Higher-order anisotropic models in phase separation

Laurence Cherfils; Alain Miranville; Shuiran Peng

doi:10.1515/anona-2016-0137

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Higher-order anisotropic models in phase separation

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Veröffentlicht/Copyright: 16. März 2017

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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,

(1.1) Ψ GL = ∫ Ω ( α 2 ⁢ | ∇ ⁡ u | 2 + F ⁢ ( u ) ) ⁢ 𝑑 x , α > 0 ,

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 ℝ 3 , with boundary Γ; we can of course also consider bounded and regular domains of ℝ and ℝ 2 ). The Allen–Cahn equation (which corresponds to an L 2 -gradient flow of the Ginzburg–Landau free energy) then reads

∂ ⁡ u ∂ ⁡ t - α ⁢ Δ ⁢ u + f ⁢ ( u ) = 0 ,

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

∂ ⁡ u ∂ ⁡ t + α ⁢ Δ 2 ⁢ u - Δ ⁢ f ⁢ ( 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:

(1.2) Ψ HOGL = ∫ Ω ( 1 2 ⁢ ∑ i = 1 k ∑ | α | = i a α ⁢ | 𝒟 α ⁢ u | 2 + F ⁢ ( u ) ) ⁢ 𝑑 x , k ∈ ℕ ,

where, for α = ( k 1 , k 2 , k 3 ) ∈ ( ℕ ∪ { 0 } ) 3 ,

| α | = k 1 + k 2 + k 3

and, for α ≠ ( 0 , 0 , 0 ) ,

𝒟 α = ∂ | α | ∂ ⁡ x 1 k 1 ⁢ ∂ ⁡ x 2 k 2 ⁢ ∂ ⁡ x 3 k 3

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

(1.3) ∂ ⁡ u ∂ ⁡ t + ∑ i = 1 k ( - 1 ) i ⁢ ∑ | α | = i a α ⁢ 𝒟 2 ⁢ α ⁢ u + f ⁢ ( u ) = 0 ,

(1.4) ∂ ⁡ u ∂ ⁡ t - Δ ⁢ ∑ i = 1 k ( - 1 ) i ⁢ ∑ | α | = i a α ⁢ 𝒟 2 ⁢ α ⁢ u - Δ ⁢ f ⁢ ( u ) = 0 .

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

∂ ⁡ u ∂ ⁡ t + P ⁢ ( - Δ ) ⁢ u + f ⁢ ( u ) = 0 ,

∂ ⁡ u ∂ ⁡ t - Δ ⁢ P ⁢ ( - Δ ) ⁢ u - Δ ⁢ f ⁢ ( u ) = 0 ,

where

P ⁢ ( s ) = ∑ i = 1 k a i ⁢ s i , a k > 0 , k ≥ 1 , s ∈ ℝ .

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 ∈ ℕ , k ≥ 2 , and

a α > 0 , | α | = k ,

and we introduce the elliptic operator A k defined by

〈 A k ⁢ v , w 〉 H - k ⁢ ( Ω ) , H 0 k ⁢ ( Ω ) = ∑ | α | = k a α ⁢ ( ( 𝒟 α ⁢ v , 𝒟 α ⁢ w ) ) ,

where H - k ⁢ ( Ω ) is the topological dual of H 0 k ⁢ ( Ω ) . Furthermore, ( ( ⋅ , ⋅ ) ) denotes the usual L 2 -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 ) ∈ H 0 k ⁢ ( Ω ) 2 ↦ ∑ | α | = k a α ⁢ ( ( 𝒟 α ⁢ v , 𝒟 α ⁢ w ) )

is bilinear, symmetric, continuous and coercive, so that

A k : H 0 k ⁢ ( Ω ) → H - k ⁢ ( Ω )

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

D ⁢ ( A k ) = H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) ,

where, for v ∈ D ⁢ ( A k ) ,

A k ⁢ v = ( - 1 ) k ⁢ ∑ | α | = k a α ⁢ 𝒟 2 ⁢ α ⁢ v .

We further note that D ⁢ ( A k 1 / 2 ) = H 0 k ⁢ ( Ω ) and, for ( v , w ) ∈ D ⁢ ( A k 1 / 2 ) 2 ,

( ( A k 1 / 2 ⁢ v , A k 1 / 2 ⁢ w ) ) = ∑ | α | = k a α ⁢ ( ( 𝒟 α ⁢ v , 𝒟 α ⁢ w ) ) .

We finally note that (see, e.g., [41]) ∥ A k ⋅ ∥ (respectively, ∥ A k 1 / 2 ⋅ ∥ ) is equivalent to the usual H 2 ⁢ k -norm (respectively, H k -norm) on D ⁢ ( A k ) (respectively, D ⁢ ( A k 1 / 2 ) ).

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

A ¯ k : H 0 k + 1 ⁢ ( Ω ) → H - k - 1 ⁢ ( Ω )

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

D ⁢ ( A ¯ k ) = H 2 ⁢ k + 2 ⁢ ( Ω ) ∩ H 0 k + 1 ⁢ ( Ω ) ,

where, for v ∈ D ⁢ ( A ¯ k ) ,

A ¯ k ⁢ v = ( - 1 ) k + 1 ⁢ Δ ⁢ ∑ | α | = k a α ⁢ 𝒟 2 ⁢ α ⁢ v .

Furthermore, D ⁢ ( A ¯ k 1 / 2 ) = H 0 k + 1 ⁢ ( Ω ) and, for ( v , w ) ∈ D ⁢ ( A ¯ k 1 / 2 ) 2 ,

( ( A ¯ k 1 / 2 ⁢ v , A ¯ k 1 / 2 ⁢ w ) ) = ∑ | α | = k a α ⁢ ( ( ∇ ⁡ 𝒟 α ⁢ v , ∇ ⁡ 𝒟 α ⁢ w ) ) .

