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On global dynamics of 2D convective Cahn–Hilliard equation
Boundary Value Problems volume 2020, Article number: 175 (2020)
Abstract
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 \(H^{4}(\Omega )\) when the initial value belongs to \(H^{1}(\Omega )\).
1 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 [1–9].
The convective Cahn–Hilliard equation [10–16], which arises naturally as a continuous model for the formation of facets and corners in crystal growth, is a typical fourth order nonlinear parabolic equation. Let \(\Omega =[0,L]\times [0,L]\), where \(L>0\), γ is a positive constant, β⃗ is a vector. We consider the convective Cahn–Hilliard equation in the 2D case:
Equation (1) is supplemented by the following boundary conditions:
and the initial condition
In this paper, we denote by \(H=L^{2}(\Omega )\), \((\cdot ,\cdot )\) the H-inner product and by \(\|\cdot \|\) the corresponding H-norm, denote \(A=-\Delta \), where Δ is the Laplace operator. Assume that the initial function has zero mean, i.e., \(\int _{\Omega }u_{0}(x)\,dx=0\), then it follows that \(\int _{\Omega }u(x,t)\,dx=0\) for \(t>0\). Here, as [3], we set
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 \(u_{0}\in \dot{H}_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{2}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then there exists a unique solution u for problem (1)–(3) such that
By Lemma 1.1, we can define the operator semigroup \(S(t)u_{0}:\dot{H}^{1}_{per}(\Omega )\times \mathbb{R}^{+} \rightarrow \dot{H}^{1}_{per}(\Omega ) \), which is \((\dot{H}^{1}_{per},\dot{H}^{1}_{per})\)-continuous. In what follows, we always assume that \(\{S(t)\}_{t\geq 0}\) is the semigroup generated by the weak solutions of problem (1). It is sufficient to see that the restriction of \(\{S(t)\}\) on the affined space \(\dot{H}^{1}_{per}(\Omega )\) is a well-defined semigroup.
Proposition 1.2
Suppose that \(\mathcal{A}\) is an \((H^{1}, H^{1})\)-global attractor for \(\{S(t)\}_{t\geq 0}\). Suppose further that \(\{S(t)\}_{t\geq 0}\) has a bounded \((H^{1}, H^{4})\)-absorbing set and \(\{S(t)\}_{t\geq 0}\) is \((H^{1}, H^{4})\)-asymptotically compact. Then \(\mathcal{A}\) is also an \((H^{1}, H^{4})\)-global attractor.
The main result of this paper will be stated in the following.
Theorem 1.3
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then there exists an \((H^{1},H^{4})\)-global attractor for the solution \(u(x,t)\) of problem (1)–(3), which is invariant and compact in \(H^{4}(\Omega )\) and attracts every bounded subset of \(H^{1}(\Omega )\) with respect to the norm topology of \(H^{4}(\Omega )\).
Remark 1.4
In the previous papers [18, 20, 21], my cooperators and I also studied the existence of global attractor for a 2D convective Cahn–Hilliard equation. There are two main differences between the previous results and Theorem 1.3. First, in [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 \(H^{1}(\Omega )\) 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.
2 Uniform estimates of solutions
First of all, we establish the uniform estimates of solutions of problem (1) as \(t\rightarrow \infty \). These estimates are necessary to prove the existence of global attractors.
Lemma 2.1
Suppose that \(u_{0}\in L^{2}(\Omega )\) and the functions \(\varphi (r)\in C^{1}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy
Then, for problem (1)–(3), we have
and
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 \(\|u_{0}\|^{2}\leq R^{2}\).
Proof
Multiplying equation (1) by u and integrating the resulting relation over Ω, 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 \(\frac{2\gamma }{(c')^{2}}-c_{4}>0\). Using Gronwall’s inequality, we deduce that
for all \(t\geq T^{*}=\frac{(c')^{2}}{2\gamma -c_{2}(c')^{2}}\ln \frac{[2\gamma -c_{2}(c')^{2}]R^{2}}{c_{3}(c')^{2}}\). Integrating (6) over \((t,t+1)\) with \(t\geq T^{*}\) yields
By using a mean value theorem for integrals, we obtain the existence of a time \(t_{0}'\in (T^{*},T^{*}+1)\) such that
holds uniformly, the proof is complete. □
Lemma 2.2
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{2}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)–(3), we have
and
Here, \(M_{1}\) is a positive constant depending on γ and \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{1}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).
