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Phase-Asymptotic Stability of Transition Front

Solutions in Cahn-Hilliard Equations and Systems

Peter Howard, Texas A&M University

University of Helsinki, September 22, 2011

Outline of the Talk

Introduction Physical motivation: spinodal decomposition Transition fronts and phase-asymptotic stability Overview of recent results The Cahn-Hilliard equation in one space dimension

Spectral analysis and the Evans function The classical semigroup framework Contour analysis The pointwise semigroup framework Local tracking Closing the argument

Multiple space dimensions Cahn-Hilliard systems Further work

Cahn-Hilliard Equations and Systems

For x ∈ Rn and u ∈ R

m, we consider systems of the form

ujt = ∇ ·

m∑

i=1

Mji(u)∇(

(−Γ∆u)i + Fui (u))

.

This is a standard model of certain phase separation processes suchas spinodal decomposition, where the components of u characterizem components of a mixture that contains m+1 components in all.

Here, F ∈ R is a measure of bulk free energy density, M ∈ Rm×m

is a measure of molecular mobility, and Γ ∈ Rm×m characterizes

interfacial energy. Based on physical considerations, we assume M

and Γ are symmetric and positive definite, M uniformly so.

We are interested in the phase-asymptotic stability of transitionfronts u(x) (for n = 1) and u(x1) (for n ≥ 2).

Example Cases

For x ∈ R and u ∈ R, we have the Cahn-Hilliard equation

ut =(

M(u)(−γuxx + F ′(u))x)

x.

For analysis, we often take M(u) ≡ 1, giving

ut = (−γuxx + F ′(u))xx .

For x ∈ Rn and u ∈ R we have respectively

ut = ∇ ·

M(u)∇(−γ∆u + F ′(u))

,

andut = ∆(−γ∆u + F ′(u)).

Spinodal Decomposition

Spinodal decomposition is a phenomenon in which the rapidcooling (quenching) of a homogeneously mixed alloy with m + 1components (e.g., iron, chromium, nickel, etc.) causes separationto occur, resolving the mixture into regions of different crystallinestructure, separated by steep transition layers.

(.25,.35,.40) High temperature

(.55,.20,.25) (.20,.35,.45) (0,.5,.5) Low temperature

Figure: Possible concentrations.

Numerical Simulation

Figure: Numerical simulation for n = 2.

Modeling Spinodal Decomposition

For binary alloys this process can be described by a single equationfor the concentration of a chosen component u. Since u isconserved, we have a conservation law

ut +∇ · J = 0,

where J denotes flux.

Let E (u) denote the total free energy associated with a system inconfiguration u. A standard phenomenological assumption, akin toFourier’s law of heat conduction, is

J = −M(u)∇δE

δu.

This says: the system tends to move from configurations in whichsmall changes in u lead to large changes in E to configurations inwhich small changes in u lead to small changes in E .



ut +∇ · J = 0,



HEAT EQ J = −M(u)∇u HOT → COLD




ut +∇ · J = 0,



J = −M(u)∇δE

δu.


Cahn-Hilliard Energy

In 1958, John W. Cahn and John E. Hilliard suggested the energyE should have the approximate form

E (u) =

ˆ

ΩF (u) +

γ

2|∇u|2dx ,

withδE

δu= F ′(u)− γ∆u.

We expect solutions to evolve toward minimizers of E . Moreprecisely, it’s easy to verify that any solution u(x , t) in anappropriate function class will satisfy

d

dtE (u) = −

ˆ

ΩM(u)

[

(δE

δu)x

]2dx ≤ 0.

Double-well Bulk Free Energy

The Helmholtz free energy is

H = U − TS ⇒ ∂H∂T

= −S .

u u

F(u)

F(u)Temperature Drops

u

u

entropy maximum

1 2

High Temperature

Low Temperature

Figure: Double-well form through quenching.

Intuition about the System

Recall that the energy is

E (u) :=

ˆ

ΩF (u) +

γ

2|∇u|2dx .

One way the system can minimize this is to approach a minimizerof F , but it’s constrained by conservation of mass.