Besides, ∥ A ¯ k ⋅ ∥ (respectively, ∥ A ¯ k 1 / 2 ⋅ ∥ ) is equivalent to the usual H 2 ⁢ k + 2 -norm (respectively, H k + 1 -norm) on D ⁢ ( A ¯ k ) (respectively, D ⁢ ( A ¯ k 1 / 2 ) ).

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

A ~ k : H 0 k - 1 ⁢ ( Ω ) → H - k + 1 ⁢ ( Ω ) ;

note that, as - Δ and A k commute, the same holds for ( - Δ ) - 1 and A k , so that A ~ k = A k ⁢ ( - 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 ) = H 2 ⁢ k - 2 ⁢ ( Ω ) ∩ H 0 k - 1 ⁢ ( Ω ) ,

where, for v ∈ D ⁢ ( A ~ k ) ,

A ~ k ⁢ v = ( - 1 ) k ⁢ ∑ | α | = k a α ⁢ 𝒟 2 ⁢ α ⁢ ( - Δ ) - 1 ⁢ v .

Furthermore, D ⁢ ( A ~ k 1 / 2 ) = H 0 k - 1 ⁢ ( Ω ) and, for ( v , w ) ∈ D ⁢ ( A ~ k 1 / 2 ) 2 ,

( ( A ~ k 1 / 2 ⁢ v , A ~ k 1 / 2 ⁢ w ) ) = ∑ | α | = k a α ⁢ ( ( 𝒟 α ⁢ ( - Δ ) - 1 / 2 ⁢ v , 𝒟 α ⁢ ( - Δ ) - 1 / 2 ⁢ w ) ) .

Besides, ∥ A ~ k ⋅ ∥ (respectively, ∥ A ~ k 1 / 2 ⋅ ∥ ) is equivalent to the usual H 2 ⁢ k - 2 -norm (respectively, H k - 1 -norm) on D ⁢ ( A ~ k ) (respectively, D ⁢ ( A ~ k 1 / 2 ) ).

Proof.

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

D ⁢ ( A ~ k ) = { v ∈ H 0 k - 1 ⁢ ( Ω ) : A ~ k ⁢ v ∈ L 2 ⁢ ( Ω ) } .

Noting that A ~ k ⁢ v = f , f ∈ L 2 ⁢ ( Ω ) , v ∈ D ⁢ ( A ~ k ) , is equivalent to A k ⁢ v = - Δ ⁢ f , where - Δ ⁢ f ∈ H 2 ⁢ ( Ω ) ′ , it follows from the elliptic regularity results of [1, 2, 3] that v ∈ H 2 ⁢ k - 2 ⁢ ( Ω ) , so that D ⁢ ( A ~ k ) = H 2 ⁢ k - 2 ⁢ ( Ω ) ∩ H 0 k - 1 ⁢ ( Ω ) . Noting then that A ~ k - 1 maps L 2 ⁢ ( Ω ) onto H 2 ⁢ k - 2 ⁢ ( Ω ) and recalling that k ≥ 2 , we deduce that A ~ k has compact inverse. We now note that, considering the spectral properties of - Δ and A k (see, e.g., [41]) and recalling that these two operators commute, - Δ and A k have a spectral basis formed of common eigenvectors. This yields that, for all s 1 , s 2 ∈ ℝ , ( - Δ ) s 1 and A k s 2 commute. Having this, we see that A ~ k 1 / 2 = ( - Δ ) - 1 / 2 ⁢ A k 1 / 2 , so that D ⁢ ( A ~ k 1 / 2 ) = H 0 k - 1 ⁢ ( Ω ) , and, for ( v , w ) ∈ D ⁢ ( A ~ k 1 / 2 ) 2 ,

( ( A ~ k 1 / 2 ⁢ v , A ~ k 1 / 2 ⁢ w ) ) = ∑ | α | = k a α ⁢ ( ( 𝒟 α ⁢ ( - Δ ) - 1 / 2 ⁢ v , 𝒟 α ⁢ ( - Δ ) - 1 / 2 ⁢ w ) ) .

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

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 k ≥ 2 (for k = 1 , the problem can be treated as in the original Allen–Cahn equation; see, e.g., [13]):

(3.1) ∂ ⁡ u ∂ ⁡ t + ∑ i = 1 k ( - 1 ) i ⁢ ∑ | α | = i a α ⁢ 𝒟 2 ⁢ α ⁢ u + f ⁢ ( u ) = 0 ,

(3.2) 𝒟 α ⁢ u = 0 on ⁢ Γ , | α | ≤ k - 1 ,

(3.3) u | t = 0 = u 0 .

Remark 3.1.

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

∂ ⁡ u ∂ ⁡ t - ∑ i = 1 3 a i ⁢ ∂ 2 ⁡ u ∂ ⁡ x i 2 + f ⁢ ( u ) = 0

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

∂ ⁡ u ∂ ⁡ t + ∑ i , j = 1 3 a i ⁢ j ⁢ ∂ 4 ⁡ u ∂ ⁡ x i 2 ⁢ ∂ ⁡ x j 2 - ∑ i = 1 3 b i ⁢ ∂ 2 ⁡ u ∂ ⁡ x i 2 + f ⁢ ( u ) = 0 .

We actually rewrite (3.1) in the equivalent form

(3.4) ∂ ⁡ u ∂ ⁡ t + A k ⁢ u + B k ⁢ u + f ⁢ ( u ) = 0 ,

where

B k ⁢ v = ∑ i = 1 k - 1 ( - 1 ) i ⁢ ∑ | α | = i a α ⁢ 𝒟 2 ⁢ α ⁢ v .

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

(3.5) f ∈ 𝒞 1 ⁢ ( ℝ ) , f ⁢ ( 0 ) = 0 ,

(3.6) f ′ ≥ - c 0 , c 0 ≥ 0 ,

(3.7) f ⁢ ( s ) ⁢ s ≥ c 1 ⁢ F ⁢ ( s ) - c 2 ≥ - c 3 , c 1 > 0 , c 2 , c 3 ≥ 0 , s ∈ ℝ ,

(3.8) F ⁢ ( s ) ≥ c 4 ⁢ s 4 - c 5 , c 4 > 0 , c 5 ≥ 0 , s ∈ ℝ ,

where F ⁢ ( s ) = ∫ 0 s f ⁢ ( ξ ) ⁢ 𝑑 ξ . In particular, the usual cubic nonlinear term f ⁢ ( s ) = s 3 - s satisfies these assumptions.

