αea(t−s)ds (4.11)

for t ≥ 0, and

‖w(t+ τ)‖α ≤ C1‖w(t)‖αebτ + C2ε∫ 0

τ(‖v(t+ s)‖α + ‖w(t+ s)‖α)eb(τ−s)ds (4.12)

for 0 ≥ τ ≥ −t. In deriving (4.11) and (4.12), use has been made of the fact that

f is Lipschitz continuous with Lipschitz constant no larger than ε, and, therefore,

the same is true of f− and f+.

26

Lemma 4.2 Let U : [−t, 0]→ [0,∞) be continuous. Suppose that for some positive

M and N ,

U(τ) ≤ MU(0) +N∫ 0

τU(s)ds

for −t ≤ τ ≤ 0. Then U(0) ≥ U(−t)e−Nt/M .

Proof. This is Gronwall’s inequality written in an unusual form. Let B(τ) =∫ 0τ U(s)ds. Then B′(τ) = −U(τ) ≥ −MU(0) − NB(τ). Adding NB(τ) to both

sides, multiplying both sides by eNτ , and integrating from −t to 0 gives

−B(−t)e−Nt ≥ MU(0)

N

(e−Nt − 1

).

Thus,

U(−t) ≤MU(0) +NB(−t) ≤MU(0)[1 +

(eNt − 1

)]= MU(0)eNt.

Dividing by MeNt produces the claimed estimate.

The following generalization of Gronwall’s inequality is a special case of a lemma

from Henry [22]. The proof presented here was obtained independently by the

author.

Lemma 4.3 Let U : [0, t]→ [0,∞) be continuous, and let 0 ≤ α < 1. Suppose that

for some positive M and N ,

U(τ) ≤MU(0) +N∫ τ

0(τ − s)−αU(s)ds (4.13)

for 0 ≤ τ ≤ t. Then

U(t) ≤ 2MU(0) exp[(2NΓ(1− α))1/(1−α)t

].

Proof. Let θ = (2NΓ(1− α))1/(1−α), and let V (t) = 2(MU(0) + ε) exp(θt), for

some small ε > 0. If τ > 0 then

∫ τ

0(τ − s)−αV (s)ds = 2(MU(0) + ε)

∫ τ

0(τ − s)−αeθsds

27

= V (τ)∫ τ

0(τ − s)−αe−θ(τ−s)ds

u=θ(τ−s)=

V (τ)

θ1−α

∫ θτ

0u−αe−udu

<V (τ)

θ1−α

∫ ∞0

u−αe−udu

=V (τ)

θ1−α Γ(1− α)

=V (τ)

2N.

Thus,

MU(0) + ε+N∫ τ

0(τ − s)−αV (s)ds

< MU(0) + ε+V (τ)

2≤ V (τ)

2+V (τ)

2= V (τ), (4.14)

for all τ > 0.

Now set

τ ∗ = sup τ ∈ [0, t] : ∀s ∈ [0, τ ], U(s) < V (s) .

Substituting τ = 0 into (4.13) gives U(0) ≤MU(0), so U(0) < V (0), which means

that this set is not empty. Applying (4.13) and (4.14) with τ = τ ∗ gives

U(τ ∗) ≤ MU(0) +N∫ τ∗

0(τ ∗ − s)−αU(s)ds

< MU(0) + ε+N∫ τ∗

0(τ ∗ − s)−αV (s)ds

≤ V (τ ∗).

Since U and V are continuous, the strict inequality U(τ ∗) < V (τ ∗) must mean that

τ ∗ = t, so

U(t) < V (t) = 2(MU(0) + ε) exp[(2NΓ(1− α))1/(1−α)t

].

Letting ε→ 0 gives the stated conclusion.

28

Now suppose ‖w(s)‖α ≥ µ‖v(s)‖α for 0 ≤ s ≤ t. Then v can be eliminated

from the right-hand side of (4.12), and an application of Lemma 4.2 with U(τ) =

‖w(t+ τ)‖αe−bτ yields

‖w(t)‖α ≥ C−11 ‖w(0)‖α exp

[(b− C2ε(1 + µ−1))t

]. (4.15)

Suppose, on the other hand, that ‖w(s)‖α ≤ µ‖v(s)‖α for 0 ≤ s ≤ t. Then w

can be eliminated from the right-hand side of (4.11), and an application of Lemma

4.3 with U(t) = ‖v(t)‖αe−at yields

‖v(t)‖α ≤ 2C3‖v(0)‖α exp[(a + (2C4ε(1 + µ)Γ(1− α))1/(1−α))t

]. (4.16)

These two estimates hold for the evolution with the modified nonlinearity f , but

note that they also hold in a neighborhood of 0 for the original nonlinearity. Note

also that no Cj is in any way dependent on ε or µ.

4.3 Invariant Manifolds

According to Theorem 3.4.4 in [22] the time-t map T (t) induced by (4.2) is

continuously differentiable from Xα to Xα and the derivative of this map at 0 is

S(t). Let a < γ < b. Then the estimates (4.3) and (4.5) imply that for t sufficiently

large S(t) is eγt-pseudo hyperbolic, using the terminology of Hirsch, Pugh, and

Shub [23]. Also, the canonical spectral decomposition of Xα corresponding to this

pseudo hyperbolic endomorphism is compatible with the decomposition X− ⊕X+

of X.

Lemma 4.4 For fixed t, the global Lipschitz constant of T (t) − S(t) can be made

arbitrarily small by making the Lipschitz constant of f sufficiently small.

Proof. Because of (4.5) and (4.6), and since X+ is finite-dimensional, there must

exist valid estimates of the form

‖S(t)u‖α ≤ C5ektt−α‖u‖ (4.17)

29

‖S(t)u‖α ≤ C6ekt‖u‖α. (4.18)

Let u1(t) and u2(t) be two solutions of (4.2). Then by (4.7) and (4.17),

‖(T (t)− S(t))u1(0)− (T (t)− S(t))u2(0)‖α

=∥∥∥∥∫ t

0S(t− s)(f(u1(s))− f(u2(s)))ds

∥∥∥∥α

≤ C5ε∫ t

0

ek(t−s)

(t− s)α‖u1(s)− u2(s)‖αds. (4.19)

Also by (4.7), (4.17), and (4.18),

‖u1(t)− u2(t)‖α ≤ C6ekt‖u1(0)− u2(0)‖α

+ C5ε∫ t

0

ek(t−s)

(t− s)α‖u1(s)− u2(s)‖αds. (4.20)

If Lemma 4.3 is applied to (4.20) with U(t) = ‖u1(t)− u2(t)‖αe−kt, then for ε < 1,

‖u1(t)− u2(t)‖α ≤ 2C6 exp(C7t)‖u1(0)− u2(0)‖α, (4.21)

with C7 independent of ε. By substituting (4.21) into (4.19) it can be seen that

there is C8 depending on t but not on ε, u1(0), or u2(0), such that

‖(T (t)− S(t))u1(0)− (T (t)− S(t))u2(0)‖α ≤ C8ε‖u1(0)− u2(0)‖α.

This completes the proof of the lemma.

Lemma 4.4 completes the verification of all the hypotheses of Theorem 5.1 and

Corollary 5.3 in [23]. Those results imply that if t is sufficiently large and if ε is

sufficiently small then the sets

W+ def= u ∈ Xα : ∀n ∈ N, T (t)−nu exists and ‖T (t)−nu‖αeγnt → 0 as n→∞

and

W− def= u ∈ Xα : ‖T (t)nu‖αe−γnt → 0 as n→∞

are, respectively, graphs of C1 maps X+ → X− ∩Xα and X− ∩Xα → X+ which

are tangent to X+ and X− ∩Xα at 0. In the definition of W+, it is only required

30

that some backward semiorbit of the map T (t) exists, and that the given estimate

holds along it. In general, it is possible for backward semiorbits to cease to exist

or to lose uniqueness. For example, consider the classical heat equation.

These are the pseudo-stable and pseudo-unstable manifolds of the time-t map of

(4.2) (corresponding to the particular decomposition of the space under considera-

tion). Similar sets can be defined for the semiflow of (4.2):

W+ def= u ∈ Xα : T (t)−1u exists for all t > 0, ‖T (t)−1u‖αeγt → 0 as t→∞

W− def= u ∈ Xα : ‖T (t)u‖αe−γt → 0 as t→∞,

where, as with W+, the estimate in the definition of W+ holds along some backward

semiorbit. Clearly W+ ⊆ W+ and W− ⊆ W−. By the proof of Lemma 4.4, there

is a maximum factor by which the norm of a solution ‖u(t)‖α can grow within a

fixed length of time. This implies that, in fact, W+ = W+ and W− = W−, so

pseudo-stable and pseudo-unstable manifolds exist for the semiflow of (4.2). The

intersection of these with a small neighborhood of 0 gives local versions of these

invariant manifolds for (4.1), but one or the other of these may be dependent on the

behavior of orbits far from the origin, so it will not have a simple characterization

in terms of the growth rates of solutions of (4.1). Which of these has this pathology

depends on the location of (a, b), the spectral gap of A. From now on, assume

b > 0 and γ > 0. With these assumptions, the local version of W+ will have a

characterization in terms of (4.1) analogous to the characterization of W+ in terms

of (4.2).

It should be noted that there are probably other ways of showing the exis-

tence of this pseudo-unstable manifold in certain circumstances. For example,

if A has an unstable subspace that is finite-dimensional then the existence of a

finite-dimensional unstable manifold is proven by Henry [22]. Furthermore, the

flow on this manifold is equivalent to the flow of a finite-dimensional system of

31

ordinary differential equations, so the pseudo-unstable manifold could be found as

a submanifold of the unstable manifold by using well-known results for ordinary

differential equations.