The system can compromise by making transitions from oneminimizer of F to another, but these transitions increase thesecond term in E . The resulting transition fronts are a balancebetween these effects.

Intuition about the System

Recall that the equation is

ut =(


x,

which we can express this as

ut =(

M(u)(−γuxxx

)

x+

(

M(u)F ′′(u))

xux +M(u)F ′′(u)uxx .

For u near a minimizer of F , F ′′(u) > 0 and we expect asymptoticdynamics to be governed by second order diffusion.

Likewise, F will have at least one local maximum, and for u nearthis maximizer F ′′(u) < 0.

Historical Remark on Terminology

The (single) Cahn-Hilliard equation first appeared in John W.Cahn’s paper from 1961, “On spinodal decomposition.”

Cahn-Hilliard systems were first studied by Didier de Fontaine inhis 1967 Northwestern thesis “A computer simulation of theevolution of coherent composition variations in solid solutions,”carried out under the direction of John Hilliard.

de Fontaine explains that Hilliard was never comfortable having hisname on the equation and referred to it himself as “the lastunnumbered equation after equation (18) in Cahn’s 1961 paper.”

Transition Fronts for m = 1

First, since F appears as F ′(u)x , we can replace F with

F (u) = F (u)− au − b,

for any constants a and b. It’s convenient to subtract off asupporting line.

u u1 2

au+b

u u1 2

Figure: Subtraction of the supporting line.

Transition Fronts for m = 1

We say u(x) is a transition front if

0 =(


x,

and

limx→−∞

u(x) = u− = u1 (respectively u2)

limx→+∞

u(x) = u+ = u2 (respectively u1)

u1 6= u2

limx→±∞

u′(x) = 0.

Integrating twice, we find

−γuxx + F ′(u) = 0.

Example Case for m = 1

For M(u) ≡ 1, γ = 1, and F (u) = u2(1− u)2.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Binary Transition Front

x

Con

cent

ratio

n

u1

u2

Figure: Binary Transition Front.

Asymptotic Behavior

Intuitively, we expect that for fairly general initial conditionsu(x , 0) = u0(x) we will find

limt→∞

u(x , t) = u(x)

for some transition front u(x).

Asymptotic Stability

We say u(x) is X → Y stable (for some Banach spaces X and Y )if given any ǫ > 0 there exists η > 0 so that

‖u(x , 0) − u(x)‖X < η ⇒ ‖u(x , t)− u(x)‖Y < ǫ

for all t ≥ 0.

We say u(x) is X → Y asymptotically stable if it is stable andthere exists η0 > 0 sufficiently small so that

‖u(x , 0) − u(x)‖X < η0 ⇒ limt→∞

u(x , t) = u(x)

in Y .

The Shift

Since concentration is conserved, perturbations of a transitionfront u(x) will not generally approach the front itself, but rather(in the case of stability) a shift of the front.

_

u(x)_

u(x)

δ(t)

x

x

u(0,x)

u(t,x)

Figure: The shifted wave.

Phase-Asymptotic Stability

We define our perturbation as

v(x , t) := u(x + δ(t), t) − u(x).

We say u(x) is X → Y phase-stable if there exists a shift functionδ(t) so that for any ǫ > 0 there exists η > 0 so that

‖v(x , 0)‖X < η ⇒ ‖v(x , t)‖Y < ǫ

for all t ≥ 0.

We say u(x) is X → Y phase-asymptotically stable if u(x) isX → Y phase-stable and there exists a shift function δ(t) and avalue η0 > 0 so that

‖v(x , 0)‖X ≤ η0 ⇒ limt→∞

‖v(x , t)‖Y = 0.

Goal

We establish L1∩L∞ → Lp phase-asymptotic stability for all p > 1.

We obtain L1 ∩ L∞ → L1 phase-stability.

