On global dynamics of 2D convective Cahn–Hilliard equation

In this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in H4(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{4}(\Omega )$\end{document} when the initial value belongs to H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}(\Omega )$\end{document}.


Introduction
The dynamic properties of diffusion equations ensure the stability of diffusion phenomena and provide the mathematical foundation for the study of diffusion dynamics. There are many studies on the existence of global attractors for diffusion equations. For the classical results, we refer the reader to [1][2][3][4][5][6][7][8][9].
Using the same method as [13], we obtain the lemma on the existence of global weak solution to problem (1)-(3). Lemma 1.1 Suppose that u 0 ∈Ḣ 1 per ( ) and the functions ϕ(r) ∈ C 2 (R), ψ(r) ∈ C 1 (R) satisfy The main result of this paper will be stated in the following.

Theorem 1.3
Suppose that u 0 ∈ H 1 per ( ) and the functions ϕ(r) ∈ C 3 (R), ψ(r) ∈ C 2 (R) satisfy  [18,20], we assumed that there exists double-well potential for the convective Cahn-Hilliard equation, which was replaced by the higher order polynomial in [21]. But, in this paper, this assumption is changed by (4), which seems more abroad than double-well potential and polynomial. Second, in [18], the existence of (H 2 , H 2 )-global attractor was obtained, and in [20,21], the existence of (H k , H k )-global attractor was proved. In this paper, we only assume that the initial data belongs to H 1 ( ) and obtain the (H 1 , H 4 )-global attractor for the 2D convective Cahn-Hilliard equation.
The remaining parts are organized as follows. We begin by giving some uniform estimates of solutions for the 2D convective Cahn-Hilliard equation in Sect. 2. Then, in Sect. 3, we prove the main results on the existence of global attractor.

Uniform estimates of solutions
First of all, we establish the uniform estimates of solutions of problem (1) as t → ∞. These estimates are necessary to prove the existence of global attractors. Lemma 2.1 Suppose that u 0 ∈ L 2 ( ) and the functions ϕ(r) ∈ C 1 (R), ψ(r) ∈ C 1 (R) satisfy Then, for problem (1)-(3), we have Here, M 0 is a positive constant depending on γ and c i (i = 0, 1). T 0 depends on γ , c i (i = 0, 1) and R, where u 0 2 ≤ R 2 .
Proof Multiplying equation (1) by u and integrating the resulting relation over , we obtain Note that Hence Applying Poincaré's inequality, we arrive at Moreover, Therefore, the following inequality holds: Summing up, we get where γ satisfies 2γ (c ) 2c 4 > 0. Using Gronwall's inequality, we deduce that By using a mean value theorem for integrals, we obtain the existence of a time t 0 ∈ (T * , T * + 1) such that holds uniformly, the proof is complete.
where k ≤ 3 is a positive constant and i = 0, 1, 2. Then, for problem (1)-(3), we have Here, M 1 is a positive constant depending on γ and c i , Proof Multiplying equation (1) byu and integrating the resulting relation over yields By Nirenberg's inequality, we obtain Thus, by Hölder's inequality and the above inequalities, we deduce that Summing up, we obtain On the other hand, Adding the above two inequalities together gives It then follows from (10) and (11) that Applying Gronwall's inequality yields Using a mean value theorem for integrals, we obtain the existence of a time t 0 ∈ (T , T + 1) such that holds uniformly. Since we consider problem (1)-(3) in the 2D case, based on Sobolev's embedding theorem, we can get Set T 1 = T , we complete the proof. Proof Multiplying equation (1) by 2 u and integrating the resulting relation over , we obtain
Using a mean value theorem for integrals, we obtain the existence of a time t 1 ∈ (T 0 + 1, T 0 + 2) such that the following estimate holds uniformly: Then the proof is complete.
Using a mean value theorem for integrals, we obtain the existence of a time t 2 ∈ (T * 1 + 1, T * 1 + 2) such that the following estimate holds uniformly: Then we complete the proof.

Lemma 2.5
Suppose that u 0 ∈ H 1 per ( ) and the functions ϕ(r) ∈ C 3 (R), ψ(r) ∈ C 2 (R) satisfy Proof Setting v = u t , differentiating (1) with respect to the time t, we deduce that Multiplying (27) by v, integrating the resulting relation over yields Using Sobolev's embedding theorem, we get Hence, A simple calculation shows that It then follows from (29) and the above inequality that where γ is sufficiently large, it satisfies c γc 30 > 0. Using Gronwall's inequality, we derive . Then the proof is complete.

Lemma 2.6
Suppose that u 0 ∈ H 1 per ( ) and the functions ϕ(r) ∈ C 3 (R), ψ(r) ∈ C 2 (R) satisfy Proof Multiplying (27) by Av, integrating the resulting relation over , we obtain Proof For equation (1), by Lemmas 2.1-2.6, we deduce that On the other hand, by Sobolev's embedding theorem, it yields that which completes the proof.
Assume that {u 0,n } ∞ n=1 is bounded inḢ 1 per ( ) and t n → ∞. In the following we prove that {S(t n )u 0,n } ∞ n=1 has a convergent subsequence inḢ 4 per ( ). Denote u n (t) = S(t)u 0,n and v n (t n ) = du n dt t=t n .