4.4 Flow near the Pseudo-unstable Manifold

4.4.1 Evolution in Cones

If µ > 0 let

Kµ = v + w ∈ Xα : µ‖v‖α ≤ ‖w‖α,

γ+µ (ε) = b− C2ε(1 + µ−1),

and

γ−µ (ε) = a + (2C4ε(1 + µ)Γ(1 + α))1/(1−α).

Also, let C9 = 2C1C3.

Lemma 4.5 Let µ > 0 be given. If the Lipschitz constant ε of f is so small that

γ+µ/C9

(ε) > γ−µ (ε) then any semiorbit starting in Kµ remains in Kµ/C9for all positive

time.

Proof. Let ε > 0 be so small that γ+µ/C9

(ε) > γ−µ (ε). Let v(t)+w(t) be a semiorbit

starting at v(0)+w(0) ∈ Kµ. If v(0)+w(0) = 0 then the conclusion of the lemma is

obvious, so assume that v(0)+w(0) 6= 0. Suppose that v(t)+w(t) eventually reaches

∂Kµ/C9. Then without loss of generality it may be assumed that v(0)+w(0) ∈ ∂Kµ,

v(t) + w(t) ∈ ∂Kµ/C9\ 0, and for 0 < τ < t, v(τ) + w(τ) ∈ Kµ/C9

\Kµ. Using

(4.15) and (4.16),

‖v(t)‖α‖w(t)‖α

≤2C1C3‖v(0)‖α exp(γ−µ (ε)t)

‖w(0)‖α exp(γ+µ/C9

(ε)t)=C9

µexp((γ−µ (ε)− γ+

µ/C9(ε))t) <

C9

µ.

This means v(t) + w(t) 6∈ ∂Kµ/C9 , contrary to assumption. This contradiction

shows that v(t) +w(t) never reaches ∂Kµ/C9 , so the positive semiorbit must remain

in Kµ/C9for all time.

32

Let Kcµ represent the closure of the complement of Kµ. Restating the preceding

lemma in terms of backward semiorbits, yields the following.

Lemma 4.6 Let µ > 0 be given. If the Lipschitz constant ε of f is so small that

γ+µ/C9

(ε) > γ−µ (ε) then any backward semiorbit starting in Kcµ/C9

remains in Kcµ as

long as it exists.

The same method that gave the estimates (4.15) and (4.16) on the growth and

decay of a solution can be used to give similar estimates on the growth and decay

of the difference of two solutions. Thus, the cones described above can be centered

at points other than the origin, and if the centers are allowed to evolve with the

flow, then the corresponding generalizations of Lemmas 4.5 and 4.6 hold.

Define the truncated cones

Kµ(r) = v + w ∈ Kµ : ‖w‖α ≤ r

and

Kcµ(r) = v + w ∈ Kc

µ : ‖v‖α ≤ r.

The following theorem says roughly that a semiorbit starting at a random point

near 0 will blow up as t→∞, and as it does so it will stay close to W+.

Theorem 4.1 Let µ > 0 be given. Suppose the Lipschitz constant ε of f is so small

that W+ exists and is tangent to X+ at 0, γ+µ/C9

(ε) > 0, and

γ+µ/(2C9)(ε)− γ−C9µ(ε) >

b− aλ

,

for some λ > 1. Then for any δ > 0 and R > 0 there exists r > 0 such that any

semiorbit v(t) + w(t) starting at v(0) + w(0) ∈ Kµ(r) \ 0 will eventually exit the

cylinder

v + w ∈ Xα : ‖w‖α ≤ R,

33

and the distance in the X− direction from the exit point to W+ is less than δ. In

particular, r will satisfy these conditions if

0 < r < min

δµ

8C3

(δµ

C9(1 + C9)R

)λγ−µ/2

(ε)/(b−a)

, R

. (4.22)

Proof. Recall that every point on W+ lies on a backward semiorbit that ap-

proaches 0 as t → −∞. Since W+ is tangent to X+ at 0, each of these backward

semiorbits intersects KC9µ. By the choice of ε,

γ+µ (ε) > γ+

µ/(2C9)(ε) > γ−C9µ(ε),

so by Lemma 4.5 it must be true that W+ ⊂ Kµ.

Let u2(0) = v2(0) + w2(0) be a point on the base of Kµ/C9(R), and let u1(0) =

v1(0) + w1(0) be the point lying on W+ such that w1(0) = w2(0). Assume

‖v1(0)− v2(0)‖α ≥ δ.

By the choice of ε,

γ+µ/(2C9)(ε) > γ−C9µ

(ε) > γ−µ/2(ε),

so the analogue of Lemma 4.6 for cones centered at u1(−t) says that as long as the

backward orbit u2(−t) exists, u2(−t) ∈ u1(−t) +Kcµ/2. Also, an estimate similar to

(4.16) follows easily:

‖v2(0)− v1(0)‖α ≤ 2C3‖v2(−t)− v1(−t)‖α exp(γ−µ/2(ε)t).

Now as long as u2(−t) exists,

‖v2(−t)− v1(−t)‖α ≥δ

2C3

exp(−γ−µ/2(ε)t),

so by the triangle inequality

‖v2(−t)‖α ≥δ

2C3exp(−γ−µ/2(ε)t)− ‖v1(−t)‖α

34

≥ δ

2C3exp(−γ−µ/2(ε)t)− 1

µ‖w1(−t)‖α

≥ δ

2C3exp(−γ−µ/2(ε)t)− C1R

µexp(−γ+

µ (ε)t).

On the other hand, as long as u2(−t) exists and remains in Kµ/C9

‖w2(−t)‖α ≤ C1R exp(−γ+µ/C9

(ε)t).

Suppose u2(−t) exists and has not left Kµ/C9. Then

‖w2(−t)‖α ≥ µ/(2C1C3)‖v2(−t)‖α,

so

C1R exp(−γ+µ/C9

(ε)t) ≥ δµ

4C1C23

exp(−γ−µ/2(ε)t)− R

2C3exp(−γ+

µ (ε)t).

Thus,

R

2C3exp(−γ+

µ (ε)t) + C1R exp(−γ+µ/C9

(ε)t) ≥ δµ

4C1C23

exp(−γ−µ/2(ε)t),

⇒(R

2C3

+ C1R)

exp(−γ+µ/C9

(ε)t) ≥ δµ

4C1C23

exp(−γ−µ/2(ε)t),

⇒ 4C1C23

δµ

(R

2C3+ C1R

)≥ exp(γ+

µ/C9(ε)t− γ−µ/2(ε)t),

⇒ C9(1 + C9)R

δµ≥ exp

(b− aλ

t

),

⇒ λ

b− a ln

(C9(1 + C9)R

δµ

)≥ t.

This gives a contradiction if t > t∗, where

t∗ =λ

b− a ln

(C9(1 + C9)R

δµ

).

Hence, u2(−t) must cease to exist or must exit Kµ/C9by this time.

35

Now suppose that u2(−t) never left Kµ/C9(so t ≤ t∗), and suppose that u2(−t) ∈

Kµ(r). Then since u2(−t) ∈ u1(−t) + Kcµ/2 and u1(−t) ∈ Kµ, it follows from

elementary analytic geometry that u1(−t) ∈ Kµ(3r). Hence,

‖v1(−t)− v2(−t)‖α ≤ ‖v1(−t)‖α + ‖v2(−t)‖α ≤3r

µ+r

µ=

4r

µ. (4.23)

But

‖v1(−t)− v2(−t)‖α ≥δ

2C3

exp(−γ−µ/2(ε)t)

≥ δ

2C3exp(−γ−µ/2(ε)t∗)

=δ

2C3

exp

[−γ−µ/2(ε)

(λ

b− a

)ln

(C9(1 + C9)R

δµ

)]

=δ

2C3

(δµ

C9(1 + C9)R

)λγ−µ/2

(ε)/(b−a)

. (4.24)

Combining (4.23) and (4.24) gives a contradiction if r satisfies (4.22). For such r,

u2(−t) cannot enter Kµ(r) without first exiting Kµ/C9 .

Now if a (forward) semiorbit starts in Kµ(r) it must exit Kµ/C9(R) through its

base, because of the choice of ε. But by the preceding argument, if r satisfies (4.22)

the distance between its exit point and W+ in the X− direction must be less than

δ.

Corollary 4.1 Let Ω be a neighborhood of the union of the ω-limit sets (under

(4.1)) of the nonzero points on the local pseudo-unstable manifold corresponding to

(4.1), and let µ > 0 be given. Then there exists r > 0 such that any semiorbit

v(t) + w(t) starting at v(0) + w(0) ∈ Kµ(r) \ 0 enters Ω.

Proof. Let ε satisfy the hypotheses of Theorem 4.1, for some λ > 1. Choose

R small enough that f and f agree on Kµ/C9(R). Let SR consist of those points

v + w ∈ X+ ⊕ X− which lie on the local pseudo-unstable manifold and satisfy

36

‖w‖α = R. Note that any semiorbit beginning on SR enters Ω. Because (4.1) has

continuous dependence on initial data [35], some open subset V of

v + w ∈ Xα : ‖w‖α = R

containing SR has this same property. Since X+ is finite-dimensional, SR is com-

pact, so there exists δ > 0 such that

⋃v+w∈SR

v + w ∈ Xα : ‖v − v‖α < δ

is contained in V . (See, e.g., [27].) The corollary now follows from an application

of Theorem 4.1.

Corollary 4.2 Let Ω be as in Corollary 4.1, and let u0 ∈ Xα have a nontrivial X+

component. Then if |k| is sufficiently small, the positive semiorbit beginning at ku0

enters Ω.

Proof. This is an immediate consequence of Corollary 4.1.

4.4.2 Regions Tangent to the Pseudo-Stable Subspace

Theorem 4.2 Let Ω be as in Corollary 4.1. Then there exists r > 0 and F :

R → R satisfying F (0) = F ′(0) = 0 such that if F (‖v0‖α) < ‖w0‖α < r then the

semiorbit of (4.1) beginning at v0 + w0 enters Ω.

