ANALYSIS AND APPROXIMATION OF A FRACTIONAL CAHN … · 2.1. Stability. The Allen-Cahn equation (α = 0) fails to preserve mass. The next results show that several properties of the

ANALYSIS AND APPROXIMATION OF A FRACTIONAL

CAHN-HILLIARD EQUATION

MARK AINSWORTH AND ZHIPING MAO

Abstract. We derive a Fractional Cahn-Hilliard Equation (FCHE) by consid-ering a gradient flow in the negative order Sobolev space H−α, α ∈ [0, 1] where

the choice α = 1 corresponds to the classical Cahn-Hilliard equation whilst thechoice α = 0 recovers the Allen-Cahn equation. It is shown that the equationpreserves mass for all positive values of fractional order α and that it indeedreduces the free energy. The well-posedness of the problem is established inthe sense that the H1-norm of the solution remains uniformly bounded. Wethen turn to the delicate question of the L∞ boundedness of the solution andestablish an L∞ bound for the FCHE in the case where the non-linearity is aquartic polynomial. As a consequence of the estimates, we are able to showthat the Fourier-Galerkin method delivers a spectral rate of convergence forthe FCHE in the case of a semi-discrete approximation scheme. Finally, wepresent results obtained using computational simulation of the FCHE for avariety of choices of fractional order α. It is observed that the nature of thesolution of the FCHE with a general α > 0 is qualitatively (and quantita-tively) closer to the behaviour of the classical Cahn-Hilliard equation than tothe Allen-Cahn equation, regardless of how close to zero be the value of α. Anexamination of the coarsening rates of the FCHE reveals that the asymptoticrate is rather insensitive to the value of α and, as a consequence, is close tothe well-established rate observed for the classical Cahn-Hilliard equation.

1. Introduction

A simple model for the phase separation of a binary alloy at a fixed temperatureis given by the Cahn-Hilliard equation [5]

∂u

∂t+ (−∆)(−ε2∆u+ F ′(u)) = 0 in Ω,

where u is an order parameter, ε is a length scale parameter and F (s) = 14 (1− s2)2

is of bistable type, admitting two local minima. The mass in the system is measuredby the quantity ∫

Ω

u(x, t)dx

and remains constant in the case of the Cahn-Hilliard equation as one would expectfrom physical considerations.

1991 Mathematics Subject Classification. 65N12, 65N30, 65N50.Key words and phrases. Fractional Cahn-Hilliard equation, Mass conservation, Stability, L∞

boundness, Fourier spectral method, Error estimates.This work was supported by the MURI/ARO on “Fractional PDEs for Conservation Laws and

Beyond: Theory, Numerics and Applications” (W911NF-15-1-0562).

1

2 MARK AINSWORTH AND ZHIPING MAO

The non-equilibrium state of the system can be expected to evolve in such a waythat the free energy functional

E(u) =

∫

Ω

(ε2

2|∇u|2 + F (u)

)(1)

decreases in time and approaches a minimum. The Cahn-Hilliard equation can beviewed as a gradient flow [12]

∂u

∂t∝ −∇V E(u).

where ∇V E denotes the gradient of E, and V is an approximately chosen Hilbertspace with associated norm (•, •)V . The gradient of E at a point u ∈ V is definedby the equation

(∇V E(u), v)V =d

dsE(u + sv)|s=0 .

Formal computations (ignoring boundary terms for the moment) reveal that theright hand is given by

d

dsE(u+ sv)|s=0 = (ε2∇u,∇v)0 + (F ′(u), v)0 = (−ε2∆u+ F ′(u), v)0

where (•, •)0 denotes the usual L2(Ω)-norm.As a consequence, if one chooses V to be the space L2(Ω), then the gradient is

given by

∇0E(u) = −ε2∆u+ F ′(u)

and the associated gradient flow is given by the Allen-Cahn equation [1]:

∂u

∂t+ (−ε2∆u+ F ′(u)) = 0 in Ω.

Unfortunately, the Allen-Cahn equation fails to preserve mass. Suppose instead,that we choose V to be the space H−1(Ω) equipped with the norm defined by

(v, w)−1 = ((−∆)−1/2v, (−∆)−1/2w)0.

The gradient with respect to the space H−1 is given by

∇−1E(u) = (−∆)(−ε2∆u+ F ′(u))

and the associated gradient flow is the Cahn-Hilliard equation given above which,as mentioned earlier, does preserve the mass.

The derivation of the Cahn-Hilliard equation as a gradient flow in the dual spaceH−1 may appear somewhat ad hoc. Why not, for instance, look at the gradientflow in the space H−α where α > 0 instead? Taking the norm on the space H−α

to be defined by

(v, w)−α = ((−∆)−α/2v, (−∆)−α/2w)0

results in a gradient given by

∇−αE(u) = (−∆)α(−ε2∆u+ F ′(u)).

The fractional order operator (−∆)α is defined below, while the justifications forthe above steps are provided in the Appendix. The associated gradient flow in thegeneral case α > 0 will be referred to as the Fractional Cahn-Hilliard Equation

(FCHE).

ANALYSIS AND APPROXIMATION FOR FCHE 3

We consider the following FCHE for 0 ≤ α ≤ 1 with periodic boundary condition:

∂u

∂t(x, t) + (−)α(−ε2u(x, t) + f(u(x, t))) = 0, (x, t) ∈ Ω× (0, T ],

u(·, t) is 2π-periodic for all t ∈ (0, T ],

u(x, 0) = u0(x), x ∈ Ω,

(2)

where Ω = (0, 2π)2, ε is a positive constant, x = (x1, x2), and f(u) = F ′(u) = u3−u.The choice α = 1 corresponds to the classical Cahn-Hilliard equation whilst thechoice α = 0 gives the Allen-Cahn equation.

In order to define the fractional Laplacian (−)α, we use the Fourier decompo-sition. For any u ∈ L2

per(Ω),

u(x) =∑

k,l∈Z

ukleikx1+ilx2 , (3)

where i2 = −1, and the Fourier coefficients are given by

ukl = 〈u, eikx1+ilx2〉 = 1

(2π)2

∫

Ω

ueikx1+ilx2dx.

The fractional Laplacian is then defined by

(−)αu =∑

k,l∈Z

(k2 + l2)αukleikx1+ilx2 . (4)

Fractional order Allen-Cahn and Cahn-Hilliard models have been considered byother authors. A fractional extension of a mass-conserving Allen-Cahn phase fieldmodel has been studied in [23]. Bosch and Stoll [2] proposed a fractional inpaint-ing model based on a fractional order vector-valued Cahn-Hilliard equation. TheFourier spectral method has been widely used to numerically approximate the solu-tion of the classical Cahn-Hilliard equation [9,10,15,16,24] and to study the coars-ening dynamics [8, 17, 26]. Alternative approaches include the Legendre spectralGalerkin [23, 25] or the Petrov Galerkin method with general Jacobi functions [20]for Riesz fractional partial differential equations. The Fourier spectral method is anatural choice for the approximation of problems where the fractional derivative isdefined by spectral decomposition and the fact that (at least for the linear terms)the operator is diagonal offers possibilities for efficient solvers [3].