Proof
Multiplying equation (1) by \(-\Delta u\) and integrating the resulting relation over Ω yields
Hence
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,
and
Adding the above two inequalities together gives
It then follows from (10) and (11) that
Applying Gronwall’s inequality yields
for all \(t\geq T'=\max \{T^{*},\frac{2}{\gamma }\ln \frac{\gamma R^{2}}{2(c_{7}+c_{8})}\}\). Integrating (10) over \((t,t+1)\) with \(t\geq T'\) gives
Using a mean value theorem for integrals, we obtain the existence of a time \(t_{0}\in (T',T'+1)\) 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. □
Lemma 2.3
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{2}(\mathbb{R})\), \(\psi (r)\in C^{1}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)–(3), we have
and
Here, \(M_{2}\) is a positive constant depending on γ and \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{2}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|^{2}_{H^{1}_{per}}\leq R^{2}\).
Proof
Multiplying equation (1) by \(\Delta ^{2}u\) and integrating the resulting relation over Ω, we obtain
Simple calculation shows that
By Sobolev’s embedding theorem, we deduce that
and
Moreover,
Summing up and setting \(\varepsilon =\frac{\gamma }{10}\) gives
By a Calderón–Zygmund type estimate, the following inequality holds:
Then, using Gronwall’s inequality, we obtain
for all \(t\geq T_{0}'=\max \{T_{0},t_{0}'+\frac{2}{\gamma c'}\ln \frac{\gamma c'R^{2}}{2c_{12}}\}\). Setting \(t\geq T'_{0}\), taking \(s\in (t,t+1)\), integrating (14) over \((s,t+1)\), we derive that
Integrating (15) with respect to s in \((t,t+1)\), we can obtain
By (14), (12), (7), and Sobolev’s embedding theorem, we conclude
Multiplying equation (1) by \(u_{t}\), integrating the resulting relation over Ω yields
that is,
Integrating (18) over \((t+1,t+2)\), using (14), we derive that
Using a mean value theorem for integrals, we obtain the existence of a time \(t_{1}\in (T_{0}^{\prime\prime}+1,T_{0}^{\prime\prime}+2)\) such that the following estimate holds uniformly:
Then the proof is complete. □
Lemma 2.4
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)–(3), we have
and
Here, \(M_{3}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{3}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}}^{2}\leq R^{2}\).
Proof
Multiplying (1) by \(\Delta ^{3}u\) and integrating the resulting relation over Ω, we obtain
It follows form (17) that
and
Summing up, we find that
Using Nirenberg’s inequality, we obtain
On the other hand,
Hence
A simple calculation shows that
By Gronwall’s inequality, we immediately obtain
for all \(t\geq T_{1}^{*}=\max \{T_{1},t_{0}+\frac{1}{c_{22}}\ln \frac{c_{22}R^{2}}{2c_{23}}\}\). Combining (23), (14), (12), and (7) together gives
Multiplying equation (1) by \(Au_{t}\), integrating the resulting relation over Ω, we obtain
Summing up, using the result of (23) gives
Then
Setting \(t\geq T_{1}^{*}\), taking \(s\in (t,t+1)\), integrating the above inequality over \((s,t+1)\), we obtain
Integrating the above inequality with respect to s in \((t,t+1)\), we have
Integrating (25) over \((t+1,t+2)\), using (26) yields
Using a mean value theorem for integrals, we obtain the existence of a time \(t_{2}\in (T_{1}^{*}+1,T_{1}^{*}+2)\) such that the following estimate holds uniformly:
Then we complete the proof. □
Lemma 2.5
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)–(3), we have
Here, \(M_{4}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{4}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).
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 Ω 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 \(c'\gamma -c_{30}>0\). Using Gronwall’s inequality, we derive that
for all \(t\geq t_{1}+\frac{1}{c'\gamma -c_{30}}\ln \frac{c_{19}(c'\gamma -c_{30})}{c_{31}}\). Then the proof is complete. □
Lemma 2.6
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)–(3), we have
Here, \(M_{5}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{5}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).
Proof
Multiplying (27) by Av, integrating the resulting relation over Ω, we obtain
By Sobolev’s embedding theorem, we get
Summing up gives
Using Nirenberg’s inequality, we obtain
Adding the above two inequalities together gives
By Gronwall’s inequality, we can obtain
for all \(t\geq t_{2}+\frac{1}{c_{32}}\ln \frac{c_{29}c_{32}}{2c_{33}}\). Then the proof is complete. □
Lemma 2.7
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for problem (1)–(3), we have
Here, \(M_{6}\) is a positive constant depending on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\). \(T_{6}\) depends on γ, \(c_{i}\), \(c'_{i}\) \((i=0,1)\) and R, where \(\|u_{0}\|_{H^{1}_{per}}^{2}\leq R^{2}\).
