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Advances in Nonlinear Analysis

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Higher-order anisotropic models in phase separation

Laurence Cherfils / Alain Miranville
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  • Laboratoire de Mathématiques et Applications, UMR CNRS 7348 – SP2MI,Université de Poitiers, Boulevard Marie et Pierre Curie – Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
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/ Shuiran Peng
  • Laboratoire de Mathématiques et Applications, UMR CNRS 7348 – SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie – Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
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Published Online: 2017-03-16 | DOI: https://doi.org/10.1515/anona-2016-0137

Abstract

Our aim in this paper is to study higher-order (in space) Allen–Cahn and Cahn–Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor. We also give, for the Allen–Cahn models, numerical simulations which illustrate the effects of the higher-order terms and the anisotropy.

Keywords: Allen–Cahn model; Cahn–Hilliard model; higher-order models; anisotropy; well-posedness; dissipativity; global attractor; numerical simulations

MSC 2010: 35K55; 35J60

1 Introduction

The Allen–Cahn (see [4]) and Cahn–Hilliard (see [8, 9]) equations are central in materials science. They both describe important qualitative features of binary alloys, namely, the ordering of atoms for the Allen–Cahn equation and phase separation processes (spinodal decomposition and coarsening) for the Cahn–Hilliard equation. These two equations have been much studied from a mathematical point of view; we refer the readers to the review papers [14, 34] and the references therein.

Both equations are based on the so-called Ginzburg–Landau free energy,

ΨGL=Ω(α2|u|2+F(u))𝑑x,α>0,(1.1)

where u is the order parameter, F is a double-well potential and Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of 3, with boundary Γ; we can of course also consider bounded and regular domains of and 2). The Allen–Cahn equation (which corresponds to an L2-gradient flow of the Ginzburg–Landau free energy) then reads

ut-αΔu+f(u)=0,

where f=F, while the Cahn–Hilliard equation (which corresponds to an H-1-gradient flow) reads

ut+αΔ2u-Δf(u)=0.

In (1.1), the term |u|2 models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones (see [9]); it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions (see [18, 19]).

Caginalp and Esenturk recently proposed in [7] (see also [11]) higher-order phase-field models in order to account for anisotropic interfaces (see also [26, 40, 45] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these authors proposed the following modified free energy, in which we omit the temperature:

ΨHOGL=Ω(12i=1k|α|=iaα|𝒟αu|2+F(u))𝑑x,k,(1.2)

where, for α=(k1,k2,k3)({0})3,

|α|=k1+k2+k3

and, for α(0,0,0),

𝒟α=|α|x1k1x2k2x3k3

(we agree that 𝒟(0,0,0)v=v). The corresponding higher-order Allen–Cahn and Cahn–Hilliard equations then read

ut+i=1k(-1)i|α|=iaα𝒟2αu+f(u)=0,(1.3)ut-Δi=1k(-1)i|α|=iaα𝒟2αu-Δf(u)=0.(1.4)

We studied in [13] (see also [12]) the corresponding higher-order isotropic models, namely,

ut+P(-Δ)u+f(u)=0,ut-ΔP(-Δ)u-Δf(u)=0,

where

P(s)=i=1kaisi,ak>0,k1,s.

In particular, these models contain sixth-order Cahn–Hilliard models. We can note that there is currently a strong interest in the study of sixth-order Cahn–Hilliard equations. Such equations arise in situations such as strong anisotropy effects being taken into account in phase separation processes (see [42]), atomistic models of crystal growth (see [5, 6, 16, 17]), the description of growing crystalline surfaces with small slopes which undergo faceting (see [39]), oil-water-surfactant mixtures (see [20, 21]) and mixtures of polymer molecules (see [15]). We refer the reader to [10, 22, 23, 25, 27, 28, 29, 30, 31, 32, 36, 35, 37, 38, 43, 44, 46] for the mathematical and numerical analysis of such models. They also contain the Swift–Hohenberg equation (see [30, 32]).

Our aim in this paper is to study the well-posedness of (1.3) and (1.4). We also prove the dissipativity of the corresponding solution operators, as well as the existence of the global attractor. We finally give, for the Allen–Cahn models, numerical simulations which show the effects of the higher-order terms and the anisotropy.

2 Preliminaries

We assume that k, k2, and

aα>0,|α|=k,

and we introduce the elliptic operator Ak defined by

Akv,wH-k(Ω),H0k(Ω)=|α|=kaα((𝒟αv,𝒟αw)),

where H-k(Ω) is the topological dual of H0k(Ω). Furthermore, ((,)) denotes the usual L2-scalar product, with associated norm . More generally, we denote by X the norm on the Banach space X; we also set -1=(-Δ)-1/2, where (-Δ)-1 denotes the inverse minus Laplace operator associated with Dirichlet boundary conditions. We can note that

(v,w)H0k(Ω)2|α|=kaα((𝒟αv,𝒟αw))

is bilinear, symmetric, continuous and coercive, so that

Ak:H0k(Ω)H-k(Ω)

is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1, 2, 3]) that Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

D(Ak)=H2k(Ω)H0k(Ω),

where, for vD(Ak),

Akv=(-1)k|α|=kaα𝒟2αv.

We further note that D(Ak1/2)=H0k(Ω) and, for (v,w)D(Ak1/2)2,

((Ak1/2v,Ak1/2w))=|α|=kaα((𝒟αv,𝒟αw)).