In the present work, we analyse the FCHE (2) arising as a gradient flow of thefree energy functional E in the negative order fractional space H−α. In particular,we show that the equation preserves mass for all positive values of fractional order αand that it indeed reduces energy. The well-posedness of the problem is establishedin the sense that the H1-norm of the solution remains uniformly bounded. We thenturn to the more interesting, and delicate, question of the L∞ boundedness of thesolution.

The uniform L∞-boundedness of the solution is of particular interest both fromphysical and computational considerations. For instance, L∞-boundedness is usedin order to prove the convergence of the Fourier-Galerkin scheme for the approxi-mation of the classical Cahn-Hilliard equation where a key step requires a uniformbound on the quantity max |F ′′(s)| where s lies between u(t) and the Fourier-Galerkin approximation uN (t). Previous work dealing with the convergence of nu-merical schemes for approximating the classical Cahn-Hilliard equation has assumed

that such an estimate holds in the analysis [11], or have simply assumed that F has


quadratic growth [22] (giving the uniform estimate). Caffarelli et. al. [4] showedthat the solution of the classical Cahn-Hilliard equation remains bounded in L∞

in the case where the non-linear term F is of quadratic type. Subsequently, He et.al. [15] weakened the hypothesis to allow the case when F is a quartic polynomialand used it to establish the convergence of the Fourier-Galerkin approxmation inthe case of the classical Cahn-Hilliard equation.

The analysis of the uniform L∞-boundness in the case of the FCHE is moredelicate because the underlying regularity of the solution is only H1+α rather thanthe H2-regularity for the classical equation. This creates technical issues whenattempting to use the Gagliardo-Nirenberg type estimate as in [15] to obtain theL∞-boundedness. Nevertheless, we succeed in establishing an L∞ bound for theFCHE in the case where F is a quartic polynomial. The same estimate holds for theFourier spectral approximation using the same arguments for the finite dimensionalsubspace. Combining the L∞ estimate for the infinite dimensional case leads to auniform estimate for max |F ′′(s)| where s lies between u(t) and the Fourier-Galerkinapproximation uN(t) without any additional assumptions on the problem or thedata.

As a consequence of the estimates, we are able to show that the Fourier-Galerkindelivers a spectral rate of convergence for the FCHE in the case of a semi-discreteapproximation scheme. We show that the same estimate holds for a simple fullydiscrete scheme. Finally, we present results obtained using computational simu-lation of the FCHE for a variety of choices of fractional order α. It is observedthat the nature of the solution of the FCHE with a general α > 0 is qualitatively(and quantitatively) closer to the behavior of the classical Cahn-Hilliard equationthan to the Allen-Cahn equation, regardless of how close to zero be the value of α.An examination of the coarsening rates of the FCHE reveals that the asymptoticrate is rather insensitive to the value of α and, as a consequence, is close to thewell-established rate observed for the classical Cahn-Hilliard equation.

2. Properties of Fractional Cahn-Hilliard Equation

Let C∞per(Ω) be the set of all restrictions onto Ω of all real-valued, 2π-periodic,

C∞-functions on R2. For each s ≥ 0, let Hs

per(Ω) be the closure of C∞per(Ω) in

the usual Sobolev norm ‖ · ‖s with ‖u‖2s =∑

k,l∈Z

u2kl(1 + k2 + l2)s. Note that

H0per(Ω) = L2

per(Ω). In particular, we use ‖ · ‖ to denote the usual L2-norm ‖ · ‖0.The weak form of (2) is obtained in the usual way by multiplying a test func-

tion v ∈ H1+αper (Ω) on both sides of (2) and integrating over Ω. Using fractional

integration by parts (see Lemma 8), we arrive at the weak formulation of (2): findu ∈ H1+α

per (Ω), such that

(∂u∂t

, v)+(ε2(−)(1+α)/2u, (−)(1+α)/2v

)+(f(u), (−)αv

)= 0, ∀v ∈ H1+α

per (Ω).

(5)

2.1. Stability. The Allen-Cahn equation (α = 0) fails to preserve mass. The nextresults show that several properties of the solution of the standard Cahn-Hilliardequation are also enjoyed by the Fractional Cahn-Hilliard equation.

Lemma 1. Let 0 < α ≤ 1, then the FCHE is mass conserving, i.e.,∫Ω u(x, t)dx is

a constant for all t ≥ 0.


Proof. Take v ≡ 1 in equation (5), then since α > 0, we have (−)αu = 0, so

d

dt(u, 1) =

d

dt

∫

Ω

udx = 0.

It follows that the mass is conserved, i.e.,∫

Ω

u(x, t)dx =

∫

Ω

u(x, 0)dx =

∫

Ω

u0(x)dx.

The next result shows that FCHE, like the Allen-Cahn and Cahn-Hilliard equa-tions, does not result in an increase in the free energy functional (1).

Lemma 2. The energy is non-increasing for all 0 ≤ α ≤ 1:

E(u(t)) ≤ E(u0), ∀ t ≥ 0. (6)

Proof. Direct computation gives

d

dtE(u) =

d

dt

∫

Ω

(ε22|∇u|2 + F (u)

)dx =

(ε2∇u,∇∂u

∂t

)+(f(u),

∂u

∂t

)

=(ε2(−)u,

∂u

∂t

)+(f(u),

∂u

∂t

)=

(ε2(−)u+ f(u),

∂u

∂t

)

=−(ε2(−)u+ f(u), (−)α(ε2(−)u+ f(u))

)

=− ‖(−)α/2(ε2(−)u+ f(u))‖2 ≤ 0.

and the result follows.

We can now establish the stability and continuous dependence of the weak solu-tion as follows:

Theorem 1. The solution of FCHE is stable and satisfies for all 0 < α ≤ 1

‖u(t)‖1 ≤ C(1 + ‖u0‖1), ∀ t > 0, (7)

where C is a positive constant independent of t.

Proof. Observe that F satisfies

s4

8− 1

4≤ F (s) ≤ 3s4

8+

3

4, s ∈ R,

and hence,

1

2ε2‖∇u(t)‖2 + 1

8

∫

Ω

|u(t)|4dx ≤ E(u(t)) +1

4, ∀ t > 0.

Using the fact that the energy is non-increasing gives

E(u(t)) +1

4≤ E(u0) +

1

4≤ 1

2ε2‖∇u0‖2 +

3

8

∫

Ω

|u0|4dx+ 1.

Thanks to the following Sobolev inequality, valid for all s > 12 ,

‖v‖L4 ≤ C‖v‖s, ∀ v ∈ Hsper(Ω),

we arrive at

1

2ε2‖∇u(t)‖2 ≤ 1

2ε2‖∇u0‖2 +

3

8‖u0‖4L4 + 1 ≤ 1

2ε2‖∇u0‖2 + C‖u0‖4s + 1.


Finally, thanks to the Poincare inequality, using conservation of mass gives

‖u(t)‖ ≤‖ 1

|Ω|

∫

Ω

u0dx‖+ ‖u(t)− 1

|Ω|

∫

Ω

u0dx‖

≤‖ 1

|Ω|

∫

Ω

u0dx‖+ C‖∇u(t)‖.

The result now follows since

‖ 1

|Ω|

∫

Ω

u0dx‖ ≤ C‖u0‖.