Overview of Recent Results, x ∈ Rn, u ∈ Rm

1. For m = 1, n = 1

J. Bricmont, A. Kupiainen, and J. Taskinen, Comm. PureAppl. Math. 1999

E. Carlen, M. Carvalho, and E. Orlandi, Comm. Math. Phys.2001

H., Comm. Math. Phys. 2007

2. For m = 1, n ≥ 2

T. Korvola, Doctoral Thesis, University of Helsinki 2003

T. Korvola, A. Kupiainen, and J. Taskinen, Comm. PureAppl. Math. 2005

H., Physica D 2007

3. For m ≥ 2, n = 1

H. and B. Kwon, Discrete and Continuous Dynamical SystemsA (spectral analysis) and 2011 Preprints

http://www.math.tamu.edu/˜phoward/mathpubs.html

Existence Theorem for m = 1, n = 1

For

ut = (M(u)(−γuxx + F ′(u))x )x

suppose γ > 0, M(u) ≥ M0 > 0, F has a double-well form andM ∈ C 2(R), F ∈ C 4(R).

Then there exist two transition fronts u(x) and u(−x), both ofwhich are strictly monotonic. Aside from tranlations of these, thereare no other transition fronts.

Stability Theorem for m = 1, n = 1

Suppose u(x) is a transition front as described in the existencetheorem. Then for Holder continuous initial conditionsu0(x) ∈ C γ(R), 0 < γ < 1, with

‖u(0, x) − u(x)‖L1 + ‖u(0, x) − u(x)‖L∞ ≤ ǫ,

for ǫ > 0 sufficiently small, there exists a unique solution of (CH)

u ∈C 4+γ,1+ γ

4 (R× (0,∞)) ∩ C γ, γ4 (R× [0,∞))

and a shift δ ∈ C 1+γ[0,∞) so that

‖u(x + δ(t), t)− u(x)‖Lp ≤Cǫ(1 + t)−12(1− 1

p)

|δ(t)− δ∞| ≤Cǫ(1 + t)−1/4.

Pointwise Stability Theorem

Suppose u(x) is a transition front as described in the existencetheorem. Then for Holder continuous initial conditionsu0(x) ∈ C γ(R), γ > 0, with

|u(0, x) − u(x)| ≤ ǫ(1 + |x |)−3/2,


u ∈C 4+γ,1+ γ

4 (R× (0,∞)) ∩ C γ, γ4 (R× [0,∞))


|u(x + δ(t), t)− u(x)| ≤Cǫ[

(1 + t)−1/2e−x2

Lt + (1 + |x |+√t)−3/2

]

|δ(t) − δ∞| ≤Cǫ(1 + t)−1/4.

Linearization

We define our perturbation as

v(x , t) := u(x + δ(t), t) − u(x),

where δ(t) tracks the shift between u(x , t) and u(x). Theperturbation equation is

vt =(

M(u)(−γvxx + F ′′(u)v)x)

x+ δ(t)(ux + vx) + Qx ,

where|Q| ≤ C

[

e−η|x ||v |2 + |v ||vx |+ |v ||vxxx |]

.

This is advantageous because we expect vx and vxxx to decayfaster than v as |x |+ t → ∞. On the other hand, we expect vxand vxxx to blow up respectively like t−1/4 and t−3/4 as t → 0.

The Associated Eigenvalue Problem

The associated linear equation is

vt = Lv :=(

M(u)(−Γvxx + F ′′(u)v)x)

x.

If we look for solutions of the form v(x , t) = eλtφ(x), we obtainthe eigenvalue problem

Lφ = λφ.

The resolvent for this problem is

R(λ; L) := (λI − L)−1,

and we say λ ∈ C is in the resolvent set of L if R(λ; L) is abounded linear operator.

Spectrum of L: σ(L) = σpt ∪ σess

We consider two (not necessarily disjoint) sets of spectrum. Thepoint spectrum is

σpt := λ ∈ C : Lφ = λφ for some φ ∈ H2.

We refer to elements of σpt as eigenvalues. By essential spectrumσess , we mean any value λ ∈ C that is not in the resolvent set andis not an isolated eigenvalue with finite multiplicity.

Roughly, eigenvalues characterize local (transitional) behavior ofthe transition front, while the essential spectrum characterizesendstate behavior.

The Essential Spectrum

The essential spectrum is determined by the asymptotic(x → ±∞) eigenvalue equations

−M(u±)γφxxxx + F ′′(u±)φxx = λφ.