3.2 A priori estimates

We multiply (3.4) by ∂ ⁡ u ∂ ⁡ t and integrate over Ω and by parts. This gives

(3.9) d d ⁢ t ⁢ ( ∥ A k 1 / 2 ⁢ u ∥ 2 + B k 1 / 2 ⁢ [ u ] + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) + 2 ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 = 0 ,

where

B k 1 / 2 ⁢ [ u ] = ∑ i = 1 k - 1 ∑ | α | = i a α ⁢ ∥ 𝒟 α ⁢ u ∥ 2

(note that B k 1 / 2 ⁢ [ u ] is not necessarily nonnegative). We can note that, owing to the interpolation inequality

(3.10) ∥ v ∥ H i ⁢ ( Ω ) ≤ c ⁢ ( i ) ⁢ ∥ v ∥ H m ⁢ ( Ω ) i m ⁢ ∥ v ∥ 1 - i m , v ∈ H m ⁢ ( Ω ) , i ∈ { 1 , … , m - 1 } , m ∈ ℕ , m ≥ 2 ,

there holds

| B k 1 / 2 ⁢ [ u ] | ≤ 1 2 ⁢ ∥ A k 1 / 2 ⁢ u ∥ 2 + c ⁢ ∥ u ∥ 2 .

This yields, employing (3.8),

∥ A k 1 / 2 ⁢ u ∥ 2 + B k 1 / 2 ⁢ [ u ] + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ≥ 1 2 ⁢ ∥ A k 1 / 2 ⁢ u ∥ 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x + c ⁢ ∥ u ∥ L 4 ⁢ ( Ω ) 4 - c ′ ⁢ ∥ u ∥ 2 - c ′′ ,

whence

(3.11) ∥ A k 1 / 2 ⁢ u ∥ 2 + B k 1 / 2 ⁢ [ u ] + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ≥ c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 ,

noting that, owing to Young’s inequality,

(3.12) ∥ u ∥ 2 ≤ ϵ ⁢ ∥ u ∥ L 4 ⁢ ( Ω ) 4 + c ⁢ ( ϵ ) for all ⁢ ϵ > 0 .

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

d d ⁢ t ⁢ ∥ u ∥ 2 + c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) ≤ c ′ ⁢ ∥ u ∥ 2 + c ′′ ,

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

(3.13) d d ⁢ t ⁢ ∥ u ∥ 2 + c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) ≤ c ′ , c > 0 .

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

(3.14) d ⁢ E 1 d ⁢ t + c ⁢ ( E 1 + ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ) ≤ c ′ , c > 0 ,

where

E 1 = ∥ A k 1 / 2 ⁢ u ∥ 2 + B k 1 / 2 ⁢ [ u ] + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x + ∥ u ∥ 2

satisfies, owing to (3.11),

(3.15) E 1 ≥ c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 .

Note indeed that

E 1 ≤ c ⁢ ∥ u ∥ H k ⁢ ( Ω ) 2 + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ≤ c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 , c ′ ≥ 0 .

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

(3.16) ∥ u ⁢ ( t ) ∥ H k ⁢ ( Ω ) 2 ≤ c ⁢ e - c ′ ⁢ t ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u 0 ) ⁢ 𝑑 x ) + c ′′ , c ′ > 0 , t ≥ 0 ,

and

(3.17) ∫ t t + r ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ⁢ 𝑑 s ≤ c ⁢ e - c ′ ⁢ t ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u 0 ) ⁢ 𝑑 x ) + c ′′ , c ′ > 0 , t ≥ 0 , r > 0 ⁢ given .

Next, we multiply (3.4) by A k ⁢ u and find, owing to the interpolation inequality (3.10),

d d ⁢ t ⁢ ∥ A k 1 / 2 ⁢ u ∥ 2 + c ⁢ ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 ≤ c ⁢ ( ∥ u ∥ 2 + ∥ f ⁢ ( u ) ∥ 2 ) .

It follows from the continuity of f and F, the continuous embedding H k ⁢ ( Ω ) ⊂ 𝒞 ⁢ ( Ω ¯ ) (recall that k ≥ 2 ) and (3.16) that

(3.18) ∥ u ∥ 2 + ∥ f ⁢ ( u ) ∥ 2 ≤ Q ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 0 ,

so that

(3.19) d d ⁢ t ⁢ ∥ A k 1 / 2 ⁢ u ∥ 2 + c ⁢ ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 ≤ e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′′ , c , c ′ > 0 , t ≥ 0 .

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

d ⁢ E 2 d ⁢ t + c ⁢ ( E 2 + ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 + ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ) ≤ e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′′ , c , c ′ > 0 , t ≥ 0 ,

where

E 2 = E 1 + ∥ A k 1 / 2 ⁢ u ∥ 2

satisfies

E 2 ≥ c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 .

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

(3.20) A k ⁢ u = - ∂ ⁡ u ∂ ⁡ t - B k ⁢ u - f ⁢ ( u ) , 𝒟 α ⁢ u = 0 on ⁢ Γ , | α | ≤ k - 1 .

Multiplying (3.20) by A k ⁢ u , we obtain, owing to the interpolation inequality (3.10),

∥ A k ⁢ u ∥ 2 ≤ c ⁢ ( ∥ u ∥ 2 + ∥ f ⁢ ( u ) ∥ 2 + ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ) ,

hence, owing to (3.18),

(3.21) ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 ≤ c ⁢ ( e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ) + c ′′ , c ′ > 0 .