2.2. L∞ boundness. In the case of one space dimension, the Sobolev inequality‖u‖∞ ≤ C‖u‖s is valid for all s > 1

2 which, thanks to Theorem 1, means that thesolution of FCHE is bounded for all t ≥ 0. Unfortunately, the same argument can-not be used in two space dimensions owing to the failure of the Sobolev inequality(which requires s > 1 in 2D). Nevertheless we will show that the boundness of‖u(t)‖∞ also holds true for the two dimensional problem by using a different argu-ment. The key estimate needed is the following Sobolev inequality. An analogousresult in the case Ω = R

d is given in [14, Exercise 6.1.2].

Lemma 3. Let v : [0 : 2π]d → R be periodic. Then for −∞ < 2µ < d < 2ν < ∞,

there holds for C = C(µ, ν, d) > 0,

‖v‖∞ ≤ C‖v‖+ ‖(−)µ/2v‖

ν−d/2ν−µ ‖(−)ν/2v‖

d/2−µν−µ

. (8)

Proof. It suffices to show that for v with vanishing average value, i.e., v = 0, thereholds

‖v‖∞ ≤ C‖(−)µ/2v‖ν−d/2ν−µ ‖(−)ν/2v‖

d/2−µν−µ , (9)

since the result would then follow for general v using

‖v‖∞ ≤ ‖v − v‖∞ + |v| ≤ ‖v − v‖∞ + C‖v‖.If v has vanishing average, then the Fourier expansion of v is given by

v(x) =∑

|k|6=0

vkeik·x,

where k = (k1, · · · , kd) denotes the multi-index and x = (x1, · · · , xd). Thus, forany N ∈ N

1

2‖v‖2∞ ≤ 1

2

( ∑

|k|6=0

|vk|)2 ≤

( ∑

0<|k|≤N

|vk|)2

+( ∑

|k|>N

|vk|)2,

and then the first term is bounded by( ∑

0<|k|≤N

|vk|)2 ≤

( ∑

0<|k|≤N

|k|−2µ)( ∑

0<|k|≤N

|k|2µ|vk|2)

≤( ∑

0<|k|≤N

|k|−2µ)‖(−)µ/2v‖2,

and the second term is bounded similarly by( ∑

|k|>N

|vk|)2 ≤

( ∑

|k|>N

|k|−2ν)‖(−)ν/2v‖2.


Now for d− 2µ > 0,

∑

0<|k|≤N

|k|−2µ ≤∫

|x|≤N

dx

|x|2µ =

∫ N

0

r−2µrd−1dr =Nd−2µ

d− 2µ,

whilst for 2ν − d > 0,

∑

|k|>N

|k|−2ν ≤∫

|x|>N

dx

|x|2ν =

∫ ∞

N

r−2νrd−1dr =Nd−2ν

2ν − d.

Hence, for all N ∈ N

1

2‖v‖2∞ ≤ Nd−2µ

d− 2µ‖(−)µ/2v‖2 + Nd−2ν

2ν − d‖(−)ν/2v‖2.

If ‖(−)µ/2v‖ = 0, then the result is trivial. Otherwise, since ν > µ, we have‖(−)ν/2v‖ ≥ ‖(−)µ/2v‖ thanks to Lemma 10 and we can choose N ∈ N to begiven by

N =⌊[‖(−)ν/2v‖

/‖(−)µ/2v‖

]1/(ν−µ)⌋,

where ⌊f⌋ denotes the integer part of f . With this choice we obtain

‖v‖2∞ ≤C(µ, ν, n)(‖(−)ν/2v‖‖(−)µ/2v‖

)n−2µν−µ ‖(−)µ/2v‖2

≤C(µ, ν, n)‖(−)µ/2v‖2ν−nν−µ ‖(−)ν/2v‖

n−2µν−µ .

and the result follows.

A consequence of Lemma 3 is the following special case in two space dimensionsd = 2, with µ = 1− α, ν = 1 + α for α > 0:

Lemma 4. For α > 0, if v ∈ H1+αper (Ω), then

‖v‖2∞ ≤ C(α)(‖v‖2 + ‖(−)(1−α)/2v‖‖(−)(1+α)/2v‖

). (10)

The main result of this section is a uniform L∞-bound on the solution of FCHEgiven in the following theorem:

Theorem 2. Let u be the solution of FCHE, then there holds for all 0 < α ≤ 1t ≥ 0:

‖u(t)‖∞ + ‖(−)(1+α)/2u(t)‖ ≤ C(u0), (11)

where C(u0) is a constant that depends on u0 but not on t.

We begin by establishing the following weaker variant of the estimate (11):

Lemma 5. Let u be the solution of (5), then for 0 < α ≤ 1, the following estimate∫ t+1

t

‖(−)(1+α)/2u(τ)‖2dτ ≤ C(u0), ∀ t ≥ 0, (12)

holds, where C(u0) depends on u0 but not on t.

Proof. Taking v = u in (5) gives

(∂u∂t

, u)+ ε2‖(−)(1+α)/2u‖2 +

(f(u), (−)αu

)= 0,


or, equally well,

ε2‖(−)(1+α)/2u(t)‖2 = −1

2

d

dt‖u(t)‖2 −

(f(u(t)), (−)αu(t)

).

Integrating from t to t+ 1, we have

ε2∫ t+1

t

‖(−)(1+α)/2u(τ)‖2dτ =1

2‖u(t)‖2 − 1

2‖u(t+ 1)‖2

−∫ t+1

t

(f(u(τ)), (−)αu(τ)

)dτ.

(13)

To obtain the estimate (12), let us first consider the last term of (13). We claimthat the following estimate holds.

(u3, (−)αu

)

≤C(α)κ‖∇u‖2(‖u‖4 + ‖(−)(1−α)/2u‖2‖(−)(1+α)/2u‖2) + 1

2κ‖∇u‖2,

(14)

where C(α) is a constant depends on α and κ is a positive constant to be determined.If 0 < α ≤ 1

2 , then we can prove (14) by arguing as follows:

(u3, (−)αu

)≤ κ

2‖u3‖2 + 1

2κ‖(−)αu‖2 ≤ κ

2‖u‖4∞‖u‖2 + 1

2κ‖(−)αu‖2.

Then using the estimate (10), we obtain(u3, (−)αu

)

≤C(α)κ‖u‖2(‖u‖4 + ‖(−)(1−α)/2u‖2‖(−)(1+α)/2u‖2) + 1

2κ‖(−)αu‖2.

Since α ≤ 12 , (14) holds for 0 < α ≤ 1

2 thanks to (39). In the case 12 < α ≤ 1 we

argue as follows: By virtue of (37) and (38), we get(u3, (−)αu

)=

((−)1/2u3, (−)α−1/2u

).

Using the Cauchy-Schwarz and Young inequalities, we obtain

((−)1/2u3, (−)α−1/2u

)≤ κ

2‖(−)1/2u3‖2 + 1

2κ‖(−)α−1/2u‖2.