It corresponds with solutions

φ(x) = e iξx ,

so thatλ = −M(u±)γξ

4 − F ′′(u±)ξ2.

By positivity of M and the double-well structure of F ,

σess = (−∞, 0].

The Point Spectrum

First, for λ 6= 0, if φ(x ;λ) solves(

M(u)(−γφxx + F ′′(u)φ)x)

x= λφ,

we can show by integrating both sides that´ +∞−∞ φ(x ;λ)dx = 0.

We set

ϕ(x ;λ) :=

ˆ x

−∞φ(y ;λ)dy ,

so that the integrated eigenvalue problem is

M(u)(−γϕxxx + F ′′(u)ϕx )x = λϕ.

Divide by M(u), multiply by ϕ and integrate to find

−〈ϕx ,−γϕxxx + F ′′(u)ϕx 〉 = λ〈 ϕ

M(u), ϕ〉.

Here, 〈·, ·〉 denotes L2 inner product.

The Eigenvalues λ 6= 0

We can express this last equation as

−〈ϕx ,Hϕx 〉 = λ〈 ϕ

M(u), ϕ〉,

where H is the Schrodinger type operator

H := −γ∂2x + F ′′(u).

NowHu′ = 0,

and u′ has a fixed sign by monotonicity, so by standard secondorder theory H is a non-negative operator. Noting also that H isself-adjoint, we conclude

λ ≤ 0.

The Eigenvalue λ = 0

First, by definition

−γuxx + F ′(u) = 0 ⇒ −γuxxx + F ′′(u)ux = 0

so that φ = ux solves

−γφxx + F ′′(u)φ = 0.

We see that(

M(u)(−γφxx + F ′′(u)φ)x)

= 0,

so that λ = 0 is an eigenvalue with associated eigenfunction ux .

The Evans Function

We characterize the eigenvalue λ = 0 in terms of the Evansfunction.

The eigenvalue problem Lφ = λφ has four linearly independentsolutions. For Arg λ 6= π, we can construct these so that twodecay as x → −∞ and two grow as x → −∞. Alternatively, wecan construct these so that two decay as x → +∞ and two growas x → +∞. We denote the associated decaying solutionsφ+1 (x ;λ), φ

+2 (x ;λ), φ

−1 (x ;λ), and φ−

2 (x ;λ).

In this context, the Evans function is the Wronskian

D(λ) := det

φ−1 φ−

2 φ+1 φ+

2

φ−′

1 φ−′

2 φ+′

1 φ+′

2

φ−′′

1 φ−′′

2 φ+′′

1 φ+′′

2

φ−′′′

1 φ−′′′

2 φ+′′′

1 φ+′′′

2

∣

∣

∣

x=0.

The Evans Function

Any eigenfunction of L must decay at both −∞ and +∞, and soany such eigenfunction must, for some constants α1, α2, β1, andβ2 satisfy

α1φ−1 (x ;λ) + α2φ

−2 (x ;λ) = φ(x ;λ) = β1φ

+1 (x ;λ) + β2φ

+2 (x ;λ).

In this case D(λ) = 0, so D serves as a characteristic function forL. This function is not analytic in λ, but is analytic in the variableρ :=

√λ. We let Da(ρ) denote this analytic function, and we find

Da(0) = 0 and D ′a(0) = 0. Using the uniqueness of u(x) (to the

extent specified by our existence theorem), we can verify

D ′′a (0) 6= 0.

The Classical Semigroup Framework

We can write our full perturbation equation as

vt = Lv + δ(t)(ux + vx) + Qx .

For an initial perturbation v(x , 0), we can express v(x , t) insemigroup formalism as

v(x , t) = eLtv0 +

ˆ t

0eL(t−s)

[

δ(s)u′(y) + δ(s)vy (y , s) + Qy

]

ds,

where by Laplace transform

eLt :=1

2πi

ˆ

ΩeλtR(λ; L)dλ.

Here, Ω denotes a contour in the resolvent set of L, entirely to theright of σ(L), so that arg λ → ±θ as |λ| → ∞ for some θ ∈ (π2 , π).In addition we can move Ω as allowed by Cauchy’s Theorem.