We finally note that (see, e.g., [41]) Ak (respectively, Ak1/2) is equivalent to the usual H2k-norm (respectively, Hk-norm) on D(Ak) (respectively, D(Ak1/2)).

Similarly, we can define the linear operator A¯k=-ΔAk,

A¯k:H0k+1(Ω)H-k-1(Ω)

which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

D(A¯k)=H2k+2(Ω)H0k+1(Ω),

where, for vD(A¯k),

A¯kv=(-1)k+1Δ|α|=kaα𝒟2αv.

Furthermore, D(A¯k1/2)=H0k+1(Ω) and, for (v,w)D(A¯k1/2)2,

((A¯k1/2v,A¯k1/2w))=|α|=kaα((𝒟αv,𝒟αw)).

Besides, A¯k (respectively, A¯k1/2) is equivalent to the usual H2k+2-norm (respectively, Hk+1-norm) on D(A¯k) (respectively, D(A¯k1/2)).

We finally consider the operator A~k=(-Δ)-1Ak, where

A~k:H0k-1(Ω)H-k+1(Ω);

note that, as -Δ and Ak commute, the same holds for (-Δ)-1 and Ak, so that A~k=Ak(-Δ)-1.

We have the following result:

Lemma 2.1.

The operator A~k is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

D(A~k)=H2k-2(Ω)H0k-1(Ω),

where, for vD(A~k),

A~kv=(-1)k|α|=kaα𝒟2α(-Δ)-1v.

Furthermore, D(A~k1/2)=H0k-1(Ω) and, for (v,w)D(A~k1/2)2,

((A~k1/2v,A~k1/2w))=|α|=kaα((𝒟α(-Δ)-1/2v,𝒟α(-Δ)-1/2w)).

Besides, A~k (respectively, A~k1/2) is equivalent to the usual H2k-2-norm (respectively, Hk-1-norm) on D(A~k) (respectively, D(A~k1/2)).

Proof.

We first note that A~k clearly is linear and unbounded. Then, since (-Δ)-1 and Ak commute, it easily follows that A~k is selfadjoint. Next, the domain of A~k is defined by

D(A~k)={vH0k-1(Ω):A~kvL2(Ω)}.

Noting that A~kv=f, fL2(Ω), vD(A~k), is equivalent to Akv=-Δf, where -ΔfH2(Ω), it follows from the elliptic regularity results of [1, 2, 3] that vH2k-2(Ω), so that D(A~k)=H2k-2(Ω)H0k-1(Ω). Noting then that A~k-1 maps L2(Ω) onto H2k-2(Ω) and recalling that k2, we deduce that A~k has compact inverse. We now note that, considering the spectral properties of -Δ and Ak (see, e.g., [41]) and recalling that these two operators commute, -Δ and Ak have a spectral basis formed of common eigenvectors. This yields that, for all s1,s2, (-Δ)s1 and Aks2 commute. Having this, we see that A~k1/2=(-Δ)-1/2Ak1/2, so that D(A~k1/2)=H0k-1(Ω), and, for (v,w)D(A~k1/2)2,

((A~k1/2v,A~k1/2w))=|α|=kaα((𝒟α(-Δ)-1/2v,𝒟α(-Δ)-1/2w)).

Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm A~k1/2 is equivalent to the norm (-Δ)-1/2Hk(Ω) and, thus, to the norm (-Δ)k-12. ∎

Throughout the paper, the same letters c, c and c′′ denote (generally positive) constants which may vary from line to line. Similarly, the same letter Q denotes (positive) monotone increasing and continuous functions which may vary from line to line.

3 The Allen–Cahn theory

3.1 Setting of the problem

We consider in this section the following initial and boundary value problem, for k2 (for k=1, the problem can be treated as in the original Allen–Cahn equation; see, e.g., [13]):

ut+i=1k(-1)i|α|=iaα𝒟2αu+f(u)=0,(3.1)𝒟αu=0on Γ,|α|k-1,(3.2)u|t=0=u0.(3.3)

Remark 3.1.

For k=1 (anisotropic Allen–Cahn equation), we have an equation of the form

ut-i=13ai2uxi2+f(u)=0

and, for k=2 (fourth-order anisotropic Allen–Cahn equation), we have an equation of the form

ut+i,j=13aij4uxi2xj2-i=13bi2uxi2+f(u)=0.

We actually rewrite (3.1) in the equivalent form

ut+Aku+Bku+f(u)=0,(3.4)

where

Bkv=i=1k-1(-1)i|α|=iaα𝒟2αv.

As far as the nonlinear term f is concerned, we assume that

f𝒞1(),f(0)=0,(3.5)f-c0,c00,(3.6)f(s)sc1F(s)-c2-c3,c1>0,c2,c30,s,(3.7)F(s)c4s4-c5,c4>0,c50,s,(3.8)

where F(s)=0sf(ξ)𝑑ξ. In particular, the usual cubic nonlinear term f(s)=s3-s satisfies these assumptions.

3.2 A priori estimates

We multiply (3.4) by ut and integrate over Ω and by parts. This gives

ddt(Ak1/2u2+Bk1/2[u]+2ΩF(u)𝑑x)+2ut2=0,(3.9)

where

Bk1/2[u]=i=1k-1|α|=iaα𝒟αu2

(note that Bk1/2[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality

vHi(Ω)c(i)vHm(Ω)imv1-im,vHm(Ω),i{1,,m-1},m,m2,(3.10)

there holds

|Bk1/2[u]|12Ak1/2u2+cu2.