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

(3.22) ∂ ∂ ⁡ t ⁢ ∂ ⁡ u ∂ ⁡ t + A k ⁢ ∂ ⁡ u ∂ ⁡ t + B k ⁢ ∂ ⁡ u ∂ ⁡ t + f ′ ⁢ ( u ) ⁢ ∂ ⁡ u ∂ ⁡ t = 0 ,

(3.23) 𝒟 α ⁢ ∂ ⁡ u ∂ ⁡ t = 0 on ⁢ Γ , | α | ≤ k - 1 ,

(3.24) ∂ ⁡ u ∂ ⁡ t | t = 0 = - A k ⁢ u 0 - B k ⁢ u 0 - f ⁢ ( u 0 ) .

We can note that, if u 0 ∈ H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) ( = D ⁢ ( A k ) ) , then ∂ ⁡ u ∂ ⁡ t ⁢ ( 0 ) ∈ L 2 ⁢ ( Ω ) and

(3.25) ∥ ∂ ⁡ u ∂ ⁡ t ⁢ ( 0 ) ∥ ≤ Q ⁢ ( ∥ u 0 ∥ H 2 ⁢ k ⁢ ( Ω ) ) .

We multiply (3.22) by ∂ ⁡ u ∂ ⁡ t and have, owing to (3.6) and the interpolation inequality (3.10),

(3.26) d d ⁢ t ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 + c ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ H k ⁢ ( Ω ) 2 ≤ c ′ ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 , c > 0 .

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

(3.27) ∥ ∂ ⁡ u ∂ ⁡ t ⁢ ( t ) ∥ 2 ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 1 ,

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

(3.28) ∥ ∂ ⁡ u ∂ ⁡ t ⁢ ( t ) ∥ 2 ≤ e c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H 2 ⁢ k ⁢ ( Ω ) ) , t ≥ 0 .

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

(3.29) ∥ u ⁢ ( t ) ∥ H 2 ⁢ k ⁢ ( Ω ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 1 ,

and

(3.30) ∥ u ⁢ ( t ) ∥ H 2 ⁢ k ⁢ ( Ω ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H 2 ⁢ k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 0 .

3.3 The dissipative semigroup

Theorem 3.2.

The following statements hold.

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

u ∈ L ∞ ⁢ ( ℝ + ; H 0 k ⁢ ( Ω ) ) ∩ L 2 ⁢ ( 0 , T ; H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) )

and

∂ ⁡ u ∂ ⁡ t ∈ L 2 ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) .
If we further assume that u 0 ∈ H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) , then

u ∈ L ∞ ⁢ ( ℝ + ; H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( 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 u 1 and u 2 be two solutions to (3.1)–(3.2) with initial data u 0 , 1 and u 0 , 2 , respectively. We set u = u 1 - u 2 and u 0 = u 0 , 1 - u 0 , 2 and have

(3.31) ∂ ⁡ u ∂ ⁡ t + A k ⁢ u + B k ⁢ u + f ⁢ ( u 1 ) - f ⁢ ( u 2 ) = 0 ,

(3.32) 𝒟 α ⁢ u = 0 on ⁢ Γ , | α | ≤ k - 1 ,

(3.33) u | t = 0 = u 0 .

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

(3.34) d d ⁢ t ⁢ ∥ u ∥ 2 + c ⁢ ∥ u ∥ H k ⁢ ( Ω ) 2 ≤ c ′ ⁢ ∥ u ∥ 2 , c > 0 .

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

(3.35) ∥ u ⁢ ( t ) ∥ 2 ≤ e c ⁢ t ⁢ ∥ u 0 ∥ 2 , t ≥ 0 .

Hence the uniqueness is proved, as well as the continuous dependence with respect to the initial data in the L 2 -norm. ∎

It follows from Theorem 3.2 that we can define the continuous (for the L 2 -norm) semigroup S ⁢ ( t ) : Φ → Φ , u 0 ↦ u ⁢ ( t ) , t ≥ 0 (i.e., S ⁢ ( 0 ) = I (identity operator) and S ⁢ ( t + τ ) = S ⁢ ( t ) ∘ S ⁢ ( τ ) , t, τ ≥ 0 ), where Φ = H 0 k ⁢ ( Ω ) . Furthermore, S ⁢ ( t ) is dissipative in Φ, owing to (3.16), in the sense that it possesses a bounded absorbing set ℬ 0 ⊂ Φ (i.e., for all B ⊂ Φ bounded, there exists t 0 = t 0 ⁢ ( B ) ≥ 0 such that t ≥ t 0 implies S ⁢ ( t ) ⁢ B ⊂ ℬ 0 ).

Remark 3.3.

We can also prove the continuous dependence with respect to the initial data in the H k - and H 2 ⁢ k -norms and it then follows from (3.30) that S ⁢ ( t ) is defined, continuous and dissipative in ( H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) ) .

Actually, it follows from (3.29) that S ⁢ ( t ) possesses a bounded absorbing set ℬ 1 such that ℬ 1 is compact in Φ and bounded in H 2 ⁢ k ⁢ ( 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 H 2 ⁢ k ⁢ ( Ω ) .

Remark 3.5.

It follows from (3.35) that we can extend S ⁢ ( t ) (by continuity and in a unique way) to L 2 ⁢ ( 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 t ≥ 0 ) 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 = Π i = 1 3 ⁢ ( 0 , L i ) , L i > 0 , i ∈ { 1 , 2 , 3 } . In that case, we consider the operator 𝐀 k = I + A k (in order to have a strictly positive operator), where A k is as above, but based on Sobolev spaces with periodic functions (see, e.g., [41]), and rewrite (3.1) in the form

∂ ⁡ u ∂ ⁡ t + 𝐀 k ⁢ u + B k ⁢ u + 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 ∈ ℕ , k ≥ 2 (the case k = 1 can be treated as in the original Cahn–Hilliard equation; see, e.g., [13]):

(4.1) ∂ ⁡ u ∂ ⁡ t - Δ ⁢ ∑ i = 1 k ( - 1 ) i ⁢ ∑ | α | = i a α ⁢ 𝒟 2 ⁢ α ⁢ u - Δ ⁢ f ⁢ ( u ) = 0 ,

(4.2) 𝒟 α ⁢ u = 0 on ⁢ Γ , | α | ≤ k ,

(4.3) u | t = 0 = u 0 .