Spectrum and Contour

Clearly, if σ(L) ⊂ (−∞,−κ] for some κ > 0 then eLt will decay atexponential rate in t.

Re λ

Im λ

Essential spectrum

Eigenvalues

Ω

Figure: Example of a clearly stable spectrum.


By shift invariance, there will be an eigenvalue at λ = 0.

Ω

Re λ

Im λ

Essential spectrum

Eigenvalues

Figure: Neutral eigenvalue at λ = 0.


We can accomodate the eigenvalue at λ = 0 by separating out aterm that does not decay in time.

Ω

Re λ

Im λ

Essential spectrum

Eigenvalues

Figure: Accomodating the neutral eigenvalue.


For Cahn-Hilliard systems σess = (−∞, 0].

Re λ

Im λ

Essential spectrum

Eigenvalues

Ω

Figure: Essential spectrum for Cahn-Hilliard systems.

The Pointwise Semigroup Framework

Let G (x , t; y) denote a Green’s function (distribution) for vt = Lv :

Gt = LG

G (x , t; y) = δy (x).

We can express our semigroup operator eLt as

eLt f =

ˆ +∞

−∞G (x , t; y)f (y)dy .

Our expression for v becomes

v(x , t) =

ˆ +∞

−∞G (x , t; y)v0(y)dy + δ(t)u′(x)

+

ˆ t

0

ˆ +∞

−∞G (x , t − s, y)

[

δ(s)v(y , s) + Q]

ydy .

Constructing G

We compute the Laplace transform of G (t → λ), writingLG = Gλ(x ; y), so that

LGλ − λGλ = −δy (x).

Inverting, we find

G (x , t; y) =1

2πi

ˆ

ΩeλtGλ(x ; y)dλ,

where Ω denotes the same contour previously described.

The Choice of Splitting

The main step in the analysis consists of deriving the splitting

G (x , t; y) = u′(x)e(t; y) + G (x , t; y),

where u′(x)e(t; y) is a leading order term associated with λ = 0that does not decay as t → ∞ and G(x , t; y) decays roughly like aheat kernel.

Intuitively, u′(x)e(t; y) captures behavior associated with λ = 0,while G captures behavior associated with essential spectrum.

Choosing the Local Shift

We find (substituting G = u′e + G and integrating by parts)

v(x , t) =

ˆ +∞

−∞G(x , t; y)v0(y)dy + δ(t)u′(x)

−ˆ t

0

ˆ +∞

−∞Gy (x , t − s; y)

[

δ(s)v(y , s) + Q]

dyds

+u′(x)

ˆ +∞

−∞e(t; y)v0(y)dy

−ˆ t

0

ˆ +∞

−∞ey (t − s; y)

[

δ(s)v(y , s) + Q]

dyds

.

Choosing the Local Shift

We take

δ(t) = −ˆ +∞

−∞e(t; y)v0(y)dy

+

ˆ t

0

ˆ +∞

−∞ey (t − s; y)

[

δ(s)v(y , s) +Q]

dyds,

which leaves

v(x , t) =

ˆ +∞

−∞G (x , t; y)v0(y)dy

−ˆ t

0

ˆ +∞

−∞Gy(x , t − s; y)

[

δ(s)v(y , s) + Q]

dyds.

We obtain similar integral equations for δ(t) and vx(x , t), making4 equations (3 since δ only appears as δ).

Green’s Function Estimates

Using our spectral information and the structure of our equation,we find for t ≥ 1

e(t; y) = c

ˆ

y√4γM−t

−∞e−z2dz + R(t; y)

|R(t; y)| ≤Ct−1/2e−y2

At ,

and for t ≥ 1 and |x − y | ≥ Kt

|G (x , t; y)| ≤ Ct−1/2e−(x−y)2

At .

Here, c , C , K , η, and A are fixed positive constants dependingonly on the spectrum of L and the structure of the equations.

