Pointwise Estimates of Solutions for the Viscous Cahn-Hilliard Equation with Inertial Term

Abstract

In this paper, we study the pointwise estimates of solutions to the viscous Cahn-Hilliard equation with the inertial term in multidimensions. We use Green’s function method. Our approach is based on a detailed analysis on the Green’s function of the linear system. And we get the solution’s convergence rate.

1. Introduction

In this paper, we study the pointwise estimates of the solution to the Cauchy problem: where . is the intrinsic chemical potential which is smooth in the small neighborhood of the origin, and when and is a positive integer. When , Eq. (1) is the well-known Cahn-Hilliard equation. When , is the inertial term. When , is the viscous term. Without loss of generality, we let and .

The classical Cahn-Hilliard equation was proposed in the sixties by Cahn and Hilliard which describes the phase separation in materials science, and it has been widely studied. The reader may see references ([1–6]) and the related references therein. The Cahn-Hilliard equations with inertial term model nonequilibrium decompositions caused by deep supercooling in certain glasses. As we know, the well-known Cahn-Hilliard equation is a parabolic equation, but the Cahn-Hilliard equation with the inertial term is a hyperbolic equation with relaxation which brings many mathematical difficulties to study. For which, without smallness assumption on initial data, [7] got the global existence of the classical solution. [8] obtained the global existence and the optimal decay rate of the classical solution by the Fourier splitting method. Wang and Wu [9] obtained the global existence and optimal decay rate of the classical solution by long wave-short wave method. Li and Mi [10] got the pointwise estimates and the () convergence rate of the solution by Green’s function method. Some other works on the Cahn-Hilliard equation with the inertial term can be seen in [11–13].

For viscous Cahn-Hilliard equation, [14] discussed the large time behavior of solutions when the dimension . For the viscous Cahn-Hilliard equation with the inertial term, it describes the early stages of spinodal decomposition in certain glasses (see [15–16]). And for which, [17] established the existence of families of exponential attractors and inertial manifolds; [18] studied the long time dynamic of the system in three-dimensional. In this paper, we are interested in the viscous Cahn-Hilliard equation with the inertial term. Under the smallness assumption on initial data, based on the detailed analysis of the Green’s function, we get the pointwise estimates of solutions. From the representation of the symbol value to the Green’s function for the linear problem of Eq. (1), we also find that the decay rate mainly depends on the lower-frequency part, i.e., the long wave part. It is shown that the solution’s decay rate is the same as [10]. Our study bases on Section 4 in [9].

To the best of our knowledge, this is the first time to obtain the pointwise estimates of the solution to Eq. (1).

Throughout this paper, C denotes the generic positive constants. denote the usual Lebesgue space with norms and the usual Sobolev space with its norm

In particular, we use .

The main result can be stated as following Theorem 1:

Theorem 1. If , , and for any multi-index , , there exists a constant , such that then for , the solution to Eq. (1) has the following estimates: where and is sufficiently small positive constants, ,

Corollary 2. Under the assumptions of Theorem 1, for , , we have that

Remark 3. We get the same decay rate of the solution as [10].

Remark 4. Our study bases on [9, 10], where the spacial dimension . Then in this paper, we have the same assumptions for the spacial dimension.

2. The Green Function

We first study the Green’s function to Eq. (1)which satisfies

We apply the Fourier transform and the inverse Fourier transform

By applying the Fourier transform with respect to the variable , we get the symbol of which is

Here, and correspond to and , By a direct calculation, we have

By Duhamel’s principle, we get the solution of the nonlinear problem (1)

Now we decompose , where

Let and be smooth cut-off functions, where , .

Set

Now we estimate the Green’s function .

2.1. Lower-Frequency Part

First, we give the following Lemma.

Lemma 5. If has compact support in the variable , is a positive integer, and there is a constant , such that for any multi-indexes with , then where is any fixed positive number, .

The proof of Lemma (9) can be seen in [10].

For is sufficiently small, from (9) and the Taylor expansion, we have

Then

Since are smooth functions to variable near , we obtain that when ,

By Lemma (9), we get

For , we have

Then, we have where .

By Lemma (9), we get

From (20)–(23), we have the following proposition:

Proposition 6. For sufficiently small , we have

2.2. Middle-Frequency Part

We can get the following proposition.

Proposition 7. For fixed and , there exist positive numbers and such that The proof of Proposition 7 is similar to that of Proposition 12 in [10], so we omit it.

2.3. Higher-Frequency Part

For is large enough, we have

Then, we have where are polynomials in with degree no more than .

Let

Because taking or , we get the following proposition:

Proposition 8. For being sufficiently large, we have where .

Combining Proposition 6–8, we obtain the following estimate of the Green’s function:

Proposition 9. For any multi-index , we have

3. The Proof of Theorem 1

In this section, we shall give the pointwise estimates of the solution to the problem (1). From (3), we have where

For and , we have the following proposition:

Proposition 10. where and , The proof of the above proposition is similar to proposition 4.1-4.2 in [10], so we omit it.

Next, we give a Lemma which is important to estimate and has been proved in [10].

Lemma 11. If , , we have We write Making use of (31), Lemma 11 and (3), we have where .
From the definition of , we have taking or , we obtain then, we get Thus, we obtain

Together with (38) and (42), we obtain the following result:

Proposition 12. If then

Combining Proposition 10–12, we have the following result:

Proposition 13. where
By the smallness of E and the continuity of , we have Thus, we complete the proof of Theorem 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgments

The authors are grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 11601036), the Science Foundation of Binzhou University (Grant No. BZXYL1402), by the Natural Foundation of Shandong Province (Grant No. ZR2018MF023), and by the Science Foundation of Binzhou University (Grant No. BZXYLG1903 and BZXYL1506).