This yields, employing (3.8),

Ak1/2u2+Bk1/2[u]+2ΩF(u)𝑑x12Ak1/2u2+ΩF(u)𝑑x+cuL4(Ω)4-cu2-c′′,

whence

Ak1/2u2+Bk1/2[u]+2ΩF(u)𝑑xc(uHk(Ω)2+ΩF(u)𝑑x)-c,c>0,(3.11)

noting that, owing to Young’s inequality,

u2ϵuL4(Ω)4+c(ϵ)for all ϵ>0.(3.12)

We then multiply (3.4) by u and have, owing to (3.7) and the interpolation inequality (3.10),

ddtu2+c(uHk(Ω)2+ΩF(u)𝑑x)cu2+c′′,

hence, proceeding as above and employing, in particular, (3.8),

ddtu2+c(uHk(Ω)2+ΩF(u)𝑑x)c,c>0.(3.13)

Summing (3.9) and (3.13), we obtain a differential inequality of the form

dE1dt+c(E1+ut2)c,c>0,(3.14)

where

E1=Ak1/2u2+Bk1/2[u]+2ΩF(u)𝑑x+u2

satisfies, owing to (3.11),

E1c(uHk(Ω)2+ΩF(u)𝑑x)-c,c>0.(3.15)

Note indeed that

E1cuHk(Ω)2+2ΩF(u)𝑑xc(uHk(Ω)2+ΩF(u)𝑑x)-c,c>0,c0.

It follows from (3.14)–(3.15) and Gronwall’s lemma that

u(t)Hk(Ω)2ce-ct(u0Hk(Ω)2+ΩF(u0)𝑑x)+c′′,c>0,t0,(3.16)

and

tt+rut2𝑑sce-ct(u0Hk(Ω)2+ΩF(u0)𝑑x)+c′′,c>0,t0,r>0 given.(3.17)

Next, we multiply (3.4) by Aku and find, owing to the interpolation inequality (3.10),

ddtAk1/2u2+cuH2k(Ω)2c(u2+f(u)2).

It follows from the continuity of f and F, the continuous embedding Hk(Ω)𝒞(Ω¯) (recall that k2) and (3.16) that

u2+f(u)2Q(uHk(Ω))e-ctQ(u0Hk(Ω))+c,c>0,t0,(3.18)

so that

ddtAk1/2u2+cuH2k(Ω)2e-ctQ(u0Hk(Ω))+c′′,c,c>0,t0.(3.19)

Summing (3.14) and (3.19), we have a differential inequality of the form

dE2dt+c(E2+uH2k(Ω)2+ut2)e-ctQ(u0Hk(Ω))+c′′,c,c>0,t0,

where

E2=E1+Ak1/2u2

satisfies

E2c(uHk(Ω)2+ΩF(u)𝑑x)-c,c>0.

We then rewrite (3.4) as an elliptic equation, for t>0 fixed,

Aku=-ut-Bku-f(u),𝒟αu=0on Γ,|α|k-1.(3.20)

Multiplying (3.20) by Aku, we obtain, owing to the interpolation inequality (3.10),

Aku2c(u2+f(u)2+ut2),

hence, owing to (3.18),

uH2k(Ω)2c(e-ctQ(u0Hk(Ω))+ut2)+c′′,c>0.(3.21)

Next, we differentiate (3.4) with respect to time and find

tut+Akut+Bkut+f(u)ut=0,(3.22)𝒟αut=0on Γ,|α|k-1,(3.23)ut|t=0=-Aku0-Bku0-f(u0).(3.24)

We can note that, if u0H2k(Ω)H0k(Ω)(=D(Ak)), then ut(0)L2(Ω) and

ut(0)Q(u0H2k(Ω)).(3.25)

We multiply (3.22) by ut and have, owing to (3.6) and the interpolation inequality (3.10),

ddtut2+cutHk(Ω)2cut2,c>0.(3.26)

It follows from (3.17) (for r=1), (3.26) and the uniform Gronwall’s lemma that

ut(t)2e-ctQ(u0Hk(Ω))+c,c>0,t1,(3.27)

and from (3.25)–(3.26) and Gronwall’s lemma that

ut(t)2ectQ(u0H2k(Ω)),t0.(3.28)

We finally deduce from (3.21) and (3.27)–(3.28) that

u(t)H2k(Ω)e-ctQ(u0Hk(Ω))+c,c>0,t1,(3.29)

and

u(t)H2k(Ω)e-ctQ(u0H2k(Ω))+c,c>0,t0.(3.30)

3.3 The dissipative semigroup

Theorem 3.2.

The following statements hold.

  • (i)

    We assume that u0H0k(Ω) . Then problem ( 3.1 )–( 3.3 ) possesses a unique weak solution u such that, for all T>0,

    uL(+;H0k(Ω))L2(0,T;H2k(Ω)H0k(Ω))

    and

    utL2(0,T;L2(Ω)).

  • (ii)

    If we further assume that u0H2k(Ω)H0k(Ω) , then

    uL(+;H2k(Ω)H0k(Ω)).

Proof.

The proofs of existence and regularity in (i) and (ii) follow from the a priori estimates derived in the previous subsection and, e.g., a standard Galerkin scheme.