Observing that,

‖(−)1/2v‖2 =∑

k,l∈Z

(k2 + l2)v2kl = ‖∇v‖2, ∀ v ∈ H1per(Ω),

we have

κ

2‖(−)1/2u3‖2 = κ

2‖∇u3‖2 =

κ

2‖3u2∇u‖2 ≤ 9κ

2‖u‖4∞‖∇u‖2,

then using the estimate (10) again and noting that α − 12 ≤ 1

2 , the estimate (14)

follows for 12 < α ≤ 1. Furthermore, to handle the linear part, we use

(u, (−)αu

)= ‖(−)α/2u‖2 ≤ ‖∇u‖2. (15)

If u is a constant, then estimate (12) holds trivially. Otherwise, we can choose κ such

that C(α)κ‖∇u‖2‖(−)(1−α)/2u‖2 = ε2

2 , i.e., κ = ε2/(2C(α)‖∇u‖2‖(−)(1−α)/2u‖2).Thanks to ‖u(t)‖1 ≤ C(1+ ‖u0‖1) ( see Theorem 1), and using (39) again, we have


that κ is a bounded constant which depends on u0 but not on t. Moreover, bycombining (13), (14) and (15), we obtain

ε2

2

∫ t+1

t

‖(−)(1+α)/2u(τ)‖2dτ ≤ C(α, ‖u0‖1).

Hence, the estimate (12) holds.

Now we can finally give the proof of Theorem 2.

Proof. (Proof of Theorem 2) Taking v = (−)1+αu in (5) and using (37) yields

0 =1

2

d

dt‖(−)(1+α)/2u‖2 + ε2‖(−)1+αu‖2 +

((−)αf(u), (−)1+αu

)

=1

2

d

dt‖(−)(1+α)/2u‖2 + ε2

2‖(−)1+αu‖2

+(ε22(−)1+αu+ (−)αf(u), (−)1+αu

).

(16)

Notice that

‖ε2(−)1+αu + (−)αf(u)‖2 − ‖(−)αf(u)‖2

=(ε2(−)1+αu, ε2(−)1+αu+ 2(−)αf(u)

)

=2ε2((−)1+αu,

ε2

2(−)1+αu+ (−)αf(u)

).

(17)

By equation (16) and (17), we obtain

1

2

d

dt‖(−)(1+α)/2u‖2 + ε2

2‖(−)1+αu‖2 + 1

2ε2‖ε2(−)1+αu + (−)αf(u)‖2

=1

2ε2‖(−)αf(u)‖2 ≤ 1

2ε2‖(−)(1+α)/2f(u)‖2

(18)thanks to Lemma 10. Now f(u) = u3 − u, and so with the help of (41), we have

‖(−)(1+α)/2f(u)‖2 ≤ C0‖u‖4∞‖(−)(1+α)/2u‖2 + ‖(−)(1+α)/2u‖2, (19)

where C0 is a constant that depends on α. By (10) we have

‖u‖2∞ ≤ C(α)(‖u‖2 + ‖(−)(1−α)/2u‖‖(−)(1+α)/2u‖). (20)

Moreover, since (1− α)/2 ≤ 1/2, then by Theorem 1, we know that

‖(−)(1−α)/2u‖ ≤ ‖u‖1 ≤ C(1 + ‖u0‖1). (21)

Hence, by (19)-(21), we obtain

‖(−)(1+α)/2f(u)‖2 ≤ C1‖(−)(1+α)/2u‖4 + C2‖(−)(1+α)/2u‖2,where C1 depends on u0 but not on t. Therefore, by (18), we get

1

2

d

dt‖(−)(1+α)/2u‖2

≤1

2

d

dt‖(−)(1+α)/2u‖2 + ε2

2‖(−)1+αu‖2 + 1

2ε2‖ε2(−)1+αu + (−)αf(u)‖2

≤ 1

2ε2

(C1‖(−)(1+α)/2u‖4 + C2‖(−)(1+α)/2u‖2

).


That is

d

dt‖(−)(1+α)/2u‖2 ≤ 1

ε2

(C1‖(−)(1+α)/2u‖4 + C2‖(−)(1+α)/2u‖2

),

and the bound on ‖(−)(1+α)/2u(t)‖ follows thanks to estimate (12) and the uni-form Gronwall Lemma 12 proved later. The bound on ‖u(t)‖∞ follows thanks tothe estimate (10) proved in Lemma 4.

3. Fourier Galerkin Semi-discretization in Space

Let XN = spaneikx1+ilx2 ,−N ≤ k, l ≤ N, then the Fourier spectral-Galerkinapproximation for (5) consists of finding uN ∈ XN , such that,

(∂uN

∂t, v)+(ε2(−)(1+α)/2uN , (−)(1+α)/2v

)+(f(uN), (−)αv

)= 0, ∀v ∈ XN .

(22)with initial condition uN (0) = ΠNu0. Here ΠN is the usual L2-projection operatorin XN , namely

(u −ΠNu, v) = 0, ∀ v ∈ XN . (23)

All arguments used for the continuous problem (5) discussed in Section 2 remainvalid for the Fourier Galerkin problem (22). In particular, we note that the Fourierspectral-Galerkin method preserves mass, i.e.

∫

Ω

uN (x, t)dx =

∫

Ω

uN (x, 0)dx =

∫

Ω

ΠNu0(x)dx =

∫

Ω

u0(x)dx,

and the monotonicity of the energy in the sense

d

dtE(uN ) = −‖(−)

s2 (ε2(−)suN + f(uN))‖20 ≤ 0, ∀ t > 0.

Moreover, we also have the same stability estimate for uN(x, t):

Corollary 1. The Fourier spectral Galerkin method (22) is stable and it holds for

all 0 < α ≤ 1

‖uN(t)‖1 ≤ C(1 + ‖u0‖1), ∀ t > 0, (24)

where C is independent of t and N .

Similarly, applying the same argument as the one used in the continuous case,we have that

Corollary 2. Let uN be the solution of weak problem (22), then there holds for all

0 < α ≤ 1 and t > 0

‖uN(t)‖∞ + ‖(−)(1+α)/2uN(t)‖ ≤ C(u0), (25)

where C(u0) is a constant depends on u0 but not on N and t.

3.1. L2-projection estimate. The operator ΠN commutates with the fractionalLaplacian operator (−)s:

Lemma 6. For any u ∈ H2sper(Ω) and s ≥ 0, we have

ΠN (−)su(x) = (−)sΠNu(x). (26)

In particular, the L2 projection is uniformly stable in H2sper(Ω),

‖ΠNu‖2s ≤ C‖u‖2s. (27)


Proof. Expanding u in the form (3), then by the definition of ΠN , we know that

ΠNu(x) =

N∑

k,l=−N

ukleikx1+ilx2 .

We have thus,

(−)sΠNu(x) =

N∑

k,l=−N

ukl(k2 + l2)seikx1+ilx2 .

On the other hand, we have

ΠN (−)su(x) = ΠN

∑

k,l∈Z

ukl(k2 + l2)seikx1+ilx2 =

N∑

k,l=−N

ukl(k2 + l2)seikx1+ilx2

as required. Finally, if u is a constant, then ‖ΠNu‖2s = ‖u‖2s, otherwise,

‖ΠNu‖2s ≤ C‖(−)sΠNu‖ = C‖ΠN (−)su‖ ≤ C‖(−)su‖ ≤ C‖u‖2s,

and hence, (27) holds.