Proof. Apply Corollary 4.1 with µ = 1. Define F as follows: F is even and for

x ≥ 0, F (x) is the supremum of all y ≤ r such that there exists v + w ∈ X− ⊕X+

satisfying ‖v‖α = x, ‖w‖α = y, and such that the semiorbit under (4.1) beginning

at v + w never enters Ω. Then from its definition, F has all the desired properties

except possibly F ′(0) = 0. This property is verified below.

Now the graph of F lies below the graph of x 7→ |x| on (−r, r). Let µ ∈ (0, 1).

Applying Corollary 4.1 again produces a secondary truncated cone Kµ(r). The

37

base of Kµ(r) hits the lateral surface of K1(r) at some positive distance d(µ) away

from the X+ axis. Then it is clear that on (−d(µ), d(µ)) the graph of F lies below

the graph of x 7→ µ|x|. Since this holds for all µ ∈ (0, 1), it must be true that

F ′(0) = 0.

4.4.3 Regions with Boundaries Satisfying a Power Law

Theorem 4.2 would be more powerful if it provided more information about the

function F . A stronger result is possible if a further assumption is made.

Lemma 4.7 If ‖Df(u)‖L(Xα,X) = O(‖u‖α) as u → 0 then the tangency of the

pseudo-unstable manifold W+ at 0 is quadratic. That is, ‖v‖α = O(‖w‖2α) for

u = v + w ∈W+ as u→ 0.

Proof. As was mentioned in the proof of Theorem 4.1, Lemma 4.5 implies that

the pseudo-unstable manifold for (4.2) is contained in Kµ provided that

γ+µ (ε) > γ−C9µ(ε).

Written out explicitly, this condition is

b− C2ε(1 + µ−1) > a + (2C4ε(1 + C9µ)Γ(1 + α))1/(1−α). (4.25)

As µ→∞, there exists a constant k1 > 0 such that ε and µ will satisfy (4.25) if

ε ≤ k1

µ. (4.26)

Also, Lemma 4.1 says that there exists a constant k2 > 0 such that f and f can be

made to agree on Kµ(R) provided that

R√

1 + µ−2 ≤ k2ε. (4.27)

Thus, for such R, the local version of W+ for (4.1) (independent of ε) exits Kµ(R)

through its base, not through its lateral surface. Combining (4.26) and (4.27), there

38

is a constant k3 > 0 such that if R ≤ k3/µ, then the pseudo-unstable manifold

exits Kµ(R) through its base. This means that if v + w is on this manifold and

‖w‖α ≤ k3/µ then µ‖v‖α ≤ ‖w‖α. In particular, if ‖w‖α = k3/µ then

‖v‖α ≤1

µ‖w‖α =

1

k3‖w‖2α.

Hence, ‖v‖α = O(‖w‖2α).

The following theorem strengthens the result of Theorem 4.2.

Theorem 4.3 Let Ω be as in Corollary 4.1. If ‖Df(u)‖L(Xα,X) = O(‖u‖α) as

u→ 0 then for any power p < (3b− 2a)/(2b− a) there exists r > 0 and k > 0 such

that if k‖v‖pα < ‖w‖α < r then the semiorbit of (4.1) beginning at v + w enters Ω.

Proof. Apply Corollary 4.1 with µ = 1, to get a finite cone of initial data that

produce orbits which enter Ω. Using this cone and Lemma 4.7 it can be seen that

for R sufficiently small the δ in Theorem 4.2 can be taken of the same order as R.

That is, there is a constant k1 > 0 such that, for R small, if

δ ≤ k1R, (4.28)

then every point on the cylinder

v + w ∈ Xα : ‖w‖α = R

whose distance from W+ in the X− direction is no bigger than δ generates a positive

semiorbit which enters Ω. Similarly, in producing the secondary cones in the proof

of Theorem 4.2, it is sufficient for ε and R to satisfy

ε ≤ k2µ, (4.29)

and

R√

1 + C29µ−2 ≤ k3ε, (4.30)

39

for some constants k2 > 0 and k3 > 0 and for every µ sufficiently small. Combining

(4.29) and (4.30) shows that (4.30) can be replaced by

R ≤ k4µ2. (4.31)

Substituting (4.28), (4.29), and (4.31) into (4.22) and simplifying gives the estimate

r ≤ k5µ((2+λ)b−2a)/(b−a) , (4.32)

for some constant k5 > 0, as the only requirement on the heights of the secondary

cones in the proof of Theorem 4.2. If x is the radius of one of these secondary cones

(in the X− direction), then µ = r/x. Making this substitution in (4.32) gives the

estimate

r ≥ kxp, (4.33)

where

p =(2 + λ)b− 2a

(1 + λ)b− a

and k is some positive constant depending on p. Applying (4.33) to the proof of

Theorem 4.2 for λ sufficiently near 1 gives the desired result.

The bound on the exponents obtained from Theorem 4.3 is 3b−2a2b−a while the

corresponding exponent in (3.2) is ba, which is larger. It is not clear whether the

difference is inherent in the more general setting of this chapter or if it is simply

an artifact of the particular methods used here.

4.4.4 Measure-theoretic Results

Theorems 4.2 and 4.3 each identify a region of initial data near the origin whose

positive semiorbits are, in some sense, controlled by the pseudo-unstable manifold.

It is natural to try to find some way of determining the significance of this region,

and a natural way of doing this is finding its size with respect to some measure. A

very general measure-theoretic restatement of the results of the previous sections

40

is presented in a theorem below. The following definition will be helpful: A subset

S of a topological vector space is balanced if, for any scalar α satisfying |α| ≤ 1,

αS ⊂ S.

Lemma 4.8 If S is balanced then λS is balanced for any scalar λ. Also, the union

of balanced sets is balanced.

Proof. Let α be a scalar satisfying |α| ≤ 1. If S is balanced then αS ⊂ S, so

α(λS) = λ(αS) ⊂ λS.

This proves the first statement. Now, if Sβ is balanced for every β in some index

set B and α is as above then

α

⋃β∈B

Sβ

=⋃β∈B

(αSβ) ⊂⋃β∈B

(Sβ).

This proves the second statement.

Theorem 4.4 Let Ω be as in Corollary 4.1, and let ϕ be a probability measure on

the Borel sets of Xα. Given ε > 0 let ϕε be the scaled probability measure defined

by ϕε(V ) = ϕ(ε−1V ). Suppose ϕ(X−∩Xα) = 0. Then ϕε(Ω′)→ 1 as ε→ 0, where

Ω′ is the set of points which produce positive semiorbits that enter Ω.

Proof. Define S to be the union of the (closed) finite cones constructed in the

proof of Theorem 4.2. Since each cone is balanced, by the second part of Lemma

4.8 S is balanced, also. Applying the first part of Lemma 4.8, ε−1S is balanced for

any ε > 0. Hence, ε−1i S ⊂ ε−1

j S if εi > εj.

Now, let (εk) be a sequence of positive real numbers satisfying εk → 0. By

Corollary 4.2,

Xα \X− ⊂∞⋃k=1

1

εk(S \ 0),

so

1 = ϕ(Xα)

41

= ϕ(Xα \X−) + ϕ(Xα ∩X−)

= ϕ(Xα \X−)

≤ ϕ

( ∞⋃k=1

1

εk(S \ 0)

)

= ϕ

( ∞⋃k=1

1

εkS

)

= limk→∞

ϕ(

1

εkS)

= limk→∞

ϕ(

1

εk(S \ 0)

)≤ lim

k→∞ϕ(

1

εkΩ′)

= limk→∞

ϕεk(Ω′)

≤ 1.

This proves the theorem.

Now consider some particular classes of measures. One of the most natural mea-

sures on Rn is n-dimensional Lebesgue measure mn. Some of the most important

characteristics of this measure are

1. if V is a nonempty open set then mn(V ) > 0;

2. if V is a bounded Borel set then mn(V ) <∞;

3. if V is a Borel set and x ∈ Rn then mn(x+ V ) = mn(V ).

One might ask whether n-dimensional Lebesgue measure can be generalized to

a measure m on an infinite-dimensional Hilbert space H while at least keeping

these three properties intact. The answer to that question is negative, and the

proof is quite simple. Let e1, e2, . . . be an orthonormal sequence in H . Let

B(x, r) represent the open ball of radius r centered at x. By properties 1,2, and

42

3, , m(B(0, 1/2)) > 0, m(B(0, 2)) < ∞, and m(B(ej , 1/2)) = m(B(0, 1/2)). But

combining these facts with countable additivity gives

∞ > m(B(0, 2)) ≥ m

∞⋃j=1

B(ej ,

1

2

) =

∞∑j=1

m(B(ej,

1

2

))=∞∑j=1

m(B(

0,1

2

))=∞.

This contradiction shows the impossibility of such a measure.

If those initial values near 0 are of particular interest, it makes sense to use some

measure that is concentrated near that point. One such measure that is commonly

referred to is “white noise.” Unfortunately, the white-noise measure corresponding

to a particular Hilbert space is really a probability measure on its algebraic dual;

furthermore, the white-noise measure of the subspace corresponding to the contin-

uous dual is 0 (See, e.g., Skorohod [34].) Using the canonical identification of a

Hilbert space with its continuous dual (provided by Riesz’s Theorem), the entire

Hilbert space has measure 0. Clearly, this measure will be of no use.

Probably the most important measure on R that is concentrated near one

particular point is one-dimensional Gaussian measure ϕ defined by

ϕ(E) =1√

2πσ2

∫E

exp

(−(x−m)2

2σ2

)dx.

This measure corresponds to the normal distribution with mean m and variance

σ2. A point mass measure may be considered a Gaussian measure with variance 0.