Let now u1 and u2 be two solutions to (3.1)–(3.2) with initial data u0,1 and u0,2, respectively. We set u=u1-u2 and u0=u0,1-u0,2 and have

ut+Aku+Bku+f(u1)-f(u2)=0,(3.31)𝒟αu=0on Γ,|α|k-1,(3.32)u|t=0=u0.(3.33)

Multiplying (3.31) by u, we obtain, owing to (3.6) and the interpolation inequality (3.10),

ddtu2+cuHk(Ω)2cu2,c>0.(3.34)

It follows from (3.34) and Gronwall’s lemma that

u(t)2ectu02,t0.(3.35)

Hence the uniqueness is proved, as well as the continuous dependence with respect to the initial data in the L2-norm. ∎

It follows from Theorem 3.2 that we can define the continuous (for the L2-norm) semigroup S(t):ΦΦ, u0u(t), t0 (i.e., S(0)=I (identity operator) and S(t+τ)=S(t)S(τ), t, τ0), where Φ=H0k(Ω). Furthermore, S(t) is dissipative in Φ, owing to (3.16), in the sense that it possesses a bounded absorbing set 0Φ (i.e., for all BΦ bounded, there exists t0=t0(B)0 such that tt0 implies S(t)B0).

Remark 3.3.

We can also prove the continuous dependence with respect to the initial data in the Hk- and H2k-norms and it then follows from (3.30) that S(t) is defined, continuous and dissipative in (H2k(Ω)H0k(Ω)).

Actually, it follows from (3.29) that S(t) possesses a bounded absorbing set 1 such that 1 is compact in Φ and bounded in H2k(Ω). It thus follows from classical results (see, e.g., [33, 41]) that we have the following result.

Theorem 3.4.

The semigroup S(t) possesses the global attractor A which is compact in Φ and bounded in H2k(Ω).

Remark 3.5.

It follows from (3.35) that we can extend S(t) (by continuity and in a unique way) to L2(Ω).

Remark 3.6.

(i) We recall that the global attractor 𝒜 is the smallest (for the inclusion) compact set of the phase space which is invariant by the flow (i.e., S(t)𝒜=𝒜 for all t0) and attracts all bounded sets of initial data as time goes to infinity; it thus appears as a suitable object in view of the study of the asymptotic behavior of the system. We refer the reader to, e.g., [33, 41] for more details and discussions on this.

(ii) We can also prove, based on standard arguments (see, e.g., [33, 41]) that 𝒜 has finite dimension, in the sense of covering dimensions such as the Hausdorff and the fractal dimensions. The finite-dimensionality means, very roughly speaking, that even though the initial phase space has infinite dimension, the reduced dynamics can be described by a finite number of parameters (we refer the interested reader to, e.g., [33, 41] for discussions on this subject).

Remark 3.7.

We can also consider periodic boundary conditions, namely, u is Ω-periodic, in which case we have Ω=Πi=13(0,Li), Li>0, i{1,2,3}. In that case, we consider the operator 𝐀k=I+Ak (in order to have a strictly positive operator), where Ak is as above, but based on Sobolev spaces with periodic functions (see, e.g., [41]), and rewrite (3.1) in the form

ut+𝐀ku+Bku+g(u)=0,

where g(s)=f(s)-s (note that g satisfies properties which are similar to (3.5)–(3.8)).

4 The Cahn–Hilliard theory

4.1 Setting of the problem

We consider the following initial and boundary value problem, for k, k2 (the case k=1 can be treated as in the original Cahn–Hilliard equation; see, e.g., [13]):

ut-Δi=1k(-1)i|α|=iaα𝒟2αu-Δf(u)=0,(4.1)𝒟αu=0on Γ,|α|k,(4.2)u|t=0=u0.(4.3)

Remark 4.1.

For k=1 (anisotropic Cahn–Hilliard equation), we have an equation of the form

ut+Δi=13ai2uxi2-Δf(u)=0

and, for k=2 (fourth-order anisotropic Cahn–Hilliard equation), we have an equation of the form

ut-Δi,j=13aij4uxi2xj2+Δi=13bi2uxi2-Δf(u)=0.

Keeping the same notation as in the previous section, we rewrite (4.1) as

ut-ΔAku-ΔBku-Δf(u)=0.(4.4)

As far as the nonlinear term f is concerned, we assume that the assumptions of the previous section hold and that f is of class 𝒞2.

4.2 A priori estimates

We multiply (4.4) by (-Δ)-1ut . This gives

ddt(Ak1/2u2+Bk1/2[u]+2ΩF(u)𝑑x)+2ut-12=0.(4.5)

We then multiply (4.4) by (-Δ)-1u and have, owing to (3.7) and the interpolation inequality (3.10) and proceeding as in the previous section,

ddtu-12+c(uHk(Ω)2+ΩF(u)𝑑x)c,c>0.(4.6)

Summing (4.5) and (4.6), we obtain a differential inequality of the form

dE3dt+c(E3+ut-12)c,c>0,(4.7)

where

E3=Ak1/2u2+Bk1/2[u]+2ΩF(u)𝑑x+u-12

satisfies

E3c(uHk(Ω)2+ΩF(u)𝑑x)-c,c>0.(4.8)

It follows from (4.7)–(4.8) and Gronwall’s lemma that

u(t)Hk(Ω)2ce-ct(u0Hk(Ω)2+ΩF(u0)𝑑x)+c′′,c>0,t0,(4.9)

and

tt+rut-12𝑑sce-ct(u0Hk(Ω)2+ΩF(u0)𝑑x)+c′′,c>0,t0,r>0 given.(4.10)

Multiplying next (4.4) by A~ku, we find, owing to the interpolation inequality (3.10) and proceeding as in the previous section,

ddtA~k1/2u2+cuH2k(Ω)2e-ctQ(u0Hk(Ω))+c′′,c,c>0,t0.(4.11)

Summing (4.7) and (4.11), we have a differential inequality of the form

dE4dt+c(E4+uH2k(Ω)2+ut-12)e-ctQ(u0Hk(Ω))+c′′,c,c>0,t0,(4.12)

where

E4=E3+A~k1/2u2

satisfies

E4c(uHk(Ω)2+ΩF(u)𝑑x)-c,c>0.