Integral Equations

We need to recall our system of integral equations:

δ(t) = −ˆ +∞

−∞et(t; y)v0(y)dy

+

ˆ t

0

ˆ +∞

−∞ety (t − s; y)

[

δ(s)v(y , s) + Q]

dyds,

v(x , t) =

ˆ +∞

−∞G (x , t; y)v0(y)dy

−ˆ t

0

ˆ +∞

−∞Gy(x , t − s; y)

[

δ(s)v(y , s) + Q]

dyds

vx(x , t) =

ˆ +∞

−∞Gx(x , t; y)v0(y)dy

−ˆ t

0

ˆ +∞

−∞Gxy(x , t − s; y)

[

δ(s)v(y , s) + Q]

dyds.

Linear Stability

Using the estimates we obtain on G , we can verify that∣

∣

∣

ˆ +∞

−∞et(t; y)v0(y)dy

∣

∣

∣≤C (1 + t)−1

∥

∥

∥

ˆ +∞

−∞G (x , t; y)v0(y)dy

∥

∥

∥

Lp≤C (1 + t)

− 12(1− 1

p)

∥

∥

∥

ˆ +∞

−∞Gx(x , t; y)v0(y)dy

∥

∥

∥

Lp≤Ct−1/4(1 + t)−

14 .

Bounding Function ζ(t)

We take these values as preliminary estimates and set

ζ(t) := sup0≤s≤t1≤p≤∞

‖v(·, s)‖Lp (1 + s)12(1− 1

p)

+ ‖vx(·, s)‖Lp s1/4(1 + s)1/4 + |δ(s)|(1 + s)

.

Clearly,

‖v(·, s)‖Lp ≤ ζ(t)(1 + s)−12(1− 1

p)

‖vx(·, s)‖Lp ≤ ζ(t)s−1/4(1 + s)−1/4

|δ(s)| ≤ ζ(t)(1 + s)−1.

Nonlinear Stability

Upon substitution of these bounds into our integral equations, wefind

ζ(t) ≤ C (ǫ+ ζ(t)2).

Here C is a new constant and we recall

‖v(·, 0)‖L1 + ‖v(·, 0)‖L∞ ≤ ǫ.

It’s straightforward to verify that this implies

ζ(t) < 2Cǫ

for all t ≥ 0. This inequality is equivalent with the statement ofour theorem.

Multiple Space Dimensions

For x ∈ Rn the Cahn-Hilliard equation is

ut = ∇ · M(u)∇(−γ∆u + F ′(u)).

In this case, we consider planar transition fronts u(x1). Since u

depends only on x1, planar transition fronts have the samestructure as transition fronts for n = 1. In this case, the shift δ can

depend on the transverse variables x := (x2, x3, . . . , xn), and wedefine our perturbation variable as

v(x , t) := u(x , t)− u(x1 − δ(x , t)).

The Shift For n = 2

x

x

1

2Layer

δ

δ

n=2Transition

(x ,t )2

(x ,t )2

Figure: Shift function δ(x2, t) for n = 2.

Linearization

Upon substitution of

v(x , t) := u(x , t)− u(x1 − δ(x , t))

into the Cahn-Hilliard equation we obtain

(∂t − L)v = (∂t − L)(δ(x , t)u′(x1)) +∇ ·Q,

where

Lv =∇ · M(u)∇(−γ∆v + F ′′(u)v)Q =Q(u, δ, v).


The relevant eigenvalue problem is Lφ = λφ. We take a Fouriertransform in the transverse variable x (transforming x → ξ):

Lξφ := −DξHξφ = λφ.

Here

Dξ := − ∂x1M(u)∂x1 + |ξ|2M(u)

Hξ := − γ∂x1x1 + (F ′′(u) + γ|ξ|2).

Using asymptotic considerations again, we can verify that theessential spectrum for Lξ is

σess =(

−∞,−M(u±)F′′(u±)|ξ|2 − γM(u±)|ξ|4

]

.

The Point Spectrum for ξ 6= 0

For ξ 6= 0 we set ϕ := D−1/2ξ φ (akin to integrating for n = 1) so

thatLξϕ := D

1/2ξ HξD

1/2ξ ϕ = −λϕ.

We can take advantage of the observation that Lξ is self-adjoint toverify that for M0 := minx1∈R M(u(x1))

σpt ⊂ (−∞,−γM0|ξ|4].

