Finite-dimensional global attractor of the Cahn–Hilliard–Brinkman system

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J. Math. Anal. Appl. ••• (••••) •••–•••

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Finite-dimensional global attractor of the Cahn–Hilliard–Brinkman system Fang Li a,∗ , Chengkui Zhong a , Bo You b a b

Department of Mathematics, Nanjing University, Nanjing, 210093, PR China School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, PR China

a r t i c l e

i n f o

Article history: Received 20 May 2015 Available online xxxx Submitted by J. Shi Keywords: Cahn–Hilliard–Brinkman system Dissipativity Asymptotical a priori estimates Global attractor Fractal dimension

a b s t r a c t Our aim in this paper is to study the asymptotic behavior of the Cahn–Hilliard– Brinkman system. We obtain the existence of a finite fractal dimensional global attractor A in H 4 (Ω) ∩ VI by asymptotical a priori estimates for the Cahn–Hilliard– Brinkman system. © 2015 Published by Elsevier Inc.

1. Introduction This paper is concerned with the following Cahn–Hilliard–Brinkman system: ∂φ + u · ∇φ = ∇ · (M ∇μ), (x, t) ∈ Ω × R+ , ∂t 1 μ = −Δφ + f (φ), (x, t) ∈ Ω × R+ , 

(1.1) (1.2)

−νΔu + ηu = −∇p − γφ∇μ, (x, t) ∈ Ω × R+ ,

(1.3)

∇ · u = 0, (x, t) ∈ Ω × R+ ,

(1.4)

where Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω, R+ = [0, +∞), ν > 0 is the viscosity, η > 0 is the fluid permeability, p is the fluid pressure, M > 0 stands for the mobility,  > 0 is related to the diffuse interface thickness, f is the derivative of a double well potential F (s) = 14 (s2 − 1)2 describing phase separation, and γ > 0 is a surface tension parameter. * Corresponding author. E-mail addresses: [email protected] (F. Li), [email protected] (C. Zhong), [email protected] (B. You). http://dx.doi.org/10.1016/j.jmaa.2015.09.026 0022-247X/© 2015 Published by Elsevier Inc.

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Equations (1.1)–(1.4) are subject to the following boundary conditions u(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,

(1.5)

∂μ ∂φ = = 0, (x, t) ∈ ∂Ω × R+ ∂n ∂n

(1.6)

φ(x, 0) = φ0 (x),

(1.7)

and initial condition

where n is the normal vector on ∂Ω. A diffuse interface variant of Brinkman equation has been proposed to model phase separation of incompressible binary fluids in a porous medium (see [14]). The coupled system consists of a convective Cahn–Hilliard equation for the phase field φ, i.e., the difference of the relative concentrations of the two phases, and a modified Darcy equation proposed by H.C. Brinkman [5] in 1947 for the fluid velocity u. This equation incorporates a diffuse interface surface force proportional to φ∇μ, where μ is the so-called chemical potential which is the variational derivative of the free energy functional  1  |∇φ|2 + F (φ) dx. E(φ) = 2  Ω