A Borel measure ϕ on a Hilbert space H is said to be a Gaussian measure if each

of its projections onto one-dimensional subspaces gives a one-dimensional Gaussian

measure; i.e., 〈·, h〉 is normally-distributed for each h ∈ H . More precisely, if Sh

is the span of h ∈ H , the measure ϕh on R given by

ϕh(E) = ϕ(Eh× (Sh)⊥

)must be a one-dimensional Gaussian measure. Such measures are easiest to deal

with when H is separable, so assume that this is true for the remainder of this

43

chapter. Also, for simplicity only Gaussian measures with mean 0 (i.e., each

projection has mean 0) will be discussed. Nontrivial Gaussian measures are easy

to construct on separable Hilbert spaces by specifying the projections onto an

orthonormal system of vectors. These projections cannot, however, be arbitrarily

chosen; the variance of the nth projection must approach 0 at a certain rate as

n→∞. See Skorohod [34] for details.

The following lemma shows how a Gaussian measure behaves with respect to an

orthogonal direct sum.

Lemma 4.9 Let H be a separable Hilbert space and ϕ a Gaussian measure on H

with mean 0. Let A⊕ B be an orthogonal decomposition of H. Define ϕA and ϕB

to be measures on A and B, respectively, satisfying

ϕA(U) = ϕ(U × B)

ϕB(V ) = ϕ(A× V ),

for every U ⊂ A and every V ⊂ B. Then ϕA and ϕB are Gaussian measures with

mean 0.

Proof. By symmetry, it suffices to prove that ϕA is a Gaussian measure with

mean 0. Let a ∈ A, and let E ⊂ R. Let A′ be the orthogonal complement of Sa

in A. Then

ϕAa (E) = ϕA(Ea× A′)

= ϕ((Ea×A′)× B)

= ϕ(Ea× (A′ × B))

= ϕ(Ea× (Sa)⊥), (4.34)

where (Sa)⊥ is the orthogonal complement of Sa in H . Since ϕ is a Gaussian

measure with mean 0, (4.34) implies that ϕAa is a one-dimensional Gaussian measure.

Since this holds for any a ∈ A, ϕA must be a Gaussian measure with mean 0.

44

The covariance operator Sϕ of a measure ϕ on a Hilbert space H is the bounded

linear operator on H defined by

〈Sϕx, y〉 =∫H〈x, z〉〈y, z〉dϕ(z).

For an arbitrary measure the covariance operator may not exist. (For example,

consider the probability measure ϕ on R defined by ϕ(n) = 3/(π2n2) for nonzero

integers n and ϕ(E) = 0 if E contains no nonzero integer.) However, it is clear

that if Sϕ exists it must be positive semidefinite and self-adjoint. Furthermore, a

result due to Prohorov found in Kuo [24] shows that if ϕ is a Gaussian measure

on a separable Hilbert space then Sϕ exists and its trace is finite. This fact allows

one to estimate certain integrals with respect to ϕ. The following result, which

estimates how fast Gaussian measures must die out at infinity, makes use of some

work by Kuo [24].

Lemma 4.10 Let ϕ be a Gaussian measure with mean 0 on the separable Hilbert

space H. Then there exist positive constants M and k such that for any R ≥ 0

ϕ (h ∈ H : ‖h‖ ≥ R) ≤Me−kR2

.

Proof. Since Sϕ is bounded, positive semidefinite, and self-adjoint, H has

an orthonormal basis given by the eigenvectors en of Sϕ, with corresponding

(nonnegative) eigenvalues αn. Note that∫H〈x, en〉2dϕ(x) = 〈Sϕen, en〉 = 〈αnen, en〉 = αn,

so 〈·, en〉 is normally distributed with mean 0 and variance αn. Choose

k ∈(

0, inf

1

2αn: n ∈ N, αn 6= 0

).

This choice is possible since the boundedness of Sϕ implies that the αn are bounded

above. Estimating, ∫Hek‖x‖

2

dϕ(x) =∫Hek∑

n〈x,en〉2dϕ(x)

45

=∫H

∏n

ek〈x,en〉2

dϕ(x)

=∏n

∫Hek〈x,en〉

2

dϕ(x). (4.35)

Now,

∫Hek〈x,en〉

2

dϕ(x) =∫ ∞

0ϕ(x ∈ H : ek〈x,en〉

2 ≥ y)dy

= 1 +∫ ∞

1ϕ(x ∈ H : ek〈x,en〉

2 ≥ y)dy

= 1 + 2∫ ∞

1ϕ

(x ∈ H : 〈x, en〉 ≥

(1

kln y

)1/2)

dy

= 1 + 2∫ ∞

1

1√2παn

∫ ∞(k−1 ln y)1/2

e−t2

2αn dtdy

= 1 +2√

2παn

∫ ∞0

∫ ekt2

1e−

t2

2αn dydt

= 1 +2√

2παn

∫ ∞0

(e−( 1

2αn−k)t2 − e−

t2

2αn

)dt

=2√

2παn

∫ ∞0

e−( 12αn−k)t2dt

=2√

2παn

√π

2√

12αn− k

= (1− 2kαn)−1/2 .

Thus, (4.35) gives

∫Hek‖x‖

2

dϕ(x) =∏n

(1− 2kαn)−1/2

=

(∏n

(1− 2kαn)

)−1/2

. (4.36)

Since ∑n

2kαn = 2k∑n

αn = 2k · tr(Sϕ) <∞,

the infinite product in (4.36) converges to a positive number. (See, e.g., Rudin

[33].)

46

Define

M =∫Hek‖x‖

2

dϕ(x),

and

HR = h ∈ H : ‖h‖ ≥ R.

Then from (4.36),

ϕ(HR) =∫HR

1dϕ(x)

≤∫HR

ek‖x‖2−kR2

dϕ(x)

= e−kR2∫HR

ek‖x‖2

dϕ(x)

≤ e−kR2∫Hek‖x‖

2

dϕ(x)

= Me−kR2

.

This completes the proof.

These results permit an estimate on the rate of convergence in Theorem 4.4

when ϕ is Gaussian.

Theorem 4.5 Let Ω be as in Corollary 4.1, and assume that

‖Df(u)‖L(Xα,X) = O(‖u‖α)

and Xα is separable. Let ϕ be a Gaussian measure on Xα with mean 0 such that

for all x ∈ X+ the induced variance of 〈·, x〉 is greater than 0. If ϕε and Ω′ are

defined as in Theorem 4.4, then

ϕε(Ω′) = 1− O

(εn(p−1)

)

for any p < (3b− 2a)/(2b− a), where n = dim(X+).

47

Proof. Choose r > 0, k > 0, and

p′ ∈(p,

3b− 2a

2b− a

)

such that

Sdef=v + w ∈ Xα : k‖v‖p

′

α < ‖w‖α < r

is contained in Ω′. This is possible because of Theorem 4.3. Note that for ε > 0

sufficiently small, S contains the set

Sdef=

v + w ∈ Xα : εp < ‖w‖α < r, ‖v‖α <

(εp

k

)1/p′.

Thus,

ϕε(Ω′) ≥ ϕε(S)

≥ ϕε(S)

= ϕ(

1

εS)

≥ 1− ϕ(v + w ∈ Xα : ‖w‖α ≤ εp−1

)− ϕ

(v + w ∈ Xα : ‖w‖α ≥

r

ε

)− ϕ

(v + w ∈ Xα : ‖v‖α ≥

1

ε

(εp

k

)1/p′)

= 1− ϕX+(w ∈ X+ : ‖w‖α ≤ εp−1

)− ϕX+

(w ∈ X+ : ‖w‖α ≥

r

ε

)− ϕX−∩Xα

(v ∈ X− ∩Xα : ‖v‖α ≥

1

ε

(εp

k

)1/p′)

. (4.37)

Now for some constant C1 > 0

ϕX+(w ∈ X+ : ‖w‖α ≤ εp−1

)≤ C1

(εp−1

)n, (4.38)

where n = dim(X+). This estimate comes from the fact that an n-dimensional

Gaussian measure for which no projection is a point mass is absolutely continuous

48

with respect to n-dimensional Lebesgue measure. By Lemma 4.10, the following

estimates obtain:

ϕX+(w ∈ X+ : ‖w‖α ≥

r

ε

)≤ C2 exp

[−k2

(r

ε

)2], (4.39)

ϕX−∩Xα

(v ∈ X− ∩Xα : ‖v‖α ≥

1

ε

(εp

k

)1/p′)

≤ C3 exp

−k3

(1

ε

(εp

k

)1/p′)2

≤ C3 exp[−k4ε

2(p/p′−1)]. (4.40)

Estimates (4.39) and (4.40) say that the corresponding quantities in (4.37) are

transcendentally small as ε→ 0. Therefore, substituting (4.38) into (4.37) gives

ϕε(Ω′) ≥ 1− O

(εn(p−1)

).

Since ϕε(Ω′) ≤ 1, the desired result follows immediately.

CHAPTER 5

SPINODAL DECOMPOSITION

5.1 The Equation in an Abstract Setting

Let M lie in the spinodal region of W so that W ′′(M) < 0. Recall that the

constant u ≡M is an equilibrium for the Cahn-Hilliard equation. The goal in this

section is to show that the results of Chapter 4 regarding the nature of a semiflow in

a neighborhood of an equilibrium can be be applied to the Cahn-Hilliard equation,

by showing that the Cahn-Hilliard equation fits into the abstract setting described

in Section 4.1.

The first thing to do is to make the change of variable u = u − M so that

the particular equilibrium of interest ends up at the origin. Simultaneously, let

β2 = −W ′′(M) > 0 (as in Chapter 3) and define the function ψ by

ψ(u) = W ′(u+M) + β2u. (5.1)

(Since W is C5, ψ is C4.) Making these substitutions into (1.1) and dropping the

caret from u gives

∂u

∂t= −∆

(ε2∆u+ β2u− ψ(u)

)x ∈ Ω (5.2)

∂u

∂ν=∂∆u

∂ν= 0 x ∈ ∂Ω.