We also multiply (4.4) by ut and obtain, noting that f is of class 𝒞2,

ddt(A¯k1/2u2+B¯k1/2[u])+ut2e-ctQ(u0Hk(Ω))+c′′,c,c>0,(4.13)

where

B¯k1/2[u]=i=1k-1|α|=iaα𝒟αu2.

Summing finally (4.12) and (4.13), we find a differential inequality of the form

dE5dt+c(E5+uH2k(Ω)2+ut2)e-ctQ(u0Hk(Ω))+c′′,c,c>0,t0,(4.14)

where

E5=E4+A¯k1/2u2+B¯k1/2[u]

satisfies

E5c(uHk+1(Ω)2+ΩF(u)𝑑x)-c,c>0.(4.15)

In particular, it follows from (4.14)–(4.15) that

u(t)Hk+1(Ω)e-ctQ(u0Hk+1(Ω))+c,c>0,t0.(4.16)

We then differentiate (4.4) with respect to time and have

tut-ΔAkut-ΔBkut-Δ(f(u)ut)=0,(4.17)𝒟αut=0on Γ,|α|k.(4.18)

We multiply (4.17) by (-Δ)-1ut and obtain, owing to (3.6) and the interpolation inequality (3.10),

ddtut-12+cutHk(Ω)2cut2,c>0,

which yields, employing the interpolation inequality

v2cv-1vH1(Ω),vH01(Ω),(4.19)

the differential inequality

ddtut-12+cutHk(Ω)2cut-12,c>0.(4.20)

In particular, this yields, owing to (4.10) and employing the uniform Gronwall’s lemma,

ut(t)-1e-ctQ(u0Hk(Ω))+c,c>0,tr,r>0 given.(4.21)

We finally rewrite (4.4) as an elliptic equation, for t>0 fixed,

Aku=-(-Δ)-1ut-Bku-f(u),𝒟αu=0on Γ,|α|k-1.(4.22)

Multiplying (4.22) by Aku, we find, owing to the interpolation inequality (3.10),

uH2k(Ω)2c(e-ctQ(u0Hk(Ω))+ut-12)+c′′,c>0.(4.23)

In particular, it follows from (4.21) (for r=1) and (4.23) that

u(t)H2k(Ω)e-ctQ(u0Hk(Ω))+c,c>0,t1.(4.24)

Remark 4.2.

If we assume that u0H2k+1(Ω)H0k(Ω), we deduce from (4.20), (4.23) and Gronwall’s lemma an H2k-estimate on u on [0,1] which, combined with (4.24), gives an H2k-estimate on u, for all times. This is however not satisfactory, in particular, in view of the study of attractors.

Remark 4.3.

We further assume that f is of class 𝒞k+1. Multiplying (4.4) by A~kut, we have

12ddt(Aku2+((Aku,Bku)))+A~k1/2ut2=-((A¯k1/2f(u),A~k1/2ut)),

which yields, noting that A¯k1/2f(u)2Q(uHk+1(Ω)) and owing to (4.16),

ddt(Aku2+((Aku,Bku)))e-ctQ(u0Hk+1(Ω))+c,c>0,t0.(4.25)

Combining (4.25) with (4.14), it follows from (4.15) and the interpolation inequality (3.10) that

u(t)H2k(Ω)Q(u0H2k(Ω)),t[0,1],

so that, owing to (4.24),

u(t)H2k(Ω)e-ctQ(u0H2k(Ω))+c,c>0,t0.(4.26)

4.3 The dissipative semigroup

Theorem 4.4.

The following statements hold.

  • (i)

    We assume that u0H0k(Ω) . Then problem ( 4.1 )–( 4.3 ) possesses a unique weak solution u such that, for all T>0,

    uL(+;H0k(Ω))L2(0,T;H2k(Ω)H0k(Ω))

    and

    utL2(0,T;H-1(Ω)).

  • (ii)

    If we further assume that u0Hk+1(Ω)H0k(Ω) , then, for all T>0,

    uL(+;Hk+1(Ω)H0k(Ω))

    and

    utL2(0,T;L2(Ω)).

  • (iii)

    If we further assume that f is of class 𝒞k+1 and u0H2k(Ω)H0k(Ω) , then

    uL(+;H2k(Ω)H0k(Ω)).

Proof.

The proofs of existence and regularity in (i), (ii) and (iii) follow from the a priori estimates derived in the previous subsection and, e.g., a standard Galerkin scheme.