Using Evans function techniques, we can show that for |ξ|sufficiently small, the leading (right-most) eigenvalue of Lξ satisfies

λ∗(ξ) = −λ3|ξ|3 +O(|ξ|4),

where

λ3 =

√2γ(M(u−) +M(u+))

|u+ − u−|2ˆ max u−,u+

min u−,u+

√

F (x)− F (u−)dx .

Notes on The Scaling

The cubic scaling for the leading eigenvalue was anticipated by thephysical observation that if S denotes the average pattern sizeduring spinodal decomposition then asymptotically

S ∼ t1/3.

The associated behavior of the leading eigenvalue seems first tohave been recognized by David Jasnow and R. K. P. Zia in 1987.

Stability Theorem for m = 1, n = 2

Suppose u(x1) is a planar transition front as described in theexistence theorem. Then for Holder continuous initial conditionsu0(x) ∈ C γ(Rn), 0 < γ < 1, with

‖u(0, x) − u(x1)‖L1x≤ ǫ(1 + |x1|)−3/2,


u ∈C 4+γ,1+ γ

4 (Rn × (0,∞)) ∩ C γ, γ4 (Rn × [0,∞))

and a shift δ ∈ C 3+γ,1+γ(

Rn−1 × [0,∞))

so that for n = 2

‖u(x , t)− u(xx − δ(x , t))‖Lpx ≤Cǫ[

(1 + t)−(1− 1

p)

+(1 + t)− 5

6(1− 1

p)− 1

6 log(e + t)]

‖δ(x , t)‖Lpx≤Cǫ(1 + t)−

13(1− 1

p).

Stability Theorem for m = 1, n ≥ 3

Suppose u(x1) is a planar transition front as described in theexistence theorem. Then for Holder continuous initial conditionsu0(x) ∈ C γ(Rn), 0 < γ < 1, with

‖u(0, x) − u(x1)‖L1 + ‖u(0, x) − u(x1)‖L∞ ≤ ǫ,


u ∈C 4+γ,1+ γ

4 (Rn × (0,∞)) ∩ C γ, γ4 (Rn × [0,∞))

and a shift δ ∈ C 3+γ,1+γ(

Rn−1 × [0,∞)

)

so that for all σ > 0

sufficiently small,

‖u(x , t) − u(x − δ(x , t))‖Lp ≤Cǫ[

(1 + t)−n2(1− 1

p)

+(1 + t)−( n−1

3+ 1

2)(1− 1

p)− 1

6+σ

]

‖δ(x , t)‖Lpx≤Cǫ(1 + t)

− n−13

(1− 1p)+σ

.

Systems for n = 1

For u ∈ Rm and x ∈ R

n, we have systems of the form

ut =(

M(u)(−Γuxx + F ′(u))x)

x,

where we expect F to have m + 1 distinct minimizers ξjm+1j=1

associated with energy-preferred phases of the mixture. A commonexample for m = 2 is

F (u) = u21u22 + u21(1− u1 − u2)

2 + u22(1− u1 − u2)2,

with minimizers (0, 0), (1, 0), and (0, 1).

In this case, existence of transition fronts is more complicated,because u(x) solves the system

−Γuxx + F ′(u) = 0.

Existence of Transition Fronts

The existence of transition fronts for Cahn-Hilliard systems hasbeen established under quite general conditions by:

N. D. Alikakos, S. I. Betelu, and X. Chen (2006): for m = 2,using complex analysis

N. D. Alikakos and G. Fusco (2008): m ≥ 2, for Γ = I

V. Stefanopoulos (2008): m ≥ 2, for Γ positive definite andsymmetric

In each of these references, the transition fronts arise as minimizersof the associated energy functional

E (u) =

ˆ +∞

−∞F (u) +

1

2〈ux , Γux 〉dx .

Example Case

For m = 2, M(u) ≡ I , Γ = I , and

F (u) = u21u22 + u21(1− u1 − u2)

2 + u22(1− u1 − u2)2.

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ternary Transition Fronts

x

Con

cent

ratio

n

u1

u2

1−u1−u

2

Figure: Ternary Transition Front.


We linearize as in the case of a single equation with

v(x , t) := u(x + δ(t), t) − u(x).