Define

Au = −∆(ε2∆u+ β2u),

and let

f(u) = ∆(ψ(u)),

50

so (5.2) is of the form

ut = Au+ f(u)

with f(0) = 0 as required. Let

X =u ∈ L2(Ω) :

∫Ωudx = 0

and

D(A) =

u ∈ X ∩H4(Ω) :

∂u

∂ν

∣∣∣∣∣∂Ω

=∂∆u

∂ν

∣∣∣∣∣∂Ω

= 0

.

Because the Cahn-Hilliard equation conserves mass, spaces with integral constraints

are the most appropriate spaces with which to work. By Rankin [32], the linear

operator −A with the given domain is a sectorial operator, so A generates an

analytic semigroup S(t). Also,

X1/2 = D((−A)1/2) =

u ∈ X ∩H2(Ω) :

∂u

∂ν

∣∣∣∣∣∂Ω

= 0

.

Note that when Ω = [0, 1], an explicit spectral representation for S(t) was derived

in the process of performing the linear analysis.

In order to be able to use the results of Chapter 4 with α = 1/2, it is necessary

to show that f is continuously differentiable from X1/2 to X with Df(0) = 0; if

such results as Theorems 4.3 and 4.5 are to be used then it must also be verified

that

‖Df(u)‖L(X1/2,X) = O(‖u‖X1/2).

These verifications will be performed here under the assumption that dim(Ω) ≤ 4.

Expanding f(u) gives

f(u) = ∆(ψ(u)) = ψ′(u)∆u+ ψ′′(u)|∇u|2.

The Sobolev imbedding theorems (see, e.g., Gilbarg and Trudinger [17]) imply that

H2(Ω) → L∞(Ω) ∩W 1,4(Ω); therefore, if u ∈ H2(Ω) then f(u) ∈ L2(Ω). Also, if

u ∈ X1/2 then ∂u∂ν

= 0 on ∂Ω, so by the divergence theorem∫Ωf(u)dx =

∫Ω

∆(ψ(u))dx =∫∂Ω

∂ψ(u)

∂νdx = 0.

51

Since f(u) ∈ L2(Ω) and has mean value 0, f(u) ∈ X. Hence, f : X1/2 → X, as was

needed.

Next consider the linear mapping h 7→ ∆(ψ′(u)h) as a candidate for Df(u).

Applying the triangle inequality to the expansion

∆(ψ(u+ h))−∆(ψ(u))−∆(ψ′(u)h) =

ψ′′(u+ h)|∇(u+ h)|2 + ψ′(u+ h)∆(u+ h)− ψ′′(u)|∇u|2 − ψ′(u)∆u

− ψ′′′(u)|∇u|2h− ψ′′(u)h∆u− 2ψ′′(u)∇u · ∇h− ψ′(u)∆h

gives

‖∆(ψ(u+ h))−∆(ψ(u))−∆(ψ′(u)h)‖L2 (5.3)

≤ ‖(ψ′(u+ h)− ψ′(u))∆h‖L2

+ 2‖(ψ′′(u+ h)− ψ′′(u))∇u · ∇h‖L2

+ ‖(ψ′(u+ h)− ψ′(u))∆u− ψ′′(u)h∆u‖L2

+ ‖(ψ′′(u+ h)− ψ′′(u))|∇u|2 − ψ′′′(u)|∇u|2h‖L2

+ ‖ψ′′(u+ h)|∇h|2‖L2

def= N1 + 2N2 +N3 +N4 +N5. (5.4)

Applying the mean value theorem to N1 and N2 and to N3 and N4 twice gives

N1 ≤ ‖ψ′′(θ1)h∆h‖L2

N2 ≤ ‖ψ′′′(θ2)h∇u · ∇h‖L2

N3 ≤ ‖ψ′′′(θ3)h2∆u‖L2

N4 ≤ ‖ψ′′′′(θ4)h2|∇u|2‖L2

where the θj are functions that are pointwise between u and u + h. Applying

Holder’s inequality yields

N1 ≤ ‖ψ′′(θ1)‖L∞‖h‖L∞‖∆h‖L2

52

N2 ≤ ‖ψ′′′(θ2)‖L∞‖h‖L∞‖∇u‖L4‖∇h‖L4

N3 ≤ ‖ψ′′′(θ3)‖L∞‖h‖2L∞‖∆u‖L2

N4 ≤ ‖ψ′′′′(θ4)‖L∞‖h‖2L∞‖∇u‖

2L4

N5 ≤ ‖ψ′′(u+ h)‖L∞‖∇h‖2L4,

so by the Sobolev imbedding theorems, (5.4) gives

‖∆(ψ(u+ h))−∆(ψ(u))−∆(ψ′(u)h)‖L2 = O(‖h‖2

H2

)(5.5)

as ‖h‖H2 → 0. Also,

‖∆(ψ′(u)h)‖L2 ≤ ‖ψ′′′(u)|∇u|2h‖L2

+ ‖ψ′′(u)h∆u‖L2

+ 2‖ψ′′(u)∇u · ∇h‖L2

+ ‖ψ′(u)∆h‖L2

≤ ‖ψ′′′(u)‖L∞‖∇u‖2L4‖h‖L∞

+ ‖ψ′′(u)‖L∞‖∆u‖L2‖h‖L∞

+ 2‖ψ′′(u)‖L∞‖∇u‖L4‖∇h‖L4

+ ‖ψ′(u)‖L∞‖∆h‖L2.

By (5.1), ψ′(0) = 0, so

‖ψ′(u)‖L∞ = O (‖u‖L∞) ;

hence,

‖∆(ψ′(u)h)‖L2 = O (‖u‖H2‖h‖H2) . (5.6)

The combination of (5.5) and (5.6) implies that f : H2 → L2 is differentiable,

Df(u)h = ∆(ψ′(u)h), and

‖Df(u)‖L(H2,L2) = O(‖u‖H2).

53

In particular, this means that Df(0) = 0. The verification of the fact that the

map u 7→ Df(u) is continuous from H2 to L(H2, L2) is essentially the same as the

derivation of (5.6), so the details will be omitted.

Now recall that all of the preceding estimates held for Ω ⊂ Rn, n ≤ 4, so the

results of Chapter 4 can be applied to the Cahn-Hilliard equation on any such Ω,

provided that an appropriate decomposition X− ⊕ X+ of X is chosen. However,

for these results to be very illuminating, something must be known about the

ω-limit set of an arbitrary point on the pseudo-unstable manifold. Such information

is difficult to obtain in spaces of high dimension, so for the rest of this chapter

consider Ω = [0, 1] ⊂ R. As was mentioned in Chapter 3, generically there is a

single fastest-growing mode, say cosn0πx, for the linearization of the Cahn-Hilliard

equation about u ≡ 0. Suppose ε > 0 is such that this generic phenomenon is

realized, and let X+ be the span of cosn0πx and X− be the orthogonal subspace

consisting of all other modes. This decomposition satisfies all of the conditions in

Chapter 4.

5.2 Properties of the ω-limit Points

Now return to the original coordinate system where the equilibrium of interest

lies at u ≡ M . Because the Cahn-Hilliard equation has a Lyapunov functional,

namely the total energy functional defined in Chapter 2, and this functional is

strictly decreasing along nonequilibrium trajectories, the ω-limit set of any point

consists entirely of equilibria. By integrating (3.1) and using the corresponding

boundary conditions, it can be seen that an equilibrium u(x) of the one-dimensional

Cahn-Hilliard equation must satisfy

−ε2u′′(x) +W ′(u(x)) = c

for some constant c and boundary conditions

u′(0) = u′(1) = 0.

54

Since the only equilibria of interest are those with the same mass as the initial data,

c is not arbitrary but must be such that∫ 10 u(x)dx = M . By defining v(x) = u′(x)

and looking at the symmetry of the phase plane of the (u, v) system about the

u-axis, it is easy to see that u must either be strictly monotone (in which case it

can be thought of as being periodic with period 2) or 2/k-periodic for some positive

integer k. Also, each period is made up of two reflection-symmetric monotone

transition layers of width 1/k.

Now consider the pseudo-unstable manifold tangent to the X+ axis, where X+

is as defined above. Consider the closed subspace Y ⊂ X consisting of elements

whose Fourier expansions contain only cosn0πx and its higher harmonics, i.e,

cos 2n0πx, cos 3n0πx, etc. This space is clearly invariant under (3.1), so the methods

of Chapter 4 could be applied to Y instead of X to obtain a pseudo-unstable

manifold tangent to X+ and contained within Y . Because of the characterization

of pseudo-unstable manifolds, they are unique, so these two manifolds must be

the same; consequently, the original pseudo-unstable manifold is contained in Y .

Since Y is closed, it also contains the ω-limit sets of every nonzero point on the

pseudo-unstable manifold. Hence, any equilibrium found in such an ω-limit set

must have minimal period equal to 2/(kn0) (and, therefore, transition layers of

width 1/(kn0)) for some positive integer k. In particular, this transition length is

O(ε) as ε→ 0.

It is not too hard to relate the periodicity of a steady-state solution for the

Cahn-Hilliard equation to its amplitude. The following proposition, which is a

special case of the Sturm comparison theorem [21], will be useful in this regard. Its

proof is included for completeness.

Proposition 5.1 Let v1 and v2 satisfy

−ε2v′′1(x) + a1(x)v1(x) = 0 (5.7)

55

−ε2v′′2(x) + a2(x)v2(x) = 0 (5.8)

on [0, 1] with initial conditions

v1(0) = v2(0) = 0

v′1(0) = v′2(0) 6= 0.

Suppose a1(x) ≥ a2(x) for all x ∈ [0, 1]. Then if v1(x1) = 0 for some x1 ∈ (0, 1]

then v2(x2) = 0 for some x2 ∈ (0, x1].