Let now u1 and u2 be two solutions to (4.1)–(4.2) with initial data u0,1 and u0,2, respectively. We set u=u1-u2 and u0=u0,1-u0,2 and have

ut-ΔAku-ΔBku-Δ(f(u1)-f(u2))=0,(4.27)𝒟αu=0on Γ,|α|k,(4.28)u|t=0=u0.(4.29)

Multiplying (4.27) by (-Δ)-1u, we obtain, owing to (3.6) and the interpolation inequalities (3.10) and (4.19),

ddtu-12+cuHk(Ω)2cu-12,c>0.(4.30)

It follows from (4.30) and Gronwall’s lemma that

u(t)-12ectu0-12,t0.(4.31)

Hence the uniqueness is proved, as well as the continuous dependence with respect to the initial data in the H-1-norm. ∎

It follows from Theorem 4.4 that we can define the family of solving operators

S(t):ΦΦ,u0u(t),t0,

where Φ=H0k(Ω). This family of solving operators forms a semigroup which is continuous with respect to the H-1-topology. Finally, the following theorem is a consequence of (4.9).

Theorem 4.5.

The semigroup S(t) is dissipative in Φ.

Remark 4.6.

(i) Actually, it follows from (4.24) that we have a bounded absorbing set 1 which is compact in Φ and bounded in H2k(Ω). This yields the existence of the global attractor 𝒜 which is compact in Φ and bounded in H2k(Ω).

(ii) It follows from (4.26) that, if f is of class 𝒞k+1, then S(t) is dissipative in H2k(Ω)H0k(Ω).

(iii) It follows from (4.31) that we can extend S(t) (by continuity and in a unique way) to H-1(Ω).

Remark 4.7.

The case of periodic boundary conditions is more delicate, since, integrating (formally) (4.1) over Ω, we have the conservation of mass, namely, u(t)=u0, t0, where =1Vol(Ω)Ωdx. As a consequence, we cannot expect to find compact attractors on the whole phase space and have to deal with the nonlocal term f(u) (see, e.g., [41]).

5 Numerical simulations

In this section, we give numerical simulations which show the effects of the anisotropy for the generalized Allen–Cahn equations when k=1, 2 and 3 in the domain Ω=(0,1)×(0,1) (see Figure 1). In particular, this shows how the coefficients of highest orders affect the solutions. Furthermore, we compare the solutions when different values of k, time steps or coefficients are taken.

Figure 1

Computational domain: Ω=(0,1)×(0,1).

The numerical method applied here is a P1-finite element in space and a forward Euler discretization in time. The numerical simulations are performed with the software Freefem++ (see [24]).

For instance, when k=2, the generalized Allen–Cahn equation reads

ut+a204ux4+a024uy4+a114ux2y2-a102ux2-a012uy2+f(u)=0,

where, here and in all the simulations, f(s)=s3-s. We further assume that u is Ω-periodic. Finally, we take as initial condition a cross in the center of the computational domain, that is, the initial value in the middle cross is -0.8, while, in the complementary set, it is equal to 0.8, as shown in the following Figure 2.

Setting 2ux2=ω and 2uy2=p, then, integrating by parts, the system which needs to be solved reads

{(ut,v)-a20(ωx,vx)-a02(py,vy)-a11(ωy,vy)+a10(ux,vx)+a01(uy,vy)+(f(u),v)=0,(ω,ξ)=-(ux,ξx),(p,ζ)=-(uy,ζy),

where the test functions v, ξ, ζ all belong to Hper1(Ω).

Next, we introduce the discretization 𝒯h of Ω¯ and set

Vh={vhC0(Ω¯):(vh)|KP1 for all K𝒯h,vh is Ω-periodic}Hper1(Ω).

As mentioned above, we use a P1-finite element for the space discretization and a forward Euler scheme for the time discretization. Let uh0Vh. Then, for n0, we look for (uhn+1,ωhn+1,phn+1)Vh×Vh×Vh such that

{1dt(uhn+1,v)-a20(ωhn+1x,vx)-a02(phn+1y,vy)-a11(ωhn+1y,vy)+a10(uhn+1x,vx)+a01(uhn+1y,vy)+(f(uhn),v)-1dt(uhn,v)=0,(ωhn+1,ξ)=-(uhn+1x,ξx),(phn+1,ζ)=-(uhn+1y,ζy),

for all v,ξ,ζVh. We proceed in a similar way for k=1 and 3. In particular, for k=3, we have to deal with a system of five second-order equations.

As far as the time step dt is concerned, when k=1, we take dt=10-7 (in Figure 3 and Figure 7), dt=10-6 (in Figure 4 ) and dt=10-5 (in Figure 8). When k=2, we take dt=10-7 and, when k=3, we take dt=10-10 (in Figure 6 and Figure 12 ) and dt=10-8 (in Figure 10) or dt=10-7 (in Figure 3 and Figure 7). Here, we use a grid with 1502 points on the domain Ω.

Figure 2 shows the initial condition.

Initial condition.
Figure 2

Initial condition.

The isotropic case

When all the coefficients are set equal to 1, then, as expected, there is no isotropy. The figures below however show the effects of higher-order terms.

Results after 40 iterations with different values of k and with the same time step.


                        k=1{k=1}, d⁢t=10-7{dt=10^{-7}}.
(a)

k=1, dt=10-7.


                        k=2{k=2}, d⁢t=10-7{dt=10^{-7}}.
(b)

k=2, dt=10-7.


                        k=3{k=3}, d⁢t=10-7{dt=10^{-7}}.
(c)

k=3, dt=10-7.

In the next figures, we take a different time step. We also note that the higher k is, the smaller the time step has to be taken, since the solution evolves faster in time.

Results for k=1, dt=10-6.

After 100 iterations.
(a)

After 100 iterations.

After 250 iterations.
(b)

After 250 iterations.