For this reason, equations (1.1)–(1.4) have been called Cahn–Hilliard–Brinkman system. Such a system belongs to a class of diffuse interface models which are used to describe the behavior of multi-phase fluids. The Cahn–Hilliard–Navier–Stokes system has been investigated from the numerical and analytical viewpoint in several papers (see, e.g., [1,2,4,6,7,9–11,13,16,19,21]). The long-time behavior and well-posedness of solutions for the Cahn–Hilliard–Hele–Shaw system were proved in [17,18]. In [12], the authors have considered the well-posedness and long-time behavior of solutions for a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth which is more complicated than the Cahn–Hilliard–Brinkman system thanks to the compressibility of the fluid. Meanwhile, they established the existence of a pullback attractor in H 2 (Ω), and proved any global weak/strong solution is convergent to a single steady state as time t → +∞ and obtained its convergence rate. The Cahn–Hilliard–Brinkman system (1.1)–(1.4) with M , ν, and η possibly depending on φ has been analyzed from the numerical viewpoint in [7,8]. From the analytical point of view, the authors in [3] have considered the well-posedness of solutions, the existence of a global attractor in H 1 (Ω) for the Cahn– Hilliard–Brinkman system (1.1)–(1.4) with positive constants M , ν, η,  and more general f (u), established the convergence of a given weak solution to a single equilibrium via the Lojasiewicz–Simon inequality and gave its convergence rate. Furthermore, the authors have also studied the behavior of the solutions as the viscosity goes to zero, i.e., the existence of a weak solution to the Cahn–Hilliard–Hele–Shaw system was proved as the limit of solutions to the Cahn–Hilliard–Brinkman system when the Cahn–Hilliard–Brinkman system approaches the Cahn–Hilliard–Hele–Shaw system. It is well-known that the asymptotical behavior of solutions of a dissipative evolution equation can be adequately described in terms of its global attractor A. In many problems, the influence of initial data has vanished after a long time has elapsed, therefore permanent regimes are of importance. The simplest permanent regimes are described by time-independent functions that are solutions of the corresponding elliptic equation. Such regimes are important but very special and it is included in the global attractor A, which implies that the regularity of solutions of elliptic equation can be obtained from the regularity of global attractor of the corresponding evolution equation if its global attractor exists. From the result of [3], we know the fact that each trajectory φ(t) of the Cahn–Hilliard–Brinkman system originating from the initial data φ0 will converge to a stationary solution w of the elliptic equation

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−Δw + f (w) = mf (w)

3

(1.8)

with respect to the norm of H 1 (Ω), where mf (w) is the mean value of the function f (w) on Ω defined by  1 mf (w) = f (w) dx. |Ω| Ω

Therefore, the regularity of solutions for the elliptic equation (1.8) can be obtained by the study of the regularity of the global attractor for the Cahn–Hilliard–Brinkman system (1.1)–(1.6). However, to the best of our knowledge, there seems to be no results about the regularity and fractal dimension of the global attractor concerning the Cahn–Hilliard–Brinkman system (1.1)–(1.6). This is the main goal of the present paper. We summarize the main results of this paper as follows. First, we give some a priori estimates of solutions for (1.1)–(1.7), which imply the existence of an absorbing set in H 4 (Ω) ∩ VI for any fixed I ∈ R, where VI is defined as VI = {φ ∈ H 1 (Ω) : mφ = I}. Then, we prove the existence of a finite fractal dimensional global attractor in H 4 (Ω) ∩ VI for the semigroup {SI (t)}t≥0 associated with equations (1.1)–(1.6) by using asymptotical a priori estimates (see Theorem 3.4 and Theorem 4.3). Throughout this paper, for the sake of simplicity, we assume M =  = γ = ν = η = 1. Denote the norm in Lp (Ω) by  · p , let C, ρi (i = 1, 2, · · · ) be constants that are independent of the initial data of φ. 2. Some a priori estimates of weak solutions 2.1. The well-posedness of weak solutions In [3], the authors proved the well-posedness of weak solutions for the Cahn–Hilliard–Brinkman system. Now, we state it as follows. Theorem 2.1. (See [3].) Assume that φ0 ∈ H 1 (Ω). Then there exists a unique weak solution φ(t) ∈ C(R+ , H 1 (Ω)) ∩ L2loc (R+ , H 3 (Ω)) of (1.1)–(1.7) such that mφ(t) = mφ0 , which depends continuously on the initial data in H 1 (Ω). By Theorem 2.1, we can define the operator semigroup {SI (t)}t≥0 on VI as SI (·) : R+ × VI → VI , which is (VI , VI )-continuous. 2.2. Some a priori estimates of weak solutions In this subsection, we give some a priori estimates of weak solutions for (1.1)–(1.6), which imply the existence of an absorbing set in H 4 (Ω) ∩ VI for the semigroup {SI (t)}t≥0 generated by (1.1)–(1.6). Remark 2.2. The steps of the following proof are formal in the sense that they can be made rigorous by proving their corresponding counterpart estimates first for the Galerkin approximation system (1.1)–(1.6). Then the estimates for the exact solution can be established by passing to the limit in the Galerkin procedure via the appropriate “compactness theorems” and the uniqueness of the strong solution.

