Abstract

In this article, we use a finite difference scheme to discretize the Cahn-Hilliard equation with the space step size . We first prove that this semidiscrete system inherits two important properties, called the conservation of mass and the decrease of the total energy, from the original equation. Then, we show that the semidiscrete system has an attractor on a subspace of . Finally, the convergence of attractors is established as the space step size of the semidiscrete Cahn-Hilliard equation tends to 0.

1. Introduction

In this paper, we are concerned with some discrete forms of the following Cahn-Hilliard equation:associated with boundary conditions:where denotes a distribution function of the concentration of one component of the binary mixture, the functional (called the Ginzburg-Landau free energy) usually denotes a local free energy, and the notation given by (1) is consistent with the variational derivative of

The Cahn-Hilliard equation was first proposed by Cahn and Hilliard [1] and is usually used to describe the complex phase separation phenomenon which occurs when two miscible substances are rapidly cooled to critical temperatures at high temperatures; see [25] for details.

Nowadays, the Cahn-Hilliard equation has attracted much attention from researchers; see [3, 613], etc. Especially, Alikakos, Bates, and Fusco [6] established the existence of some extremely slowly evolving solutions of the Cahn-Hilliard equation. Later, Cholewa and Dlotko [7] proved the existence of global attractors for the Cahn-Hilliard equation. Based on this result, Li and Zhong [10] further established the existence of global attractors for the Cahn-Hilliard equation with fast growing nonlinearity. Recently, Furihata [3] proposed a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation. Specifically, they designed a new difference scheme which inherits some characteristic properties from the equation, including the conservation of mass and the decrease of the total energy.

Inspired by these works mentioned above, we use a finite difference scheme (with respect to ) to discretize the Cahn-Hilliard equations (1) and (2) with the space step size , which can generate nodes. Thus, one can obtain the following equations:whereassociated with the initial conditions

Moreover, the boundary conditions (3) can be discretized as

We are interested in the existence and the convergence of attractors for this semidiscrete Cahn-Hilliard equation. It is well-known that semidiscretization is a very effective technique in the finite element analysis of solid bodies and computational fluid mechanics; see [14, 15]. The semidiscretization of a partial differential equation means that the space is discretized and the time is continuous, which is widely studied by many researchers. For example, Castro and Micu [16] studied the controllability of a semidiscrete system with a boundary control at one extremity. Mai, Qin, and Zhang [17] studied the Turing instability of a two-dimensional semidiscrete Gierer–Meinhardt system and carried out a series of simulations. Concerning this topic of semidiscrete systems, the interested reader is referred to [18, 19, 20, 21, 22, 23] for some concrete examples.

In this paper, we discuss the attractor of the semidiscrete Cahn-Hilliard system (5)–(8). We first develop some ideas in [3, 7] to give some properties of solutions and prove the existence of attractors. Then, we establish the convergence of the attractor with respect to the space step size .

This paper is organized as follows. In Section 2, we present two well-known properties of the system (5)–(8) called the conservation of mass and the decrease of the total energy. In Section 3, we prove the existence of attractors for the system (5)–(8) on some affined subspaces. In Section 4, we further establish the convergence of the attractor with respect to the space step size ; that is, the attractor of the system (5)–(8) tends to the attractor of the original system (1)–(3) as .

2. Two Well-Known Properties

In this section, we discuss two well-known properties of the semidiscrete Cahn-Hilliard system (5-8). For the purpose, we let and and define the linear operators by

Then, one can rewrite the system (5) as

Without loss of generality, in the following, we first assume that the space step size , hoping that there is no confusion.

Let and . Then, the system (5)–(8) can be simply written as the following systemwith the initial valueand boundary conditionswhere and are linear operators defined by

In the following, we verify that the solution of the semidiscrete Cahn-Hilliard system (11)–(13) satisfieswhich corresponds to the following two properties:(called the conservation of mass and the decrease of the total energy, respectively.) of the solutions to the Cahn-Hilliard system (1)–(3); see [3, 9] for details.

In (16) and (17), is a summation operator (see [3]), given byis the discrete local free energy, where

Remark 1. These two properties (18) and (19) play an important role in studying the Cahn-Hilliard equation and are widely investigated by researchers; see e.g., [24, 25]. On the other hand, it is clear to see from (17) that (u(x, t)) can be employed as a Lyapunov function of the Cahn-Hilliard system (see e.g., [26]). Similarly, one can regard as a Lyapunov function of (11) and (12).

Proposition 1. The solution of the semidiscrete Cahn-Hilliard system (11)–(13) satisfies the properties (16) and (17).

Proof. First, we show that the solution of (11)–(13) satisfies the conservation of mass (16). Indeed, by the definition of the summation operator and (5), we easily see thatwhere .
Now, let us prove that the solution of (11)–(13) satisfies (17). By the definition of , one hasThanks to the general identity (summation by parts), we havewhereThus, it follows from (24) and (25) thatwhich completes the proof of the proposition.