Remark 4.1.

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

∂ ⁡ u ∂ ⁡ t + Δ ⁢ ∑ i = 1 3 a i ⁢ ∂ 2 ⁡ u ∂ ⁡ x i 2 - Δ ⁢ f ⁢ ( u ) = 0

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

∂ ⁡ u ∂ ⁡ t - Δ ⁢ ∑ i , j = 1 3 a i ⁢ j ⁢ ∂ 4 ⁡ u ∂ ⁡ x i 2 ⁢ ∂ ⁡ x j 2 + Δ ⁢ ∑ i = 1 3 b i ⁢ ∂ 2 ⁡ u ∂ ⁡ x i 2 - Δ ⁢ f ⁢ ( u ) = 0 .

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

(4.4) ∂ ⁡ u ∂ ⁡ t - Δ ⁢ A k ⁢ u - Δ ⁢ B k ⁢ u - Δ ⁢ f ⁢ ( u ) = 0 .

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 𝒞 2 .

4.2 A priori estimates

We multiply (4.4) by ( - Δ ) - 1 ⁢ ∂ ⁡ u ∂ ⁡ t . This gives

(4.5) d d ⁢ t ⁢ ( ∥ A k 1 / 2 ⁢ u ∥ 2 + B k 1 / 2 ⁢ [ u ] + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) + 2 ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 = 0 .

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

(4.6) d d ⁢ t ⁢ ∥ u ∥ - 1 2 + c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) ≤ c ′ , c > 0 .

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

(4.7) d ⁢ E 3 d ⁢ t + c ⁢ ( E 3 + ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 ) ≤ c ′ , c > 0 ,

where

E 3 = ∥ A k 1 / 2 ⁢ u ∥ 2 + B k 1 / 2 ⁢ [ u ] + 2 ⁢ ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x + ∥ u ∥ - 1 2

satisfies

(4.8) E 3 ≥ c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 .

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

(4.9) ∥ u ⁢ ( t ) ∥ H k ⁢ ( Ω ) 2 ≤ c ⁢ e - c ′ ⁢ t ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u 0 ) ⁢ 𝑑 x ) + c ′′ , c ′ > 0 , t ≥ 0 ,

and

(4.10) ∫ t t + r ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 ⁢ 𝑑 s ≤ c ⁢ e - c ′ ⁢ t ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u 0 ) ⁢ 𝑑 x ) + c ′′ , c ′ > 0 , t ≥ 0 , r > 0 ⁢ given .

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

(4.11) d d ⁢ t ⁢ ∥ A ~ k 1 / 2 ⁢ u ∥ 2 + c ⁢ ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 ≤ e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′′ , c , c ′ > 0 , t ≥ 0 .

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

(4.12) d ⁢ E 4 d ⁢ t + c ⁢ ( E 4 + ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 + ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 ) ≤ e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′′ , c , c ′ > 0 , t ≥ 0 ,

where

E 4 = E 3 + ∥ A ~ k 1 / 2 ⁢ u ∥ 2

satisfies

E 4 ≥ c ⁢ ( ∥ u ∥ H k ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 .

We also multiply (4.4) by ∂ ⁡ u ∂ ⁡ t and obtain, noting that f is of class 𝒞 2 ,

(4.13) d d ⁢ t ⁢ ( ∥ A ¯ k 1 / 2 ⁢ u ∥ 2 + B ¯ k 1 / 2 ⁢ [ u ] ) + ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ≤ e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′′ , c , c ′ > 0 ,

where

B ¯ k 1 / 2 ⁢ [ u ] = ∑ i = 1 k - 1 ∑ | α | = i a α ⁢ ∥ ∇ ⁡ 𝒟 α ⁢ u ∥ 2 .

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

(4.14) d ⁢ E 5 d ⁢ t + c ⁢ ( E 5 + ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 + ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 ) ≤ e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′′ , c , c ′ > 0 , t ≥ 0 ,

where

E 5 = E 4 + ∥ A ¯ k 1 / 2 ⁢ u ∥ 2 + B ¯ k 1 / 2 ⁢ [ u ]

satisfies

(4.15) E 5 ≥ c ⁢ ( ∥ u ∥ H k + 1 ⁢ ( Ω ) 2 + ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ) - c ′ , c > 0 .

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

(4.16) ∥ u ⁢ ( t ) ∥ H k + 1 ⁢ ( Ω ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k + 1 ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 0 .

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

(4.17) ∂ ∂ ⁡ t ⁢ ∂ ⁡ u ∂ ⁡ t - Δ ⁢ A k ⁢ ∂ ⁡ u ∂ ⁡ t - Δ ⁢ B k ⁢ ∂ ⁡ u ∂ ⁡ t - Δ ⁢ ( f ′ ⁢ ( u ) ⁢ ∂ ⁡ u ∂ ⁡ t ) = 0 ,

(4.18) 𝒟 α ⁢ ∂ ⁡ u ∂ ⁡ t = 0 on ⁢ Γ , | α | ≤ k .

We multiply (4.17) by ( - Δ ) - 1 ⁢ ∂ ⁡ u ∂ ⁡ t and obtain, owing to (3.6) and the interpolation inequality (3.10),

d d ⁢ t ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 + c ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ H k ⁢ ( Ω ) 2 ≤ c ′ ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ 2 , c > 0 ,

which yields, employing the interpolation inequality

(4.19) ∥ v ∥ 2 ≤ c ⁢ ∥ v ∥ - 1 ⁢ ∥ v ∥ H 1 ⁢ ( Ω ) , v ∈ H 0 1 ⁢ ( Ω ) ,

the differential inequality

(4.20) d d ⁢ t ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 + c ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ H k ⁢ ( Ω ) 2 ≤ c ′ ⁢ ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 , c > 0 .

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

(4.21) ∥ ∂ ⁡ u ∂ ⁡ t ⁢ ( t ) ∥ - 1 ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ r , r > 0 ⁢ given .