Proof. Let x∗ be the largest x ∈ (0, 1] such that v1v2 > 0 on (0, x∗). Notice that

the initial conditions imply that v1 and v2 have the same sign in some neighborhood

of 0, so x∗ is well-defined. Multiply (5.7) by v2(x), multiply (5.8) by v1(x), subtract

the two resulting equations and integrate x from 0 to s, where s ∈ (0, x∗), to get

ε2∫ s

0[v′′1(x)v2(x)− v′′2(x)v1(x)]dx =

∫ s

0(a1(x)− a2(x))v1(x)v2(x)dx.

Integrate the left-hand side by parts and use the boundary conditions to get

ε2[v′1(s)v2(s)− v′2(s)v1(s)] =∫ s

0(a1(x)− a2(x))v1(x)v2(x)dx.

Notice that this can be written as

ε2

(v1(s)

v2(s)

)′(v2(s))2 =

∫ s

0(a1(x)− a2(x))v1(x)v2(x)dx.

This equation says that (v1(s)/v2(s))′ ≥ 0 for s ∈ (0, x∗), while the initial conditions

and L’Hopital’s rule imply that (v1(s)/v2(s))→ 1 as s→ 0+, so v1/v2 ≥ 1 on (0, x∗).

Now either v1(x∗)v2(x∗) = 0 or v1(x∗)v2(x∗) 6= 0. If v1(x∗)v2(x∗) = 0 then the

fact that v1/v2 ≥ 1 on (0, x∗) implies that v2(x∗) = 0, so since v1 6= 0 on (0, x∗) the

proposition holds in this case. If v1(x∗)v2(x∗) 6= 0 then x∗ = 1, so neither v1 nor v2

has a zero on (0, 1]; therefore, the proposition holds in this case as well.

56

Now given a fixed equilibrium u(x) that is an ω-limit point of a point on the

pseudo-unstable manifold, let v(x) = u′(x). Clearly v satisfies

−ε2v′′(x) +W ′′(u(x))v(x) = 0

with boundary conditions

v(0) = v(1) = 0.

Since energy decreases along trajectories of the Cahn-Hilliard equation, it is impos-

sible for u to be the homogeneous equilibrium u ≡ M . This means that v′(0) 6= 0

since, otherwise, the uniqueness of solutions to initial value problems would imply

that v ≡ 0. Hence, Proposition 5.1 can be applied with either v = v1 or v = v2 if a

suitable complementary comparison function can be found. Applying Proposition

5.1 with a1(x) ≡ sups∈[0,1]W′′(u(s)), a2(x) ≡ W ′′(u(x)), and v′1(0) = v′2(0) = v(0)

gives the estimate that the first positive zero of v2(x) ≡ v(x) comes no later than

the first positive zero of

v1(x) ≡ εv′(0)√− sups∈[0,1]W

′′(u(s))sin

√− sups∈[0,1]W

′′(u(s))

εx

if sups∈[0,1]W

′′(u(s)) < 0. This means that the transition width 1/(kn0) of u is less

than or equal to

πε√− sups∈[0,1]W

′′(u(s))

if sups∈[0,1]W′′(u(s)) < 0. Written another way,

−π2ε2k2n20 ≤ sup

s∈[0,1]W ′′(u(s)), (5.9)

and this clearly holds if sups∈[0,1]W′′(u(s)) ≥ 0, as well. On the other hand,

applying Proposition 5.1 with a1(x) ≡ W ′′(u(x)), a2(x) ≡ infs∈[0,1]W′′(u(s)), and

the same initial conditions as before shows that the first positive zero of

v2(x) ≡ εv′(0)√− infs∈[0,1]W ′′(u(s))

sin

√− infs∈[0,1]W ′′(u(s))

εx

57

comes no later than the first positive zero of v1(x) ≡ v(x); thus, the transition

width of u is greater than or equal to

πε√− infs∈[0,1]W ′′(u(s))

.

This can be rewritten as

infs∈[0,1]

W ′′(u(s)) ≤ −π2ε2k2n20. (5.10)

Since (5.9) and (5.10) estimate the range of W ′′ u, they give bounds on the

amplitude of u. To see whether these bounds are meaningful, it helps to recall the

value of n0. In Section 3.1 it was shown that∣∣∣∣∣n0 −β

επ√

2

∣∣∣∣∣ < 1,

so

n0 =β

επ√

2+O(1)

as ε→ 0. Therefore,

−π2ε2k2n20 = −β

2k2

2+O(ε) =

k2W ′′(M)

2+O(ε).

Substituting this into (5.9) and (5.10) gives

infs∈[0,1]

W ′′(u(s)) ≤ k2W ′′(M)

2+O(ε) ≤ sup

s∈[0,1]W ′′(u(s)). (5.11)

Now since the average value of u is M ,

infs∈[0,1]

W ′′(u(s)) ≤W ′′(M) ≤ sups∈[0,1]

W ′′(u(s)). (5.12)

No matter what the value of k, (5.11) and (5.12) together give a lower bound on the

amplitude of u, which does not go to 0 as ε→ 0. The closer M is to the center of

the spinodal region, the better that bound is. Also, since W ′′ is bounded below, the

first estimate in (5.11) will give a contradiction for k sufficiently large. In fact, if M

58

is sufficiently close to the center (and ε is not too large) it will give a contradiction

for any k > 1, so, in this case, each relevant equilibrium u has exactly the same

minimal period as the fastest-growing mode.

In any case, the calculations of this section have shown that all ω-limit points of

nonzero points on the pseudo-unstable manifold are equilibria with period O(ε) and

amplitude bounded away from 0. Anything close to one of these equilibria in the L∞

norm (a fortiori in the H2 norm) will exhibit the small-wavelength large-amplitude

oscillations characteristic of the experimentally-observed spinodal decomposition.

5.3 Orbits Approaching the ω-limit Points

In the previous two sections, the foundation has been laid for the application

of the abstract results in Chapter 4 to the Cahn-Hilliard equation to show the

predominance of spinodal decomposition. The following theorems are now imme-

diate consequences of Corollary 4.2, Theorem 4.3, and Theorem 4.5. Several of

the hypotheses are essentially the same for all theorems, so they will first be listed

separately.

Let M lie in the spinodal region of W , and let cosn0πx be the unique fastest-

growing mode of the linearization of the one-dimensional Cahn-Hilliard equation

about M . Consider the linear manifold consisting of all H2 functions on [0, 1] that

have average value M and satisfy Neumann boundary conditions. Let Ω be any

neighborhood (in this manifold) of the collection of equilibria of the Cahn-Hilliard

equation having average value M and minimal period of the form 2/(kn0) for some

positive integer k. Let Ω′ consist of all initial data that produce orbits entering Ω

in positive time.

Theorem 5.1 Let u0 be any point on the linear manifold of functions with mass M

which is not H2-orthogonal to cosn0πx. If |ρ| is sufficiently small then the solution

of the Cahn-Hilliard equation with initial data M + ρ(u0 −M) enters Ω in positive

time; i.e., M + ρ(u0 −M) ∈ Ω′.

59

Theorem 5.2 Let P be the projection defined by

P

(M +

∞∑n=0

an cos nπx

)= M + an0 cos n0πx.

Let λn0 be the growth rate of the fastest-growing mode of the linearization about

u ≡ M (as described in Chapter 3), and let λn1 be the growth rate of the next

fastest-growing mode. Then if

p <3λn0 − 2λn1

2λn0 − λn1

,

there exists r > 0 and C > 0 such that if

C‖u0 − Pu0‖pH2 < ‖Pu0 −M‖H2 < r

then u0 ∈ Ω′.

Theorem 5.3 Let ϕ be any Gaussian measure of mean M on the manifold of mass

M (i.e., a measure induced by a Gaussian measure of mean 0 on the parallel linear

space) such that the induced distribution of 〈·, cosn0πx〉 has nonzero variance. Let

ϕδ be the scaled Gaussian measure on the manifold defined by

ϕδ(E) = ϕ(

1

δ(E −M) +M

)

for any Borel set E contained in the manifold. If p is as in Theorem 5.2 then

ϕδ(Ω′) = 1− O(δp−1)

as δ → 0.

All of these theorems are precise ways of stating that most choices of initial data

near a constant in the spinodal region produce solutions that eventually exhibit

spinodal decomposition.

CHAPTER 6

THE BIFURCATION DIAGRAM

To this point, this dissertation has provided a description of the initial stages

of phase separation that is mathematically precise. A more complete picture of

the qualitative behavior of solutions of the one-dimensional Cahn-Hilliard equation

would result if all of the steady-state solutions and the connecting orbits between

these equilibria could be found. A typical solution would probably be attracted to a

small neighborhood of an equilibrium and then would remain close to the network

of connecting orbits except for fairly quick transitions between orbits. The first

step in this analysis would be to identify the bifurcation diagram of steady-state

solutions with respect to the parameter ε, i.e., the locus of solutions (ε, u) to

0 = (−ε2u′′ + W ′(u))′

u′(0) = u′(1) = 0∫ 1

0u(x)dx = M (6.1)

in the space R×H2([0, 1]). The mass constraint is important because conservation

of mass implies that connections can only occur between steady-state solutions with

identical masses. In this chapter, certain properties of the bifurcation diagram will

be discussed, and the bifurcation diagrams of some finite-dimensional approxima-

tions to the Cahn-Hilliard equation will be described. At this time, the branching

structure of the bifurcation diagram of the full partial differential equation has not

been rigorously determined. It is hoped that some of the ideas discussed here will be

of help in this regard. With sufficient information about the bifurcation diagram,

61

energy methods or index theory methods may be able to determine which pairs of

equilibria have connecting orbits.

6.1 Self-similarity and Energy

Since the total energy depends on the value of ε, this dependence will be made

explicit in this section by the notation

Eε(u) =∫ 1

0

[W (u(x)) +

ε2

2(u′(x))2

]dx.