The associated eigenvalue problem is

Lφ :=(

M(u)(−Γφxx + F ′′(u)φ)x)

x= λφ.

The fact that u(x) minimizes E can be used to verify

σ(L) ⊂ (−∞, 0].

The primary difficulty involves the Evans function condition atλ = 0.

The Evans Function

Our eigenvalue problem(

M(u)(−Γφxx + F ′(u)φ)x)

x= λφ

has 2m solutions that decay as x → −∞, φ−j 2mj=1 and 2m

solutions that decay as x → +∞, φ+j 2mj=1. We will set

Φ±j :=

φ±j

φ±′

j

φ±′′

j

φ±′′′

j

, Φ± := (Φ±1 , . . . ,Φ

±2m).

The Evans function is

D(λ) = det(Φ+(0;λ),Φ−(0;λ)).

Characteristics of the Evans Function

Recall that any eigenfunction φ must be in H2, and so it mustdecay at both ±∞. In this way, there must exist constants αj2mj=1

and βj2mj=1 so that

2m∑

j=1

αjφ−j (x ;λ) = φ(x ;λ) =

2m∑

j=1

βjφ+j (x ;λ).

By linear dependence D(λ) = 0. We know λ = 0 is an eigenvalue,and it follows that D(0) = 0.

We would like to characterize this eigenvalue further by computingD ′(0), D ′′(0), etc. However, D is not differentiable at λ = 0.

Analyticity of the Evans Function

The solutions φ−j 2mj=1 and φ+

j 2mj=1 have the form

φ−j (x ;λ) = e

µ−2m+j

(λ)x (r−2m+j +O(e−η|x |))

φ+j (x ;λ) = e

µ+j(λ)x (r+j +O(e−η|x |)),

where for j = 1, . . . ,m

µ±j (λ) = −

√

ν±m+1−j +O(|λ|)

µ±m+j(λ) = −

√

λ

β±j

+O(|λ|3/2)

µ±2m+j(λ) =

√

λ

β±m+1−j

+O(|λ|3/2)

µ±3m+j(λ) =

√

ν±j +O(|λ|).

The Stability Condition

We can view the Evans function as an analytic function of ρ =√λ.

We denote this function Da(ρ). It is straightforward to verify that

Da(0) = D ′a(0) = · · · = D

(m)a (0) = 0.

Our stability condition is

dm+1Da

dρm+1(0) 6= 0.

For m = 1 (the case of a single equation) it’s easy to verify thatthis holds under standard physical assumptions. For m ≥ 2, wehave developed a framework for verifying this condition on acase-by-case basis.

Spectral Stability

It’s relatively easy to verify that

σess = (−∞, 0].

If u minimizes the Cahn-Hilliard energy, it’s easy to show that

σpt(L)\0 ⊂ (−∞,−κ]

for some κ > 0. If these conditions both hold, along with ourstability condition

dm+1Da

dρm+1(0) 6= 0.

we say u(x) is spectrally stable.

Stability Theorem

Suppose u(x) is a spectrally stable transition front. Then forHolder continuous initial conditions u0(x) ∈ C γ(R), 0 < γ < 1,with

‖u(0, x) − u(x)‖L1 + ‖u(0, x) − u(x)‖L∞ ≤ ǫ,


u ∈C 4+γ,1+ γ

4 (R× (0,∞)) ∩ C γ, γ4 (R× [0,∞))


‖u(x + δ(t), t)− u(x)‖Lp ≤Cǫ(1 + t)−12(1− 1

p)

|δ(t)− δ∞| ≤Cǫ(1 + t)−1/4.

Further Work

Extension to the case x ∈ Rn, u ∈ R

m, n ≥ 2, m ≥ 2

Periodic and pulse-type stationary solutions for m ≥ 2

Genuinely multidimensional stationary solutions for n ≥ 2

Coarsening dynamics

Navier-Stokes-Cahn-Hilliard systems

The functionalized Cahn-Hilliard equation

ut = ∆

(ǫ2∆−F ′′(u)−ǫη1)(ǫ2∆u−F ′(u))+ǫ(η2−η1)F

′(u)

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