The following approximation result is standard in the case of integer order norms:

Lemma 7. Suppose that u ∈ Hνper(Ω), then the following estimate holds for all

0 ≤ µ ≤ ν,

‖u−ΠNu‖µ ≤ cNµ−ν |u|ν . (28)

Proof. We begin with the case where Ω = (0, 2π) so that u is a univariate function.Let Πx

N be the L2 projection in the one dimensional case. In this case we haveu−Πx

Nu =∑

|k|>N

ukeikx, and so,

‖u−ΠxNu‖2µ =

∑

|k|>N

(1 + |k|2)µu2k

≤N−2(ν−µ)∑

|k|>N

(1 + |k|2)ν u2k ≤ cN−2(ν−µ)|u|2ν .

Now, in the case Ω = (0, 2π)2, ΠN = Πx1

N Πx2

N , where Πx1

N and Πx2

N are the L2

projections in the x1 and x2 variables respectively, and the error estimate (28) canbe obtained using the following triangle inequality

‖u−ΠNu‖µ ≤ ‖u−Πx2

N u‖µ + ‖Πx2

N (u−Πx1

N u)‖µ.

along with stability of the L2 projection in Hµ (Lemma 6).

3.2. Error estimate. In order to analyze the error in the Fourier Galerkin scheme,we follow the usual approach and consider the function eN(t) ∈ XN given by

eN (t) = ΠNu(t)− uN(t),

so that

eN (0) = ΠNu0 −ΠNu0 = 0.


Thus, for vN ∈ XN , with the help of (23), we have

(∂eN∂t

, vN)+(ε2(−)(1+α)/2eN , (−)(1+α)/2vN

)

=(ΠN

∂u

∂t, vN

)+(ε2ΠN (−)(1+α)/2u, (−)(1+α)/2vN

)

−(∂uN

∂t, vN

)−(ε2(−)(1+α)/2uN , (−)(1+α)/2vN

)

=(∂u∂t

, vN)+(ε2(−)(1+α)/2u, (−)(1+α)/2vN

)+(f(uN), (−)αvN

)

=(f(uN)− f(u), (−)αvN

).

Taking vN = eN in above equation, and using the Cauchy-Schwarz and Younginequalities, we get

1

2

d

dt‖eN‖2 + ε2‖(−)(1+α)/2eN‖2 ≤ 1

2ε2‖f(u)− f(uN)‖2 + ε2

2‖(−)αeN‖2

≤ 1

2ε2‖f(u)− f(uN)‖2 + ε2

2‖(−)(1+α)/2eN‖2,

which implies that

d

dt‖eN‖2 + ε2‖(−)(1+α)/2eN‖2 ≤ 1

ε2‖f(u)− f(uN )‖2. (29)

Let t be fixed and note that

|f(u)− f(uN)| ≤ |u− uN | ·MN(t),

where MN (t) = max|f ′(s)|, s is between u(t) and uN (t). Then, thanks to Theo-rem 2 and Corollary 2, we have

MN(t) ≤ L, (30)

where L is a constant independent of N and t (but depending on ‖u0‖1). Thus,|f(u(t))− f(uN(t))| ≤ L|u(t)− uN(t)|, ∀ t > 0,

which gives for all t > 0

‖f(u(t))− f(uN (t))‖ ≤ L‖u(t)− uN (t)‖ ≤ L(‖u(t)−ΠNu(t)‖ + ‖eN(t)‖).Inserting this estimate into the earlier estimate (29), we can get

d

dt‖eN‖2 + ε2‖(−)(1+α)/2eN‖2 ≤ L

ε2

‖u(t)−ΠNu(t)‖2 + ‖eN‖2

.

It follows thatd

dt

e−Lt/ε2‖eN‖2

≤ L

ε2e−Lt/ε2‖u(t)−ΠNu(t)‖2,

which in turn gives

‖eN‖2 ≤ L

ε2

∫ t

0

eL(t−τ)/ε2‖u(τ)−ΠNu(τ)‖2dτ,

since eN (0) = 0. Finally, with the help of (28), we have

‖eN‖2 ≤ L

ε2N−2reLt/ε2

∫ t

0

‖u(τ)‖2rdτ, t ≥ 0.

By the triangle inequality and (28), we obtain the following error estimate forthe Fourier Galerkin scheme:


Theorem 3. Let u(t) and uN (t) be the solutions of problem (5) and (22), if u ∈H1+α

per (Ω) ∩Hr(Ω), then the error estimate holds for 0 < α ≤ 1:

‖u(t)− uN(t)‖ ≤ CeT/ε2N−r‖u(t)‖r +

( ∫ t

0

‖u(τ)‖2r)1/2

. (31)

Observe that the rate of convergence is independent of α. Moreover, if u ∈ Hr

for arbitrary r, then the usual spectral rate of convergence holds.

4. Numerical examples

4.1. Implementation. For the time integration, we use a first-order semi-implicitscheme in time [22]. For a given partition 0 = t0 < t1 < · · · < tK = T , tn =nδt, n = 0, 1, · · · ,K, where δt is the time step, and K = T/δt, we consider theFourier Galerkin scheme (22) in space, and the first-order time semi-implicit schemegiven by

1

δt(un+1

N − unN , v) + ((−)α/2(ε2(−)un+1

N + f(unN )), (−)α/2v) = 0, ∀ v ∈ XN .

(32)where un

N denotes the numerical approximation to uN (t) at time tn. The stabilityof this scheme follows standard arguments given in Appendix C.

Writing

unN =

N/2∑

k,l=−N/2

unkle

ikx1+ilx2 , n = 0, 1, · · · ,K,

and taking v = eipx1+iqx2 , p, q = −N/2, · · · , N/2 we get, thanks to the definition offractional Laplacian,

un+1kl − un

kl

δt+ (k2 + l2)1+αun+1

kl + ε2(k2 + l2)αfnkl = 0, k, l = −N/2, · · · , N/2, (33)

where fnkl, k, l = −N/2, · · · , N/2 are the Fourier coefficients of f(un

N). In order touse the scheme (33), we need only compute the non-linear term involving f . To dothis, we use the De-aliasing technique described in [18, Section 4.3.2] along with theFFT. More precisely, the De-aliasing technique removes the aliasing errors arisingfrom dealing with the product of two functions.

We briefly describe the main idea in the univariate setting. Let v, w ∈ L2per(0, 2π),

the basic idea of De-aliasing is to represent the product vw by an interpolantq = IM (vw) of sufficiently high order, M , such that there are no aliasing errors.We compute the mesh point values of the interpolant and its coefficients efficientlywith the FFT.

Let xj = 2πj/N, j = 0, 1, · · · , N − 1 and

v(x) =

N/2∑

k=−N/2

vkeikx, w(x) =

N/2∑

k=−N/2

wkeikx.

Define new padded coefficients

ˆvk =

vk, |k| ≤ N/2,0, |k| > N/2,

ˆwk =

wk, |k| ≤ N/2,0, |k| > N/2.


It is easy to see that

ˆ(vw)m =

N/2∑

k=−N/2

vkwm−k =

N/2∑

k=−N/2

ˆvk ˆwm−k.

We then compute a set of mesh point values at M ≥ N mesh points yj = 2πj/M ,

vj =

M/2−1∑

k=−M/2

ˆvkeikyj , wj =

M/2−1∑

k=−M/2

ˆwkeikyj , j = 0, 1, · · · ,M − 1,

and obtain the Fourier coefficients of the product vw given by qj = vjwj , j =0, 1 · · · ,M − 1, using the discrete Fourier transform

qk =1

M

M−1∑

j=0

qje−ikyj , k = −M/2, · · · ,M/2− 1.