Let (ε, u) be a solution pair to (6.1) and suppose that the width of the monotone

transition layers of u is 1/k. Let

v(x)def= u

(x

k

),

and note that if x = ks then

d

dx

(−(kε)2 d

2v

dx2+W ′(v(x))

)=

1

k

d

ds

(−(kε)2 1

k2

d2u

ds2+W ′(u(s))

)=

1

k

d

ds

(−ε2d

2u

ds2+W ′(u(s))

)= 0,

and ∫ 1

0v(x)dx = k

∫ 1/k

0u(s)ds =

∫ 1

0u(s)ds = M.

Thus, (kε, v) is also a solution pair to (6.1). Furthermore,

Ekε(v) =∫ 1

0

W (v(x)) +(kε)2

2

(dv(x)

dx

)2 dx

= k∫ 1/k

0

W (u(s)) +(kε)2

2

(1

k

du(s)

ds

)2 ds

=∫ 1

0

W (u(s)) +ε2

2

(du(s)

ds

)2 ds

62

= Eε(u).

Thus, the bifurcation diagram and the energy along all branches is completely

determined by the locus of monotone solutions to (6.1) and the energy on this

locus.

Now, suppose there is a path (ε, uε) in R × H2([0, 1]) of solutions to (6.1),

parametrized by ε. Recall that uε is a critical point of Eε with respect to the H−1

norm in the set of functions with mass M , since it is an equilibrium of a gradient

flow. Thus, there is a constant C such that for any δ > 0,

|Eε(uε)− Eε(v)| ≤ δ‖uε − v‖H−1 ≤ Cδ‖uε − v‖H2 ,

if uε and v have mass M and are sufficiently close in the H2 norm. This means

that DuεEε(uε), the H2 derivative of Eε at uε, is 0. So

d

dεEε(uε) =

∂

∂εEε(uε) +

⟨DuεEε(uε),

d

dεuε

⟩

= ε∫ 1

0(u′ε(x))2dx+ 0

≥ 0,

with equality only if uε ≡M . In words, the total energy is a monotone nondecreas-

ing function of ε along continua and is strictly increasing if uε 6≡ M .

Because of the scaling properties of the bifurcation diagram, the relation between

the total energies of two points on an ε-parametrized continuum may imply a

relation between the energies of points which do not lie on such a continuum.

Suppose, for example, that u1 and u2 are two solutions of (6.1) for a fixed ε = ε.

For i ∈ 1, 2, let

vi(x)def= ui

(x

ki

),

63

where ki is the number of monotone transition layers in ui. If k1 < k2 and there

is an ε-parametrized path from (k1ε, v1) to (k2ε, v2) then the results of this section

imply that

Eε(u1) = Ek1ε(v1) ≤ Ek2ε(v2) = Eε(u2).

Thus, the steady-state solution with more monotone transition layers has higher

energy.

One last property of the bifurcation diagram to consider is how the number of

monotone transition layers changes along a continuum. Because each nontrivial

steady-state solution of the one-dimensional Cahn-Hilliard equation is made up of

an integer number of such layers, and these layers are identical up to translation

and reflection, it is not hard to see that the only place along a continuum that

this number could change is at the trivial solution u ≡ M , where the number is

undefined. Hence, the periodic structure of equilibria is invariant along nontrivial

branches of the bifurcation diagram.

6.2 Finite-dimensional Approximations

In this section, certain systems of ordinary differential equations will be derived

as approximations of the one-dimensional Cahn-Hilliard equation, and their bifur-

cation diagrams will be described. The first thing to do is to write the partial

differential equation as an infinite system of ordinary differential equations. If

u = M +∑∞k=1 ak cos(kπx), where each ak is a function of t, and the nonlinearity in

the Cahn-Hilliard equation is taken to be W ′(u) = u3− u then u will be a solution

of the Cahn-Hilliard equation satisfying Neumann and no-flux boundary conditions

if

∞∑k=1

a′k cos(kπx) =

−∞∑k=1

ε2(kπ)4ak cos(kπx) +∞∑k=1

(kπ)2ak cos(kπx)

64

−∞∑k=1

3M2(kπ)2ak cos(kπx) +∞∑p=1

∞∑q=1

3Mapaq(cos(pπx) cos(qπx))xx

+∞∑p=1

∞∑q=1

∞∑r=1

apaqar(cos(pπx) cos(qπx) cos(rπx))xx, (6.2)

where ′ denotes differentiation with respect to time. If the formulas

cosA cosB =1

2

2∑j=1

cos(A + (−1)jB)

and

cosA cosB cosC =1

4

2∑j=1

2∑k=1

cos(A + (−1)jB + (−1)kC)

are applied to (6.2) and the equation is then projected onto the linear subspace

spanned by cos(kπx) the resulting equation is

a′k = [−ε2(kπ)4 + (kπ)2 − 3M2(kπ)2]ak

− 3M

2(kπ)2

∑p+q=k

apaq +∑|p−q|=k

apaq

− 1

4(kπ)2

∑p+q+r=k

apaqar +∑

|p+q−r|=kapaqar

− 1

4(kπ)2

∑|p−q+r|=k

apaqar +∑

|p−q−r|=kapaqar

(6.3)

for each positive integer k, where the indices of the sums run over positive integers

only. If time is rescaled as τ = π2t and the definition N = 1− 3M2 is made then,

upon simplification, (6.3) becomes

1

k2a′k = (N − k2ε2π2)ak −

3M

2

∑p+q=k

apaq +∑|p−q|=k

apaq

− 1

4

∑p+q+r=k

apaqar

− 3

4

∑|p+q−r|=k

apaqar

, (6.4)

where the time differentiation is now with respect to τ .

6.2.1 Reduced Equations on a Center Manifold

To study the behavior of (6.4) near a particular bifurcation value of the pa-

rameter ε, an asymptotic analysis of the flow on the center-unstable manifold can

65

be performed. Because of the scaling properties of the bifurcation diagram, the

nature of the bifurcation of any branch from the trivial solution totally describes the

nature of all such bifurcations. Therefore, consider what happens as the bifurcation

parameter ε passes through the first bifurcation value, π−1√N , as the dimension

of the unstable subspace changes from 0 to 1. Define ν = N − ε2π2, and add the

differential equation ν ′ = 0 to the system of equations defined by (6.4), so that at

the origin in this expanded system the center-unstable subspace, which is just the

center subspace in this case, is two-dimensional. In terms of the new parameter,

(6.4) can be written as

1

k2a′k =

[k2ν − (k2 − 1)N

]ak

− 3M

2

∑p+q=k

apaq +∑|p−q|=k

apaq

− 1

4

∑p+q+r=k

apaqar

− 3

4

∑|p+q−r|=k

apaqar

. (6.5)

The center manifold should be locally representable as the graph of a func-

tion from the two-dimensional subspace corresponding to a1, ν to its orthogonal

complement. By implementing the procedure used by Armbruster, Guckenheimer,

and Holmes in their study of the Kuramoto-Sivashinsky equation in [2], a locally

valid analysis of the evolution of a1 on this center manifold can be performed.

This analysis will be presented here. Since the branch of steady-state solutions

bifurcating from the trivial solution at ν = 0 lies on the center manifold, local

information about this branch will be obtained, and the accuracy of this infor-

mation and the neighborhood on which it is valid should increase as the order of

truncation increases. On the center manifold ak = hk(a1, ν) for k ≥ 2, where hk is

a function that is at least second-order in its arguments. (Occasionally it will also

be convenient to define h1 = a1.) The goal here is to derive an evolution equation

for a1 that is valid to fifth order in a1, ν; that is, for k = 1 the right-hand side

66

of (6.5) should be calculated accurate to fifth order in these quantities. For ease of

notation, let O(k) represent a quantity that is O((|ν|+ |a1|)k).

Now the chain rule applied to hk says that

∂hk∂a1

a′1 = a′k. (6.6)

One important consequence of (6.6) is that

hk = O(k) (6.7)

for k ≥ 1. The verification of (6.7) can be done by double induction. For j ≥ k

let Pj,k be the proposition that hj = O(k). Then it is obvious that Pj,k is true for

k ∈ 1, 2 and j ≥ k. Given m ≥ n > 2, suppose that Pj,k holds for all (j, k) in

(j, k) ∈ Z× Z : 1 ≤ k < n, k ≤ j ∪ (j, k) ∈ Z× Z : n = k ≤ j < m,

and suppose that Pm,n is not true, i.e., hm 6= O(n). Let r = r(m) be the smallest

r such that hm has nontrivial monomials of degree r in its expansion in terms of

a1 and ν. By assumption, r < n. Note that the left-hand side of (6.6) with m

substituted for k is O(r+ 1) since a′1 = O(2) and ∂hm/∂a1 = O(r− 1). Hence, the

only monomials of degree r that might appear in the expansion of hm are those of

the same form as those that come from the sums on the right-hand side of (6.5).

The terms in these sums are of the form

hj1hj2

with j1 + j2 ≥ m or of the form

hj1hj2hj3

with j1 + j2 + j3 ≥ m. By the induction hypothesis, hji = O(ji) if ji < n and

hji = O(n− 1) if ji ≥ n. These estimates together with the fact that O(a)O(b) =

O(a+ b) imply that all of the terms in the sums are O(n). Since r < n this means

67

that no monomials of degree r come from the sums, and, therefore, hm has no such

monomials in its expansion. This contradicts the choice of r, and the contradiction

implies the validity of Pm,n. By double induction, Pm,n holds for all m ≥ n ≥ 1. In

particular, (6.7) holds.

The estimates for the reduced equation for a′1 on the center manifold with

increasing orders of accuracy are most easily obtained by an inductive process.

Substituting k = 1 into (6.5) immediately gives

a′1 = νa1 +O(3). (6.8)

By (6.7) and (6.5),

a′1 = νa1 − 3Ma1h2 −3

4a3

1 +O(4), (6.9)

so to increase the truncation order in (6.8) by one it is necessary to approximate

h2 to second order. Using (6.6) with k = 2 and repeating part of the argument in

the verification of (6.7) yields

h2 = Ca21 +O(3)

where C satsifies

−3NC − 3M

2= 0.