After 1,000{1{,}000} iterations.
(c)

After 1,000 iterations.

After 4,000{4{,}000} iterations.
(d)

After 4,000 iterations.

Results for k=2, dt=10-7.

After 40 iterations.
(a)

After 40 iterations.

After 400 iterations.
(b)

After 400 iterations.

After 2,000{2{,}000} iterations.
(c)

After 2,000 iterations.

After 5,000{5{,}000} iterations.
(d)

After 5,000 iterations.

Results for k=3, dt=10-10.

After 40 iterations.
(a)

After 40 iterations.

After 500 iterations.
(b)

After 500 iterations.

After 1,000{1{,}000} iterations.
(c)

After 1,000 iterations.

After 2,0002{,}000 iterations.
(d)

After 2,000 iterations.

Anisotropy in the x-direction

We consider the following situations:

  • (i)

    k=1, a10=1 and a01=0.01.

  • (ii)

    k=2, a20=1 and the other coefficients are set equal to 0.01.

  • (iii)

    k=3, a30=1 and the other coefficients are set equal to 0.01.

We first investigate the anisotropy in the x-direction after 40 iterations, comparing different values of k when the time step is the same. We then illustrate the case when k, as well as the time step, remain unchanged, but the number of iterations increases. We can note that we would have similar results in the y-direction.

Anisotropy in the x-direction after 40 iterations.


                        k=1{k=1}, d⁢t=10-7{dt=10^{-7}}.
(a)

k=1, dt=10-7.


                        k=2{k=2}, d⁢t=10-7{dt=10^{-7}}.
(b)

k=2, dt=10-7.


                        k=3{k=3}, d⁢t=10-7{dt=10^{-7}}.
(c)

k=3, dt=10-7.

Results for k=1, dt=10-5.

After 50 iterations.
(a)

After 50 iterations.

After 1,000{1{,}000} iterations.
(b)

After 1,000 iterations.

After 2,000{2{,}000} iterations.
(c)

After 2,000 iterations.

After 5,000{5{,}000} iterations.
(d)

After 5,000 iterations.

Results for k=2, dt=10-7.

After 40 iterations.
(a)

After 40 iterations.

After 400 iterations.
(b)

After 400 iterations.

After 2,000{2{,}000} iterations.
(c)

After 2,000 iterations.

After 5,000{5{,}000} iterations.
(d)

After 5,000 iterations.

Results for k=3, dt=10-8.

After 40 iterations.
(a)

After 40 iterations.

After 500 iterations.
(b)

After 500 iterations.

After 1,000{1{,}000} iterations.
(c)

After 1,000 iterations.

After 2,000{2{,}000} iterations.
(d)

After 2,000 iterations.

Influence of the off-diagonal terms

We first note that, when k=1, there is no cross term. We thus consider the following two cases:

  • (i)

    k=2, a11=1 and the other coefficients are set equal to 0.01.

  • (ii)

    k=3, a21=1 and the other coefficients are set equal to 0.01.

Results for k=2, dt=10-7.

After 40 iterations.
(a)

After 40 iterations.

After 400 iterations.
(b)

After 400 iterations.

After 2,000{2{,}000} iterations.
(c)

After 2,000 iterations.

After 5,000{5{,}000} iterations.
(d)

After 5,000 iterations.

Results for k=3, dt=10-10.

After 40 iterations.
(a)

After 40 iterations.

After 250 iterations.
(b)

After 250 iterations.

After 1,000{1{,}000} iterations.
(c)

After 1,000 iterations.

After 5,000{5{,}000} iterations.
(d)

After 5,000 iterations.

References

  • [1]

    S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Stud. 2, Van Nostrand, New York, 1965.  Google Scholar

  • [2]

    S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations. I, Comm. Pure Appl. Math. 12 (1959), 623–727.  CrossrefGoogle Scholar

  • [3]

    S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations. II, Comm. Pure Appl. Math. 17 (1964), 35–92.  CrossrefGoogle Scholar

  • [4]

    S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1085–1095.  CrossrefGoogle Scholar

  • [5]

    J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E 77 (2008), Article ID 061506.  Web of ScienceGoogle Scholar

  • [6]

    J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E 73 (2006), Article ID 031609.  Google Scholar

  • [7]

    G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), 311–350.  Google Scholar

  • [8]

    J. W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961), 795–801.  CrossrefGoogle Scholar

  • [9]

    J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 2 (1958), 258–267.  Google Scholar

  • [10]

    F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn–Hilliard systems, Commun. Comput. Phys. 13 (2013), 1189–1208.  Web of ScienceCrossrefGoogle Scholar

  • [11]

    X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Ration. Mech. Anal. 202 (2011), 349–372.  CrossrefWeb of ScienceGoogle Scholar

  • [12]

    L. Cherfils, A. Miranville and S. Peng, Higher-order Allen–Cahn models with logarithmic nonlinear terms, Advances in Dynamical Systems and Control, Stud. Syst. Decis. Control 69, Springer, Cham (2016), 247–263.  Web of ScienceGoogle Scholar

  • [13]

    L. Cherfils, A. Miranville and S. Peng, Higher-order models in phase separation, J. Appl. Anal. Comput. 7 (2017), 39–56.  Google Scholar

  • [14]

    L. Cherfils, A. Miranville and S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), 561–596.  Web of ScienceCrossrefGoogle Scholar

  • [15]

    P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys. 72 (1980), 4756–4763.  CrossrefGoogle Scholar