The coefficients of the interpolant and the exact coefficients of the product arerelated by the aliasing formula

qk = ˆ(vw)k +∑

l∈Z

l 6=0

ˆ(vw)k+lM = ˆ(vw)k +∑

l∈Z

l 6=0

N/2∑

p=−N/2

ˆvp ˆwk+lM−p. (34)

It is well-known that, with a proper choice ofM , we can eliminate the last (aliasing)

term in (34) in which qk = ˆ(vw)k. Note that ˆwm = 0 for m > N/2. Thus, if Mis chosen such that |k + M − p| > N/2 for |k|, |p| ≤ N/2, then the aliasing termvanishes for all l 6= 0. The worst case occurs when k = −N/2 and p = N/2, whichgives

−N

2− N

2+M >

N

2⇒ M >

3N

2.

So if the product in physical space is computed with at least 3N/2 + 2 (M has tobe even), the interpolant q(x) using the values at those points matches the product

vw exactly, and the Fourier interpolation coefficients qk match ˆ(vw)k. Therefore,

the computation of ˆ(vw)k using the De-aliasing technique requires two FFTs eachof which is of order M logM . For a product of three functions (which we need forthe FCHE), one can simply apply the De-aliasing technique twice.

4.2. Numerical examples. We present several numerical examples to demon-strate the accuracy of the Fourier-Galerkin scheme and to illustrate the behaviorof the solutions to problem (2).

Example 1. Verification of spectral convergence rate. Let us first test theconvergence with respect to N . We choose ε2 = 1/10 with initial condition

u0(x, y) = sinx cos y. (35)

The L2 errors at time T = 1 for different values of fractional order α are shownin Figure 1 on a semi-log scale. The true solution is unknown and we therefore usethe Fourier Galerkin approximation in the case N = 128 as a reference solution,with the time step is set to be small enough to ensure the time discretization errorcan be ignored.


As we can see in Figure 1, the spectral convergence with respect to N is observed,and the rate of convergence is independent of the fractional order α as predicted inTheorem 3.

20 30 40 50 60 7010

−16

10−14

10−12

10−10

10−8

10−6

10−4

L2 error, u

0=sinxcosy, T=1

Spectral order N

Err

or in

logs

cale

α=0.1α=0.5α=0.8

Figure 1. Spatial L2 errors at time T = 1 for FCHE with ε2 =1/10 using initial condition (35).

Example 2. Coalescence of two kissing bubbles [19]. We simulate the coa-lescence of two kissing bubbles, i.e., the initial condition is

u0(x, y) =

1, (x + 1)2 + y2 ≤ 1 or (x − 1)2 + y2 ≤ 1,−1, otherwise.

(36)

Set ε2 = 1/100, N = 128, δt = 0.01.

In order to illustrate the effect of changing the value of fractional derivativeα on the character of the solution of FCHE, we present plots of the solution fordifferent values of fractional order α at various times in Figure 2. The solutionswere obtained using the Fourier Galerkin method with N = 128 and a time stepsize δt = 0.01. Theorem 3 shows that the scheme is convergent and we assume thatthe numerical solutions provides an accurate approximation of the true solution.We observe that in all cases, as time evolves, the two bubbles coalesce into a singlebubble. The classical Allen-Cahn equation does not conserve mass, which meansthat, (see Figure 2 (A)) in the case α = 0, the bubble shrinks and finally disappears.On the other hand, when α = 1, i.e., the classical Cahn-Hilliard equation (whichdoes preserve mass), we see that the shape of the bubble reaches a steady circularshape. More interestingly, we observe that, as expected from the mass conservationfor α > 0, the volume of the bubble is also preserved for the cases α = 0.1, 0.5.However, the rate at which the solution approaches the steady asymptotic statevaries with the fractional order α. This is to be expected since, as predicted inLemma 2, the smaller the value of α, the slower the energy decays.


(a) α = 0 (b) α = 0.1 (c) α = 0.5 (d) α = 1.0

Figure 2. Evolution of solutions of FCHE with initial condition(36) for different values of fractional order α. From top to bottomt = 0, 12, 72, 100, 200.

Example 3. Coarsening dynamics of FCHE. Coarsening is a widely observedphenomenon in materials systems involving microstructures. The coarsening ratesfor the Cahn-Hilliard equation with phase-dependent diffusion mobility have beenstudied theoretically [6, 7] and numerically [8, 17]. We study the coarsening dy-namics of FCHE as the fractional order α is varied, in the case ε2 = 16/10000 andN = 512. The initial condition is set to be uniform state u = 2φ−1 augmented witha random perturbation uniformly distributed in [−0.2, 0.2]. We apply a stabilizedsecond order semi-implicit time scheme [22], which reads, ∀ v, ϕ ∈ XN ,

1

δt(3un+1

N − 4unN + un−1

N , v) + ((−)α/2wn+1, (−)α/2v) = 0,

ε2(∇un+1N ,∇ϕ) + S(un+1

N − 2unN + un−1

N , ϕ) + (2f(unN )− f(un−1

N ), ϕ) = (wn+1, ϕ),

where S is a constant. It is known [22] that this scheme is stable in the case α = 1and has the advantage of allowing a large time step. Here the time step is set tobe δt = 10−3.


(a) φ = 0.75 (b) φ = 0.5 (c) φ = 0.25

Figure 3. Configurations at time T = 20 with random initialcondition for different values of fractional order α and φ. Fromtop to bottom α = 0.1, 0.5, 1.0.

The configurations at time T = 20 with the same random initial condition fordifferent α = 0.1, 0.5, 1.0 and volume fractions φ = 0.75, 0.5, 0.25 are shown inFigure 3. To study the coarsening behavior of FCHE, we plot the log2 E(t) vslog2 t for t ∈ [1, 250] with two sets of initial data in Figure 4-6. We can see that,the coarsening dynamics for φ = 0.25 and φ = 0.75 are almost the same. Moreover,while the coarsening is initially slower for small values of α, the ssymptotic ratesare roughly the same in all cases, and agree with that of the standard Cahn-Hilliardequation (α = 1).

Appendix

Appendix A: Technical Results. We collect a number of elementary technicalresults that were used earlier in the article.

Lemma 8. Let µ, ν ≥ 0 then for any u, v ∈ Hµ+νper (Ω), it holds

((−)µ+νu, v

)=

((−)µu, (−)νv

). (37)


0 1 2 3 4 5 6 7 8−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

log2t

log 2E

(t)

φ = 0.75

α=0.1α=0.5α=1.0

1

3

0 1 2 3 4 5 6 7 8−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

log2t

log 2E

(t)

φ = 0.75

α=0.1α=0.5α=1.0

3

1

Figure 4. Coarsening dynamics for different values of fractionalorder α, φ = 0.75 (left: initial data 1, right: initial data 2).