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

(4.22) A k ⁢ u = - ( - Δ ) - 1 ⁢ ∂ ⁡ u ∂ ⁡ t - B k ⁢ u - f ⁢ ( u ) , 𝒟 α ⁢ u = 0 on ⁢ Γ , | α | ≤ k - 1 .

Multiplying (4.22) by A k ⁢ u , we find, owing to the interpolation inequality (3.10),

(4.23) ∥ u ∥ H 2 ⁢ k ⁢ ( Ω ) 2 ≤ c ⁢ ( e - c ′ ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + ∥ ∂ ⁡ u ∂ ⁡ t ∥ - 1 2 ) + c ′′ , c ′ > 0 .

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

(4.24) ∥ u ⁢ ( t ) ∥ H 2 ⁢ k ⁢ ( Ω ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 1 .

Remark 4.2.

If we assume that u 0 ∈ H 2 ⁢ k + 1 ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) , we deduce from (4.20), (4.23) and Gronwall’s lemma an H 2 ⁢ k -estimate on u on [ 0 , 1 ] which, combined with (4.24), gives an H 2 ⁢ k -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 𝒞 k + 1 . Multiplying (4.4) by A ~ k ⁢ ∂ ⁡ u ∂ ⁡ t , we have

1 2 ⁢ d d ⁢ t ⁢ ( ∥ A k ⁢ u ∥ 2 + ( ( A k ⁢ u , B k ⁢ u ) ) ) + ∥ A ~ k 1 / 2 ⁢ ∂ ⁡ u ∂ ⁡ t ∥ 2 = - ( ( A ¯ k 1 / 2 ⁢ f ⁢ ( u ) , A ~ k 1 / 2 ⁢ ∂ ⁡ u ∂ ⁡ t ) ) ,

which yields, noting that ∥ A ¯ k 1 / 2 ⁢ f ⁢ ( u ) ∥ 2 ≤ Q ⁢ ( ∥ u ∥ H k + 1 ⁢ ( Ω ) ) and owing to (4.16),

(4.25) d d ⁢ t ⁢ ( ∥ A k ⁢ u ∥ 2 + ( ( A k ⁢ u , B k ⁢ u ) ) ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H k + 1 ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 0 .

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

∥ u ⁢ ( t ) ∥ H 2 ⁢ k ⁢ ( Ω ) ≤ Q ⁢ ( ∥ u 0 ∥ H 2 ⁢ k ⁢ ( Ω ) ) , t ∈ [ 0 , 1 ] ,

so that, owing to (4.24),

(4.26) ∥ u ⁢ ( t ) ∥ H 2 ⁢ k ⁢ ( Ω ) ≤ e - c ⁢ t ⁢ Q ⁢ ( ∥ u 0 ∥ H 2 ⁢ k ⁢ ( Ω ) ) + c ′ , c > 0 , t ≥ 0 .

4.3 The dissipative semigroup

Theorem 4.4.

The following statements hold.

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

u ∈ L ∞ ⁢ ( ℝ + ; H 0 k ⁢ ( Ω ) ) ∩ L 2 ⁢ ( 0 , T ; H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) )

and

∂ ⁡ u ∂ ⁡ t ∈ L 2 ⁢ ( 0 , T ; H - 1 ⁢ ( Ω ) ) .
If we further assume that u 0 ∈ H k + 1 ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) , then, for all T > 0 ,

u ∈ L ∞ ⁢ ( ℝ + ; H k + 1 ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) )

and

∂ ⁡ u ∂ ⁡ t ∈ L 2 ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) .
If we further assume that f is of class 𝒞 k + 1 and u 0 ∈ H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( Ω ) , then

u ∈ L ∞ ⁢ ( ℝ + ; H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( 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 u 1 and u 2 be two solutions to (4.1)–(4.2) with initial data u 0 , 1 and u 0 , 2 , respectively. We set u = u 1 - u 2 and u 0 = u 0 , 1 - u 0 , 2 and have

(4.27) ∂ ⁡ u ∂ ⁡ t - Δ ⁢ A k ⁢ u - Δ ⁢ B k ⁢ u - Δ ⁢ ( f ⁢ ( u 1 ) - f ⁢ ( u 2 ) ) = 0 ,

(4.28) 𝒟 α ⁢ u = 0 on ⁢ Γ , | α | ≤ k ,

(4.29) u | t = 0 = u 0 .

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

(4.30) d d ⁢ t ⁢ ∥ u ∥ - 1 2 + c ⁢ ∥ u ∥ H k ⁢ ( Ω ) 2 ≤ c ′ ⁢ ∥ u ∥ - 1 2 , c > 0 .

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

(4.31) ∥ u ⁢ ( t ) ∥ - 1 2 ≤ e c ⁢ t ⁢ ∥ u 0 ∥ - 1 2 , t ≥ 0 .

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 ) : Φ → Φ , u 0 ↦ u ⁢ ( t ) , t ≥ 0 ,

where Φ = H 0 k ⁢ ( 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 ℬ 1 which is compact in Φ and bounded in H 2 ⁢ k ⁢ ( Ω ) . This yields the existence of the global attractor 𝒜 which is compact in Φ and bounded in H 2 ⁢ k ⁢ ( Ω ) .