  • [16]

    H. Emmerich, H. Löwen, R. Wittkowski, T. Gruhn, G. I. Tóth, G. Tegze and L. Gránásy, Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: An overview, Adv. Phys. 61 (2012), 665–743.  Web of ScienceCrossrefGoogle Scholar

  • [17]

    P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift–Hohenberg equations with fast dynamics, Phys. Rev. E 79 (2009), Article ID 051110.  Web of ScienceGoogle Scholar

  • [18]

    G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction. I. Macroscopic limits, J. Stat. Phys. 87 (1997), 37–61.  CrossrefGoogle Scholar

  • [19]

    G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction. II. Interface motion, SIAM J. Appl. Math. 58 (1998), 1707–1729.  CrossrefGoogle Scholar

  • [20]

    G. Gompper and M. Kraus, Ginzburg–Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E 47 (1993), 4289–4300.  CrossrefGoogle Scholar

  • [21]

    G. Gompper and M. Kraus, Ginzburg–Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E 47 (1993), 4301–4312.  CrossrefGoogle Scholar

  • [22]

    M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation, Math. Models Methods Appl. Sci. 24 (2014), 2743–2783.  CrossrefGoogle Scholar

  • [23]

    M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation, Discrete Contin. Dyn. Syst. 35 (2015), 2539–2564.  Web of ScienceGoogle Scholar

  • [24]

    F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), 251–265.  Web of ScienceGoogle Scholar

  • [25]

    Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable finite difference, nonlinear multigrid simulation of the phase field crystal equation, J. Comput. Phys. 228 (2009), 5323–5339.  CrossrefGoogle Scholar

  • [26]

    R. Kobayashi, Modelling and numerical simulations of dendritic crystal growth, Phys. D 63 (1993), 410–423.  CrossrefGoogle Scholar

  • [27]

    M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn–Hilliard type equation, SIAM J. Math. Anal. 44 (2012), 3369–3387.  Web of ScienceCrossrefGoogle Scholar

  • [28]

    M. Korzec and P. Rybka, On a higher order convective Cahn–Hilliard type equation, SIAM J. Appl. Math. 72 (2012), 1343–1360.  CrossrefWeb of ScienceGoogle Scholar

  • [29]

    A. Miranville, Asymptotic behavior of a sixth-order Cahn–Hilliard system, Central Europ. J. Math. 12 (2014), 141–154.  Google Scholar

  • [30]

    A. Miranville, Sixth-order Cahn–Hilliard equations with logarithmic nonlinear terms, Appl. Anal. 94 (2015), 2133–2146.  CrossrefGoogle Scholar

  • [31]

    A. Miranville, Sixth-order Cahn–Hilliard systems with dynamic boundary conditions, Math. Methods Appl. Sci. 38 (2015), 1127–1145.  Web of ScienceCrossrefGoogle Scholar

  • [32]

    A. Miranville, On the phase-field-crystal model with logarithmic nonlinear terms, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110 (2016), 145–157.  Google Scholar

  • [33]

    A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations. Vol. 4, Elsevier, Amsterdam (2008), 103–200.  Web of ScienceGoogle Scholar

  • [34]

    A. Novick-Cohen, The Cahn–Hilliard equation, Handbook of Differential Equations: Evolutionary Equations. Vol. 4, Elsevier, Amsterdam (2008), 201–228.  Web of ScienceGoogle Scholar

  • [35]

    I. Pawlow and G. Schimperna, A Cahn–Hilliard equation with singular diffusion, J. Differential Equations 254 (2013), 779–803.  CrossrefWeb of ScienceGoogle Scholar

  • [36]

    I. Pawlow and G. Schimperna, On a Cahn–Hilliard model with nonlinear diffusion, SIAM J. Math. Anal. 45 (2013), 31–63.  CrossrefGoogle Scholar

  • [37]

    I. Pawlow and W. Zajaczkowski, A sixth order Cahn–Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal. 10 (2011), 1823–1847.  Web of ScienceCrossrefGoogle Scholar

  • [38]

    I. Pawlow and W. Zajaczkowski, On a class of sixth order viscous Cahn–Hilliard type equations, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 517–546.  Web of ScienceGoogle Scholar

  • [39]

    T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E 67 (2003), Article ID 021606.  Google Scholar

  • [40]

    J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater. 40 (1992), 1475–1495.  CrossrefGoogle Scholar

  • [41]

    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci. 68, Springer, New York, 1997.  Google Scholar

  • [42]

    S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), 1337–1359.  CrossrefGoogle Scholar

  • [43]

    C. Wang and S. M. Wise, Global smooth solutions of the modified phase field crystal equation, Methods Appl. Anal. 17 (2010), 191–212.  Google Scholar

  • [44]

    C. Wang and S. M. Wise, An energy stable and convergent finite difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal. 49 (2011), 945–969.  CrossrefWeb of ScienceGoogle Scholar

  • [45]

    A. A. Wheeler and G. B. McFadden, On the notion of ξ-vector and stress tensor for a general class of anisotropic diffuse interface models, Proc. R. Soc. Lond. Ser. A 453 (1997), 1611–1630.  CrossrefGoogle Scholar

  • [46]

    S. M. Wise, C. Wang and J. S. Lowengrub, An energy stable and convergent finite difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), 2269–2288.  CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2016-06-16

Accepted: 2017-02-08

Published Online: 2017-03-16


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 278–302, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0137.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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