0 1 2 3 4 5 6 7 8−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

log2t

log 2E

(t)

φ = 0.5

α=0.1α=0.5α=1.0

1

3

0 1 2 3 4 5 6 7 8−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

log2t

log 2E

(t)

φ = 0.5

α=0.1α=0.5α=1.0

3

1


0 1 2 3 4 5 6 7 8−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

log2t

log 2E

(t)

φ = 0.25

α=0.1α=0.5α=1.0

3

1

0 1 2 3 4 5 6 7 8−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

log2t

log 2E

(t)

φ = 0.25

α=0.1α=0.5α=1.0

3

1



Proof. Since u, v ∈ Hµ+νper (Ω), then u can be expanded by Fourier series (3), and

similarly, v(x) can be expanded as

v(x) =∑

k,l∈Z

vkleikx1+ilx2 .

Then, with the help of (4), we have

(−)µ+νu =∑

k,l∈Z

uklλµ+νkl eikx1+ilx2 ,

where λkl = k2 + l2, thus,((−)µ+νu, v

)=( ∑

k,l=∈Z

uklλµ+νkl eikx1+ilx2 ,

∑

k,l∈Z

vkleikx1+ilx2

)

=∑

k,l∈Z

uklvklλµ+νkl =

∑

k,l∈Z

(uklλµkl)(vklλ

νkl)

=( ∑

k,l∈Z

uklλµkle

ikx1+ilx2 ,∑

k,l∈Z

vklλνkle

ikx1+ilx2

)

=((−)µu, (−)νv

).

Another important property of fractional Laplacian is the semigroup propertywhich follows easily using the definition (4):

Lemma 9. For any u ∈ Hµ+νper (Ω), it holds

(−)µ+νu = (−)µ(−)νu = (−)ν(−)µu. (38)

Likewise, using the definition (4), it is easy to see that the following Sobolevinequality holds:

Lemma 10. If µ ≤ ν and v ∈ Hνper(Ω), then

‖(−)µv‖ ≤ ‖(−)νv‖. (39)

To prove Theorem 2, the following lemmas will be needed.

Lemma 11. If w, v ∈ H2νper(Ω) for all ν ≥ 0, then

‖(−)ν(wv)‖2 ≤ C(ν)[‖v‖2∞‖(−)νw‖2 + ‖w‖2∞‖(−)νv‖2

], (40)

and

‖(−)νv3‖2 ≤ 3C(ν)‖v‖4∞‖(−)νv‖2, (41)

where

C(ν) = max(1, 22ν−1). (42)

Proof. Let w, v ∈ H2νper(Ω), so that

w =∑

k,l∈Z

wkleikx1+ilx2 , v =

∑

k,l∈Z

vkleikx1+ilx2 .

Then the product of w and v can be written as a discrete convolution

wv =∑

p,q∈Z

( ∑

k,l∈Z

wklvp−k,q−l

)eipx1+iqx2 ,


and by (4), we obtain

(−)ν(wv) =∑

p,q∈Z

(p2 + q2)ν( ∑

k,l∈Z

wklvp−k,q−l

)eipx1+iqx2 ,

and hence,

‖(−)ν(wv)‖2 =∑

p,q∈Z

[(p2 + q2)ν

( ∑

k,l∈Z

wklvp−k,q−l

)]2.

The following inequality holds for all ν ≥ 0,

|a+ b|2ν ≤ C(ν)(|a|2ν + |b|2ν), ∀a, b ∈ R2,

where C(ν) is given in (42). Choosing a = (k, l) and b = (p− k, q − l), we obtain

(p2 + q2)ν ≤ C(ν)((k2 + l2)ν + ((p− k)2 + (q − l)2)ν

).

Hence, we obtain

‖(−)ν(wv)‖2 ≤C(ν)[ ∑

p,q∈Z

( ∑

k,l∈Z

(k2 + l2)νwklvp−k,q−l

)2

+∑

p,q∈Z

( ∑

k,l∈Z

((p− k)2 + (q − l)2)νwklvp−k,q−l

)2]

≤C(ν)[‖v(−)νw)‖2 + ‖w(−)νv‖2

]

≤C(ν)[‖v‖2∞‖(−)νw‖2 + ‖w‖2∞‖(−)νv‖2

],

and the estimate (40) follows.In order to get estimate (41), by virtue of (40), we obtain

‖(−)νv3‖2 ≤ C(ν)[‖v2‖2∞‖(−)νv‖2 + ‖v‖2∞‖(−)νv2‖2

].

Using (40) again, we have

‖(−)νv2‖2 ≤ 2C(ν)‖v‖2∞‖(−)νv‖2.Combining above two equations, we can get the estimate (41).

Appendix B: Uniform Gronwall Lemma. To establish the L∞ estimate, weuse the following uniform Gronwall lemma which is similar to the results obtainedin [13, 21].

Lemma 12. Let Q(t) be a locally integrable positive functions on (0,∞) satisfying

d

dtQ(t) ≤ (aQ(t) + b)(cQ(t) + d), ∀ t > 0,

and ∫ t+1

t

Q(τ)dτ ≤ e, ∀ t > 0,

where a, b, c, d and e are non-negative constants, c 6= 0. Then we have

Q(t) ≤ CeC ∀ t ≥ 0, (43)

where C and C are positive constants that are independent of t.


Proof. Let

g(t) = c(aQ(t) + b), and G(r, t) =

∫ t

r

g(τ)dτ for 0 ≤ r ≤ t ≤ r + 1,

then we have∂G

∂t(r, t) = g(t); (44)

G(r, t) −G(r, r + 1) ≥ −∫ r+1

r

g(τ)dτ ≥ −C1, (45)

where C1 is a positive constant which is independent of t;

G(r, s)−G(r, t) =

∫ s

t

g(τ)dτ ≥ 0, for r ≤ t ≤ s ≤ 1. (46)

Then by (44), we get

d

ds

(Q(s)e−G(r,s)

)= e−G(r,s)

(dQ(s)

ds− g(s)Q(s)

)≤ e−G(r,s) d

cg(s). (47)

Integrate over s : r ≤ t ≤ s ≤ r + 1 to get

Q(r + 1)e−G(r,r+1) ≤ Q(t)e−G(r,t) +d

c

∫ r+1

t

e−G(r,s)g(s)ds,

which gives

Q(r + 1)eG(r,t)−G(r,r+1) ≤ Q(t) +d

c

∫ r+1

t

eG(r,t)−G(r,s)g(s)ds.

Hence, with the help of (45)-(46), for r ≤ t, we obtain

Q(r + 1)e−C1 ≤ Q(t) +d

c

∫ r+1

t

g(s)ds ≤ Q(t) +d

cC2,

where C2 is a positive constant that is independent of t. Integrate over (t, t+1) toget

Q(r + 1)e−C1 ≤∫ t+1

t

Q(s)ds+d

cC2 ≤ C3.

So

Q(r + 1) ≤ C3eC1 , ∀ r ≥ 0.

This indicate that

Q(t) ≤ C3eC1 , ∀ t ≥ 1. (48)

We want the result holds true for 0 ≤ t < 1 as well. Integrate (47) from 0 ≤ s ≤ tand put r = 0, we have

Q(t) ≤ eG(0,t)Q(0) +

d

c

∫ t

0

e−G(0,s)g(s)dsfor 0 ≤ t ≤ 1,

where

G(0, t) =

∫ t

0

g(s)ds ≤∫ 1

0

g(s)ds ≤ C4

and −G(0, s) ≤ 0. Therefore,

Q(t) ≤ C5eC4, ∀ 0 ≤ t ≤ 1. (49)

Combining (48) and (49) gives the estimate (43).