(ii) It follows from (4.26) that, if f is of class 𝒞 k + 1 , then S ⁢ ( t ) is dissipative in H 2 ⁢ k ⁢ ( Ω ) ∩ H 0 k ⁢ ( 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 ) 〉 = 〈 u 0 〉 , t ≥ 0 , where 〈 ⋅ 〉 = 1 Vol ⁢ ( Ω ) ⁢ ∫ Ω ⋅ d ⁢ x . 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 P ⁢ 1 -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

∂ ⁡ u ∂ ⁡ t + a 20 ⁢ ∂ 4 ⁡ u ∂ ⁡ x 4 + a 02 ⁢ ∂ 4 ⁡ u ∂ ⁡ y 4 + a 11 ⁢ ∂ 4 ⁡ u ∂ ⁡ x 2 ⁢ ∂ ⁡ y 2 - a 10 ⁢ ∂ 2 ⁡ u ∂ ⁡ x 2 - a 01 ⁢ ∂ 2 ⁡ u ∂ ⁡ y 2 + f ⁢ ( u ) = 0 ,

where, here and in all the simulations, f ⁢ ( s ) = s 3 - 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 ∂ 2 ⁡ u ∂ ⁡ x 2 = ω and ∂ 2 ⁡ u ∂ ⁡ y 2 = p , then, integrating by parts, the system which needs to be solved reads

{ ( ∂ ⁡ u ∂ ⁡ t , v ) - a 20 ⁢ ( ∂ ⁡ ω ∂ ⁡ x , ∂ ⁡ v ∂ ⁡ x ) - a 02 ⁢ ( ∂ ⁡ p ∂ ⁡ y , ∂ ⁡ v ∂ ⁡ y ) - a 11 ⁢ ( ∂ ⁡ ω ∂ ⁡ y , ∂ ⁡ v ∂ ⁡ y ) + a 10 ⁢ ( ∂ ⁡ u ∂ ⁡ x , ∂ ⁡ v ∂ ⁡ x ) + a 01 ⁢ ( ∂ ⁡ u ∂ ⁡ y , ∂ ⁡ v ∂ ⁡ y ) + ( f ⁢ ( u ) , v ) = 0 , ( ω , ξ ) = - ( ∂ ⁡ u ∂ ⁡ x , ∂ ⁡ ξ ∂ ⁡ x ) , ( p , ζ ) = - ( ∂ ⁡ u ∂ ⁡ y , ∂ ⁡ ζ ∂ ⁡ y ) ,

where the test functions v, ξ, ζ all belong to H per 1 ⁢ ( Ω ) .

Next, we introduce the discretization 𝒯 h of Ω ¯ and set

V h = { v h ∈ C 0 ⁢ ( Ω ¯ ) : ( v h ) | K ∈ P 1 ⁢ for all ⁢ K ∈ 𝒯 h , v h ⁢ is Ω -periodic } ⊂ H per 1 ⁢ ( Ω ) .

As mentioned above, we use a P 1 -finite element for the space discretization and a forward Euler scheme for the time discretization. Let u h 0 ∈ V h . Then, for n ≥ 0 , we look for ( u h n + 1 , ω h n + 1 , p h n + 1 ) ∈ V h × V h × V h such that

{ 1 d ⁢ t ⁢ ( u h n + 1 , v ) - a 20 ⁢ ( ∂ ⁡ ω h n + 1 ∂ ⁡ x , ∂ ⁡ v ∂ ⁡ x ) - a 02 ⁢ ( ∂ ⁡ p h n + 1 ∂ ⁡ y , ∂ ⁡ v ∂ ⁡ y ) - a 11 ⁢ ( ∂ ⁡ ω h n + 1 ∂ ⁡ y , ∂ ⁡ v ∂ ⁡ y ) + a 10 ⁢ ( ∂ ⁡ u h n + 1 ∂ ⁡ x , ∂ ⁡ v ∂ ⁡ x ) + a 01 ⁢ ( ∂ ⁡ u h n + 1 ∂ ⁡ y , ∂ ⁡ v ∂ ⁡ y ) + ( f ⁢ ( u h n ) , v ) - 1 d ⁢ t ⁢ ( u h n , v ) = 0 , ( ω h n + 1 , ξ ) = - ( ∂ ⁡ u h n + 1 ∂ ⁡ x , ∂ ⁡ ξ ∂ ⁡ x ) , ( p h n + 1 , ζ ) = - ( ∂ ⁡ u h n + 1 ∂ ⁡ y , ∂ ⁡ ζ ∂ ⁡ y ) ,

for all v , ξ , ζ ∈ V h . 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 d ⁢ t = 10 - 7 (in Figure 3 and Figure 7), d ⁢ t = 10 - 6 (in Figure 4 ) and d ⁢ t = 10 - 5 (in Figure 8). When k = 2 , we take d ⁢ t = 10 - 7 and, when k = 3 , we take d ⁢ t = 10 - 10 (in Figure 6 and Figure 12 ) and d ⁢ t = 10 - 8 (in Figure 10) or d ⁢ t = 10 - 7 (in Figure 3 and Figure 7). Here, we use a grid with 150 2 points on the domain Ω.

Figure 2 shows the 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.

Figure 3

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

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

(a)

k = 1 , d ⁢ t = 10 - 7 .

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

(b)

k = 2 , d ⁢ t = 10 - 7 .

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

(c)

k = 3 , d ⁢ t = 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.

Figure 4

Results for k = 1 , d ⁢ t = 10 - 6 .

(a)

After 100 iterations.

(b)

After 250 iterations.

(c)
After

1
,
000

{1{,}000}

iterations.

(c)

After 1 , 000 iterations.

(d)
After

4
,
000

{4{,}000}

iterations.

(d)

After 4 , 000 iterations.

Figure 5

Results for k = 2 , d ⁢ t = 10 - 7 .

(a)

After 40 iterations.

(b)

After 400 iterations.

(c)
After

2
,
000

{2{,}000}

iterations.

(c)

After 2 , 000 iterations.

(d)
After

5
,
000

{5{,}000}

iterations.

(d)

After 5 , 000 iterations.

Figure 6

Results for k = 3 , d ⁢ t = 10 - 10 .

(a)

After 40 iterations.

(b)

After 500 iterations.

(c)

After 1 , 000 iterations.

(d)
After

2
,
000

2{,}000

iterations.

(d)

After 2 , 000 iterations.

Anisotropy in the x-direction

We consider the following situations:

k = 1 , a 10 = 1 and a 01 = 0.01 .
k = 2 , a 20 = 1 and the other coefficients are set equal to 0.01.
k = 3 , a 30 = 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.

Figure 7

Anisotropy in the x-direction after 40 iterations.

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

(a)

k = 1 , d ⁢ t = 10 - 7 .

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

(b)

k = 2 , d ⁢ t = 10 - 7 .