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. □
3 Proof of Theorem 1.3
Suppose that \(M_{1}\) and \(M_{6}\) are the constants in Lemma 2.2 and Lemma 2.7, respectively. Denote
Using Lemmas 2.2 and 2.7, we easily obtain that \(B_{1}\) is a bounded \((\dot{H}^{1}_{per},\dot{H}_{per}^{1})\)-absorbing set for \(\{S(t)\}_{t\geq 0}\) and \(B_{2}\) is a bounded \((\dot{H}_{per}^{1},\dot{H}_{per}^{4})\)-absorbing set for \(\{S(t)\}_{t\geq 0}\). Note that the embedding \(\dot{H}^{4}_{per}\hookrightarrow \dot{H}^{1}_{per}\) is compacted. Applying Lemma 2.3, we obtain \(\{S(t)\}_{t\geq 0}\) is \((\dot{H}^{1}_{per},\dot{H}^{1}_{per})\)-asymptotically compact. Hence, \(\{S(t)\}_{t\geq 0}\) has an \((\dot{H}_{per}^{1}, \dot{H}_{per}^{1})\)-global attractor \(\mathcal{A}\). In the following, we show that \(\mathcal{A}\) is actually an \((\dot{H}_{per}^{1}, \dot{H}^{4}_{per})\)-global attractor for \(\{S(t)\}_{t\geq 0}\).
Lemma 3.1
Suppose that \(u_{0}\in H_{per}^{1}(\Omega )\) and the functions \(\varphi (r)\in C^{3}(\mathbb{R})\), \(\psi (r)\in C^{2}(\mathbb{R})\) satisfy
where \(k\leq 3\) is a positive constant and \(i=0,1,2\). Then, for the solution \(u(x,t)\) of problem (1)–(3), the dynamical system \(\{S(t)\}_{t\geq 0}\) is \((\dot{H}^{1}_{per}, \dot{H}^{4}_{per})\)-asymptotically compact.
Proof
For (1), we have
Assume that \(\{u_{0,n}\}_{n=1}^{\infty }\) is bounded in \(\dot{H}^{1}_{per}(\Omega )\) and \(t_{n}\rightarrow \infty \). In the following we prove that \(\{S(t_{n})u_{0,n}\}_{n=1}^{\infty }\) has a convergent subsequence in \(\dot{H}^{4}_{per}(\Omega )\). Denote
Note that \(\{u_{0,n}\}_{n=1}^{\infty }\) is bounded in \(\dot{H}_{per}^{1}\). Then there exists \(R>0\) such that
By Lemmas 2.6 and 2.7, there exists \(T>0\) such that
Since \(t_{n}\rightarrow \infty \), there exists \(N>0\) such that \(t_{n}\geq T\) for all \(n\geq N\). Therefore, by (36), we get
Note that the embedding \(D(A^{\frac{1}{2}})\hookrightarrow H\) and \(D(A^{2})\hookrightarrow D(A)\) are compacted. Hence, by (36), there exist \(v\in D(A^{\frac{1}{2}})\), \(\Delta u\in D(A)\), \(\nabla u\in \dot{H}^{3}_{per}\), and \(u\in \dot{H}_{per}^{4}\) such that, up to a subsequence,
By (37) and Sobolev’s embedding theorem, we obtain
It then follows from (36) and (38) that
and
where \(\theta _{1},\theta _{2}\in (0,1)\). Using the same method as above, we also have
Therefore
that is, \(\{u_{n}(t_{n})\}_{n=1}^{\infty }\) converges to \(A^{-2}( -v+\Delta \varphi (u)+\beta \cdot \nabla \psi (u))\) in \(\dot{H}_{per}^{4}(\Omega )\). Then we complete the proof. □
Now we give the proof of the main result.
Proof of Theorem 1.3
Note that \(\{S(t)\}_{t\geq 0}\) has an \((\dot{H}^{1}_{per}, \dot{H}^{1}_{per})\)-global attractor \(\mathcal{A}\). By Lemma 2.7, \(B_{2}\) is a bounded \((\dot{H}^{1}_{per}, \dot{H}^{4}_{per})\)-absorbing set for \(\{S(t)\}_{t\geq 0}\). On the other hand, by Lemma 3.1, we can obtain \(\{S(t)\}_{t\geq 0}\) is \((\dot{H}_{per}^{1},\dot{H}_{per}^{4})\)-asymptotically compact. Then, by Proposition 1.2, \(\mathcal{A}\) is actually an \((\dot{H}^{1}_{per}, \dot{H}_{per}^{4})\)-global attractor for \(\{S(t)\}_{t\geq 0}\). The proof of Theorem 1.3 is complete. □
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Acknowledgements
The author would like to thank Dr. Ning Duan and Ms. Haichao Meng for their helpful suggestions.
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This paper was supported by the Fundamental Research Funds for the Central Universities (grant No. N2005031).
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The main idea of this paper was proposed by XZ. XZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Zhao, X. On global dynamics of 2D convective Cahn–Hilliard equation. Bound Value Probl 2020, 175 (2020). https://doi.org/10.1186/s13661-020-01477-3
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DOI: https://doi.org/10.1186/s13661-020-01477-3