Hence,

h2 = −M2N

a21 +O(3). (6.10)

Substituting (6.10) into (6.9) gives

a′1 = νa1 +

(3M2

N− 3

4

)a3

1 +O(4). (6.11)

Now a closer look at (6.5) shows that, in fact,

a′1 = νa1 − 3Ma1h2 −3

4a3

1 +O(5), (6.12)

so the truncation order in (6.11) can be increased to O(5) if h2 is calculated to

third order. Again applying (6.6) with k = 2 implies that the only nontrivial term

68

in the expansion of h2 which is exactly third-order is of the form νa21. Equating

coefficients of this term on each side of (6.6) and solving the resulting equation

gives

h2 = −M2N

a21 −

7M

12N2νa2

1 +O(4). (6.13)

Substituting (6.13) into (6.12) yields

a′1 = νa1 +

(3M2

N− 3

4

)a3

1 +7M2

4N2νa3

1 +O(5). (6.14)

This process can be continued indefinitely to obtain reduced equations for a′1 of

arbitrarily high truncation order. To some extent the method can be automated

using a computer algebra system, such as MAPLE [16]. The next iteration yields

a′1 = νa1 − 3M [a1h2 + h2h3]− 3

4[a3

1 + a21h3 + 2a1h

22] +O(6),

where

h3 =

(3M2

16N2− 1

32N

)a3

1 +O(4)

and

h2 = −M2N

a21 −

7M

12N2νa2

1 +

(7M

32N2− M3

16N3

)a4

1 −49M

72N3ν2a2

1 +O(5).

Hence,

a′1 = νa1 +

(3M2

2N− 3

4

)a3

1 +7M2

4N2νa3

1 +49M2

24N3ν2a3

1

+

(15M4

32N3− 39M2

32N2+

3

128N

)a5

1 +O(6). (6.15)

By ignoring the O(6) terms (6.15) becomes an ordinary differential equation

with a polynomial right-hand side:

a′1 = νa1 +

(3M2

2N− 3

4

)a3

1 +7M2

4N2νa3

1 +49M2

24N3ν2a3

1

+

(15M4

32N3− 39M2

32N2+

3

128N

)a5

1. (6.16)

The bifurcation diagram for (6.16) (with respect to ν) can be computed exactly. In

Figure 6.1 those equilibria which lie on curves connected to the trivial branch are

69

ν

ν

ν

M = 0.5

M = 0.4

M = 0

1a

1

1

a

a

Figure 6.1. Center manifold bifurcation diagrams.

70

plotted for three different values of M and for values of ν corresponding to ε > 0.

Note that as M becomes large (i.e., when the mass nears the edge of the spinodal

region) the local bifurcation at ν = 0 becomes subcritical, but the bifurcating

curve bends back around in the positive ν direction. This agrees with the results

of numerical calculations done by Eilbeck, Furter, and Grinfeld [11]. There are

curves of equilibria for (6.16) that do not arise through bifurcations from the trivial

solution, but these are probably an artifact of the truncation.

If the locus of monotone solutions to (6.1) has one of the two forms depicted

in Figure 6.1 then the results of Section 6.1 imply that the following is true: For

any fixed ε > 0 the corresponding set of nonconstant solutions to (6.1) can be

partitioned into either one or two subsets such that for any two equilibria in a

subset, the one with more transition layers has a higher energy, and equilibria with

the same number of transition layers have the same energy.

6.2.2 Galerkin Approximations

Another method of obtaining a finite system from (6.4) is to simply discard the

equations for a′j and any terms on the right-hand sides of the remaining equations

that involve aj for all j sufficiently large. The solutions of the remaining system

are Galerkin approximations to the original system, since they satisfy

∂u

∂t= −P∆(ε2∆u−W ′(u)) = −∆P (ε2∆u−W ′(u)), (6.17)

where P is a projection onto a finite-dimensional subspace corresponding to, say,

a1, . . . , ak. Since some of the existence proofs for the full partial differential

equation are based on the convergence of these Galerkin approximations in an

appropriate function space (see, e.g., [12]), these finite-dimensional systems should,

in some sense, be good models for the full system.

A precise sense in which these Galerkin approximations are well-behaved can be

seen by recalling the derivation of the Cahn-Hilliard equation in Chapter 2 as the

71

gradient flow of the total energy on a linear manifold in H−1. If that manifold V

is replaced by its submanifold corresponding to the first k Fourier modes then it

is easy to check that (6.17) will be the gradient flow of the total energy on this

submanifold. The main idea is that for a path γ(t) confined to this submanifold

the formula (2.6) for the evolution of energy along γ can be rewritten as

d

dtE(γ(t))

∣∣∣∣∣t=t0

=∫

Ω∇P (ε2∆u0 −W ′(u0)) · ∇z0dx.

This preservation of the energy gradient structure is a very important aspect of the

Galerkin approximations.

Letting ν = N − ε2π2 as in the previous section, the two-dimensional Galerkin

system is

a′1 = νa1 −3

4a3

1 − 3Ma1a2 −3

2a1a

22 (6.18)

a′2 = (16ν − 12N)a2 − 6Ma21 − 6a2

1a2 − 3a32. (6.19)

This system is simple enough that exact calculation of the bifurcation diagram

(with respect to ν) is possible. In Figure 6.2 the three-dimensional bifurcation

diagram for (6.18) and (6.19) is plotted for the same three values of M used in

Figure 6.1. Schematic two-dimensional versions are given in Figure 6.3, and the

local stability of each equilibrium is shown. Because of the small size of this system,

it was possible to calculate the stabilities exactly using MAPLE. As with the plots

for (6.16), the bifurcation becomes subcritical as M becomes large. Note that the

a2-curve bifurcating from the trivial solution develops secondary bifurcations for

certain choices of M . This is not unexpected, since the system (6.18) and (6.19)

should only be a reasonable model for the Cahn-Hilliard equation for a parameter

range near the first two bifurcations from u ≡M . For example, the system clearly

does not capture the later bifurcations from the trivial solution.

72

2

2

2

1

1

1

a

a

a

a

a

a

M = 0

M = 0.4

M = 0.5

ν

ν

ν

Figure 6.2. Two-variable Galerkin bifurcation diagrams.

73

M = 0.5

M = 0.4

M = 0

us

us

usus

us

us

us

us

us

usus

s

s

s

s

s

s

s

s

||u||

||u||

||u||

ν

ν

ν

Figure 6.3. Two-variable Galerkin schematic diagrams; s = stable; us = unstable.

CHAPTER 7

OPEN PROBLEMS

As was shown in Chapter 5, the abstract results from Chapter 4 can be applied

to the Cahn-Hilliard equation on any domain Ω such that ∂Ω is sufficiently smooth

and dim(Ω) ≤ 4. The difficulty is in obtaining sufficient information about the

ω-limit sets of the points on the pseudo-unstable manifold. Suppose, for example,

that

Ω = [0, x0]× [0, y0] ⊂ R2.

The linearization of (1.1) about a constant M in the spinodal region is

ut = −ε2 (uxxxx + 2uxxyy + uyyyy)− β2 (uxx + uyy) (x, y) ∈ Ω (7.1)

ux = uxxx + uyyx = 0 x ∈ 0, x0

uy = uyyy + uxxy = 0 y ∈ 0, y0.

Substituting

u = an,m cosnπx

x0cos

mπy

y0,

a typical eigenfunction of the Laplacian on Ω, into (7.1) yields

a′n,man,m

= −ε2

(nπx0

)4

+ 2(nπ

x0

)2(mπ

y0

)2

+

(mπ

y0

)4+ β2

(nπx0

)2

+

(mπ

y0

)2 .

Thus, an,m(t) = an,m(0) exp(λn,mt), where

λn,m = rn,m(β2 − ε2rn,m),

and

rn,m =(nπ

x0

)2

+

(mπ

y0

)2

= π2

(n2

x20

+m2

y20

).

If x20/y

20 is irrational then, as when Ω = [0, 1], for all but a discrete set of ε, there

will be a unique fastest-growing mode for (7.1). Also, because the Cahn-Hilliard

75

equation has symmetry-preserving properties on rectangles analogous to those on

intervals, the relevant ω-limit points will all have a small-wavelength periodic

structure with characteristic length scale O(ε). However, some information on

the amplitude of the ω-limit points must be obtained in order for the abstract

results to imply spinodal decomposition. It is not obvious that these amplitudes

could not be arbitrarily small, and if that happens then the appearance of spinodal

decomposition has not been justified, since solutions with small amplitude have not

decomposed at all.

If x20/y

20 is rational then the set of ε for which the pseudo-unstable manifold is

one-dimensional may no longer be everywhere dense. When the pseudo-unstable

manifold and subspace are of higher dimension then the smallest subspace that is

invariant under (1.1) and contains the pseudo-unstable subspace is, in general, the

entire space. Thus, for such ε the ω-limit points may not be periodic with small

wavelength. The same problem occurs at a discrete set of ε in the one-dimensional

case. It may be possible to explain the appearance of spinodal decomposition in

these pathological cases by some other methods, such as phase plane analysis on

the pseudo-unstable manifold.

In Chapter 6, finite-dimensional approximations of the Cahn-Hilliard equation

were derived and analyzed. Some results were presented that established a con-

nection between these approximations and the full partial differential equation.

It would be helpful if more rigorous results along these lines could be obtained.

In particular, it would be good to know if the bifurcation diagram for such a

finite-dimensional system is a good representation of the bifurcation diagram for

the full system, at least for a limited parameter range. Such questions have been

addressed for certain equations by Hale, Lin, and Raugel [19], and by Hale and

Raugel [20]. The extent to which such results apply to the Cahn-Hilliard equation

is worth investigating.

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