Appendix C: Stability of Scheme (32). The scheme (32) can be alternativelyformulated in the following mixed form:

1

δt(un+1

N − unN , v) + ((−)α/2wn+1, (−)α/2v) = 0, ∀ v ∈ XN ,

ε2(∇un+1N ,∇φ) + (f(un

N ), φ) = (wn+1, φ), ∀φ ∈ XN .(50)

The next result is similar to the result proved for the standard Cahn-Hilliardequation in [22], and shows that the scheme (50) is stable provided the step size issufficiently small:

Lemma 13. Under the condition

δt ≤ 4ε2

L2, (51)

where L = max|f ′(s)|, the solution of (50) satisfies

E(un+1N ) ≤ E(un

N ), ∀n ≥ 0. (52)

Proof. Taking v = δtwn+1 and φ = un+1N − un

N in (50) and using the identity(a− b, 2a) = |a|2 − |b|2 + |a− b|2, we obtain

(un+1N − un

N , wn+1) + δt‖(−)α/2wn+1‖2 = 0, (53)

and

ε2

2(‖∇un+1

N ‖2−‖∇unN‖2+‖∇(un+1

N −unN)‖2)+(f(un

N), un+1N −un

N ) = (wn+1, un+1N −un

N).

(54)For the last term of the left side in (54), by Taylor expansion

F (un+1N )− F (un

N ) = f(unN )(un+1

N − unN ) +

f ′(ζn)

2(un+1

N − unN )2,

where ζn is between unN and un+1

N . Therefore, we can get

ε2

2(‖∇un+1

N ‖2 − ‖∇unN‖2 + ‖∇(un+1

N − unN )‖2) + (F (un+1

N )− F (unN), 1)

=f ′(ζn)

2‖un+1

N − unN‖2 + (wn+1, un+1

N − unN).

(55)

On the other hand, taking v =√δtε(un+1

N − unN) in (50), we have

ε√δt‖un+1

N − unN‖2 =−

√δtε((−)α/2wn+1, (−)α/2(un+1

N − unN ))

≤δt

2‖(−)α/2wn+1‖2 + ε2

2‖(−)α/2(un+1

N − unN)‖2

≤δt

2‖(−)α/2wn+1‖2 + ε2

2‖∇(un+1

N − unN)‖2.

(56)

Then by (53), (55) and (56), we arrive at

ε√δt‖un+1

N − unN‖2 + δt

2‖(−)α/2wn+1‖2 + ε2

2(‖∇un+1

N ‖2 − ‖∇unN‖2)

+(F (un+1N )− F (un

N ), 1) =f ′(ζn)

2‖un+1

N − unN‖2 ≤ L

2‖un+1

N − unN‖2.

We then conclude that the desired result holds true under the condition (51).


References

[1] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its

application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), pp. 1085–1095.[2] J. Bosch and M. Stoll, A fractional inpainting model based on the vector-valued Cahn-

Hilliard equation, SIAM J. Imaging Sci., 8 (2015), pp. 2352–2382.[3] A. Bueno-Orovio, D. Kay, and K. Burrage, Fourier spectral methods for fractional-in-

space reaction-diffusion equations, BIT, 54 (2014), pp. 937–954.[4] L. A. Caffarelli and N. E. Muler, An L∞ bound for solutions of the Cahn-Hilliard

equation, Archive for Rational Mechanics and Analysis, 133 (1995), pp. 129–144.[5] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free

energy, The Journal of Chemical Physics, 28 (1958), pp. 258–267.[6] S. Dai and Q. Du, Motion of interfaces governed by the Cahn-Hilliard equation with highly

disparate diffusion mobility, SIAM J. Appl. Math., 72 (2012), pp. 1818–1841.[7] , Coarsening mechanism for systems governed by the Cahn-Hilliard equation with de-

generate diffusion mobility, Multiscale Model. Simul., 12 (2014), pp. 1870–1889.[8] , Computational studies of coarsening rates for the Cahn–Hilliard equation with phase-

dependent diffusion mobility, J. Comput. Phys., 310 (2016), pp. 85–108.[9] W. M. Feng, P. Yu, S. Y. Hu, Z. K. Liu, Q. Du, and L. Q. Chen, Spectral implementation

of an adaptive moving mesh method for phase-field equations, J. Comput. Phys., 220 (2006),pp. 498–510.

[10] , A Fourier spectral moving mesh method for the Cahn-Hilliard equation with elasticity,Commun. Comput. Phys., 5 (2009), pp. 582–599.

[11] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation

for mean curvature flows, Numer. Math., 94 (2003), pp. 33–65.[12] P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential

Equations, (2000), pp. No. 48, 26 pp. (electronic).[13] C. Foias, O. Manley, and R. Temam, Attractors for the benard problem: existence and

physical bounds on their fractal dimension, Nonlinear Analysis: Theory, Methods & Appli-cations, 11 (1987), pp. 939–967.

[14] L. Grafakos, Modern Fourier analysis, vol. 250, Springer, 2009.[15] Y. He and Y. Liu, Stability and convergence of the spectral Galerkin method for the Cahn-

Hilliard equation, Numer. Methods Partial Differential Equations, 24 (2008), pp. 1485–1500.[16] Y. He, Y. Liu, and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation,

Appl. Numer. Math., 57 (2007), pp. 616–628.[17] L. Ju, J. Zhang, and Q. Du, Fast and accurate algorithms for simulating coarsening dynam-

ics of Cahn–Hilliard equations, Computational Materials Science, 108 (2015), pp. 272–282.[18] D. A. Kopriva, Implementing spectral methods for partial differential equations: Algorithms

for scientists and engineers, Springer Science & Business Media, 2009.[19] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its

approximation by a Fourier-spectral method, Phys. D, 179 (2003), pp. 211–228.[20] Z. Mao, S. Chen, and J. Shen, Efficient and accurate spectral method using generalized

Jacobi functions for solving Riesz fractional differential equations, Applied Numerical Math-ematics, (2016).

[21] M. Marion and R. Temam, Nonlinear Galerkin methods, SIAM Journal on Numerical Anal-ysis, 26 (1989), pp. 1139–1157.

[22] J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equa-

tions, Discrete Contin. Dyn. Syst., 28 (2010), pp. 1669–1691.[23] F. Song, C. Xu, and G. E. Karniadakis, A fractional phase-field model for two-phase flows

with tunable sharpness: Algorithms and simulations, Comput. Methods Appl. Mech. Engrg.,305 (2016), pp. 376–404.

[24] X. Ye and X. Cheng, The Fourier spectral method for the Cahn-Hilliard equation, Appl.Math. Comput., 171 (2005), pp. 345–357.

[25] F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, and V. Anh, A Crank-Nicolson ADI

spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion

equation, SIAM J. Numer. Anal., 52 (2014), pp. 2599–2622.


[26] J. Zhu, L.-Q. Chen, J. Shen, and V. Tikare, Coarsening kinetics from a variable-mobility

Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method, PhysicalReview E, 60 (1999), p. 3564.

Division of Applied Mathematics, Brown University, 182 George St, Providence RI02912, USA.

E-mail address: Mark [email protected], zhiping [email protected]

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