3. Attractors of the Semidiscrete System

In this section, we aim to prove the existence of attractors for the semidiscrete Cahn-Hilliard system. We first give some properties of the solutions to (11)–(13). To this end, we use to denote the Euclid space and equip it with the norm defined by

We denote by a positive constant depending on , whose value may change from line to line.

Note that the equation (11) is finite-dimensional. By the basic knowledge on ODEs, one can easily see that the Cauchy problem of (11)–(13) is well-posed in . Specifically, for each , the system (11)–(13) has a unique solution on for some with .

Let denote the local semiflow generated by (11)–(13). Then, we have the following fundamental facts on the solutions.

3.1. The Boundedness of the Solution

Theorem 1. Let be a solution of (11)–(13) with the initial data . Then,

Proof. We define a norm as follows:Then, by (20) and a simple calculation, we haveHere, we used the inequality (see [3]). Thereby, we deduce from (17) thatOn the other hand, it is trivial to show thatThus, by the equivalence of norms, (32) and (33), we can conclude that (29) holds true.

3.2. Properties of the Stationary Solution

As noted in the Remark 1, one can see is a Lyapunov function. Next, we present some basic results on .

Lemma 1. If is a solution of the system (11)–(13) satisfyingthen is a stationary (time independent) solution.

Proof. By (27), we havefrom which it can be seen thatThereby,which together with (11) gives thatHence, the result of Lemma 1 holds true.

Lemma 2. Let be a solution of (11)–(13). Then, is a stationary solution if and only if there is a constant such thatwhere .

Proof. Let be a solution of (11)–(13). Then, if is a stationary solution, we see from (6) and (11) thatBy the boundary conditions (13), one hasConversely, if there is a constant such that , thenwhich completes the proof of what we desired.
Let be the constant given by Lemma 2. Assume is a stationary solution of (11)–(13) such that (39) holds. Then,which shows that the can be expressed asLet denote the spatial average of , i.e.,Then, one can see from (44) that . Writewhere . Then, one can deduce that is actually the set of all stationary solutions of the system (11)–(13). In the following, we show that is bounded for each , which plays an important part in constructing the attractor.

Theorem 2. For each fixed , is a bounded set of .

Proof. Let . We infer from Lemma 2 thatwhere is given by (44).
By (14), one hasThus, it follows from (47) and (48) thatHere, we have used the definition of (see (44)).
Now, by Young inequality, one can deduce that for every , there exists such thatIt is trivial to pick positive constants and satisfyingSubstituting (50) and (51) into (49), we obtain thatTake in the above estimate, thenNow, we define a new norm as follows:Then, by (53), we find Therefore, by the equivalence of the norms, one can immediately conclude thatwhich completes the proof of the theorem.

3.3. The Existence of the Attractors

Inspired by much literature (see e.g., [7, 10]) on the global attractor for the Cahn-Hilliard dynamical system (1)–(3), we consider the attractor on the space , where .

Lemma 3. There exists a bounded subset of attracting each point of .

Proof. Define the set bywhere denotes the -limit set of .
For each , there is such that . Let be the family of operators which form a continuous semigroup satisfying on the space , where is the solution of (11) with the initial value . By (17), we know that is decreasing along the solution and bounded below. Thus, there exists such thatThen, one can deduce thatwhere . Hence, by (16) and Lemma 1, we see that is a stationary solution of the system (11)–(13) belonging to . Consequently, one can conclude from Theorem 2 that is bounded.
On the other hand, by Theorem 1, one finds that the semiflow generated by the system (11)–(13) takes bounded sets into bounded sets. Moreover, by the definition of , we see that is compact. Hence, attracts each point of .
According to [14], Theorem 4.2.4, we can obtain the existence of the attractor for the semifolw on .

Theorem 4. The semiflow generated by the system (11)–(13) possesses a connected global attractor on a metric space .

4. The Convergence of Attractors

Notice that we considered a special case where the space step size equals 1 in the above arguments. By using the same arguments as Theorem 3.6, one can easily verify that the system (11)–(13) has an attractor with respect to the general space step size .

Thanks to [11, Theorem 3], we deduce that the solution of (11)–(13) converges to the solution of (1)–(3) as the space step size . Based on this fact, in what follows we further show that the attractor in tends to the attractor of the original system (1-3) as .

Theorem 3. Let be the attractor of the original system (1)–(3) given by [7]. Then,where denotes the Hausdorff semidistance in .

Proof. We argue by contradiction. If the conclusion is not true, then there exists and a sequence with (as ) such thatOn the other hand, we infer from the invariance of that there exists a bounded complete solution of in such that . By a standard argument, it can be easily shown that has a subsequence converging some bounded full trajectory of in uniformly on any compact interval of . Hence,Especially, , which contradicts (60) and completes the proof of the theorem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11871368).