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

(c)

k = 3 , d ⁢ t = 10 - 7 .

Figure 8

Results for k = 1 , d ⁢ t = 10 - 5 .

(a)

After 50 iterations.

(b)
After

1
,
000

{1{,}000}

iterations.

(b)

After 1 , 000 iterations.

(c)

After 2 , 000 iterations.

(d)

After 5 , 000 iterations.

Figure 9

Results for k = 2 , d ⁢ t = 10 - 7 .

(a)

After 40 iterations.

(b)

After 400 iterations.

(c)

After 2 , 000 iterations.

(d)

After 5 , 000 iterations.

Figure 10

Results for k = 3 , d ⁢ t = 10 - 8 .

(a)

After 40 iterations.

(b)

After 500 iterations.

(c)

After 1 , 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:

k = 2 , a 11 = 1 and the other coefficients are set equal to 0.01.
k = 3 , a 21 = 1 and the other coefficients are set equal to 0.01.

Figure 11

Results for k = 2 , d ⁢ t = 10 - 7 .

(a)

After 40 iterations.

(b)

After 400 iterations.

(c)

After 2 , 000 iterations.

(d)

After 5 , 000 iterations.

Figure 12

Results for k = 3 , d ⁢ t = 10 - 10 .

(a)

After 40 iterations.

(b)

After 250 iterations.

(c)

After 1 , 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. Suche in 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. 10.1002/cpa.3160120405Suche in Google 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. 10.1002/cpa.3160170104Suche in Google 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. 10.1016/0001-6160(79)90196-2Suche in Google 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. 10.1103/PhysRevE.77.061506Suche in Google Scholar PubMed

[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. 10.1103/PhysRevE.73.031609Suche in Google Scholar PubMed

[7] G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), 311–350. 10.3934/dcdss.2011.4.311Suche in Google Scholar

[8] J. W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961), 795–801. 10.1002/9781118788295.ch11Suche in Google 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. 10.1002/9781118788295.ch4Suche in 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. 10.4208/cicp.101111.110512aSuche in Google 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. 10.1007/s00205-011-0429-8Suche in Google 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. 10.1007/978-3-319-40673-2_12Suche in Google Scholar

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

[14] L. Cherfils, A. Miranville and S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), 561–596. 10.1007/s00032-011-0165-4Suche in Google Scholar

[15] P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys. 72 (1980), 4756–4763. 10.1063/1.439809Suche in Google 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. 10.1080/00018732.2012.737555Suche in Google 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. 10.1103/PhysRevE.79.051110Suche in Google Scholar PubMed

[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. 10.1007/BF02181479Suche in Google 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. 10.1137/S0036139996313046Suche in Google 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. 10.1103/PhysRevE.47.4289Suche in Google 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. 10.1103/PhysRevE.47.4301Suche in Google Scholar PubMed

[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. 10.1142/S0218202514500365Suche in Google 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. 10.3934/dcds.2015.35.2539Suche in Google Scholar

[24] F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), 251–265. 10.1515/jnum-2012-0013Suche in Google 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. 10.1016/j.jcp.2009.04.020Suche in Google Scholar

[26] R. Kobayashi, Modelling and numerical simulations of dendritic crystal growth, Phys. D 63 (1993), 410–423. 10.1016/0167-2789(93)90120-PSuche in Google 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. 10.1137/100817590Suche in Google Scholar

[28] M. Korzec and P. Rybka, On a higher order convective Cahn–Hilliard type equation, SIAM J. Appl. Math. 72 (2012), 1343–1360. 10.1137/110834123Suche in Google Scholar

[29] A. Miranville, Asymptotic behavior of a sixth-order Cahn–Hilliard system, Central Europ. J. Math. 12 (2014), 141–154. 10.2478/s11533-013-0322-9Suche in Google Scholar

[30] A. Miranville, Sixth-order Cahn–Hilliard equations with logarithmic nonlinear terms, Appl. Anal. 94 (2015), 2133–2146. 10.1080/00036811.2014.972384Suche in Google Scholar

[31] A. Miranville, Sixth-order Cahn–Hilliard systems with dynamic boundary conditions, Math. Methods Appl. Sci. 38 (2015), 1127–1145. 10.1002/mma.3134Suche in Google 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. 10.1007/s13398-015-0227-5Suche in 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. 10.1016/S1874-5717(08)00003-0Suche in Google Scholar

[34] A. Novick-Cohen, The Cahn–Hilliard equation, Handbook of Differential Equations: Evolutionary Equations. Vol. 4, Elsevier, Amsterdam (2008), 201–228. 10.1016/S1874-5717(08)00004-2Suche in Google Scholar

[35] I. Pawlow and G. Schimperna, A Cahn–Hilliard equation with singular diffusion, J. Differential Equations 254 (2013), 779–803. 10.1016/j.jde.2012.09.018Suche in Google Scholar

[36] I. Pawlow and G. Schimperna, On a Cahn–Hilliard model with nonlinear diffusion, SIAM J. Math. Anal. 45 (2013), 31–63. 10.1137/110835608Suche in Google 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. 10.3934/cpaa.2011.10.1823Suche in Google 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. 10.3934/dcdss.2013.6.517Suche in Google 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. 10.1103/PhysRevE.67.021606Suche in Google Scholar

[40] J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater. 40 (1992), 1475–1495. 10.1016/0956-7151(92)90091-RSuche in Google Scholar

[41] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci. 68, Springer, New York, 1997. 10.1007/978-1-4612-0645-3Suche in 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. 10.1098/rspa.2008.0385Suche in Google 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. 10.4310/MAA.2010.v17.n2.a4Suche in 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. 10.1137/090752675Suche in Google 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. 10.6028/NIST.IR.5848Suche in Google 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. 10.1137/080738143Suche in Google Scholar

Received: 2016-06-16

Accepted: 2017-02-08

Published Online: 2017-03-16

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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https://doi.org/10.1515/anona-2016-0137

Schlagwörter für diesen Artikel

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

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