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2.2.1. VI estimates of φ Multiplying (1.1), (1.2) and (1.3) by μ, φ and u, respectively, and integrating by parts, we get ⎞ ⎛   d ⎝ 1 2 2 2 2 2 ⎠ |∇φ| + F (φ) dx + ∇μ2 + ∇u2 + u2 + ∇φ2 + f (φ)φ dx dt 2 

Ω



∇ · (uφ)μ dx −

= − Ω

 =

Ω

 (uφ) · ∇μ dx +

Ω

φμ dx Ω

φμ dx,

(2.1)

Ω

where we use the following equality  −

 ∇ · (uφ)μ dx −

Ω

(uφ) · ∇μ dx = 0. Ω

Note that 1 4 φ − φ2 ≤ f (φ)φ 2

2F (φ) = and

(2.2)





(μ − mμ + mμ)φ dx

φμ dx = Ω

Ω

≤ μ − mμ2 φ2 + I|Ω|mμ  ≤ C∇μ2 φ2 + I φ3 dx Ω

⎛



≤ C∇μ2 ⎝

⎞ 14

⎞ 34 ⎛  1 1 F (φ) + dx⎠ + C|I| ⎝ F (φ) + dx⎠ . 2 2

Ω

(2.3)

Ω

Combining (2.1)–(2.3) with Young inequality, we obtain ⎞ ⎛   d ⎝ 2 2 2 2 ⎠ |∇φ| + 2F (φ) dx + ∇μ2 + 2uH 1 (Ω) + 2∇φ2 + 2 F (φ) dx dt Ω

Ω

≤ C.

(2.4)

It follows from the classical Gronwall inequality that ⎛ ⎞   |∇φ|2 + 2F (φ) ≤ ⎝ |∇φ0 |2 + 2F (φ0 )⎠ e−t + C. Ω

Ω

From the fact that

 φ2H 1 (Ω)

− 2|Ω| ≤

|∇φ|2 + 2F (φ) ≤ φ4H 1 (Ω) + 1, Ω

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we deduce that there exist two positive constants ρi (i = 1, 2) satisfying that for any bounded subset B ⊂ H 1 (Ω), there exists a time T = TB such that φ(t)2H 1 (Ω) ≤ ρ1 , t+1 ∇μ(s)22 + 2u(s)2H 1 (Ω) ds ≤ ρ2

(2.5) (2.6)

t

for any t ≥ T . For brevity, we omit writing out explicitly these bounds here and we also omit writing out other similar bounds in our future discussion for all other uniform a priori estimates. 2.2.2. H 2 (Ω) ∩ VI estimates of φ Taking the inner product of (1.1) with Δ2 φ in L2 (Ω) and combining (1.6), we get 1 d Δφ22 + Δ2 φ22 2 dt     = − ∇ · (uφ)Δ2 φ dx + 6φ|∇φ|2 + 3φ2 Δφ − Δφ Δ2 φ Ω

Ω

≤ u6 ∇φ3 Δ φ2 + 6φ6 ∇φ26 Δ2 φ2 + 3φ26 Δφ6 Δ2 φ2 + Δφ2 Δ2 φ2 2

1

1

≤ CuH 1 (Ω) (∇φ2 + ∇φ22 Δφ22 )Δ2 φ2 + CφH 1 (Ω) (∇φ22 + Δφ22 )Δ2 φ2 + Cφ2H 1 (Ω) (Δφ2 + ∇Δφ2 )Δ2 φ2 + Δφ2 Δ2 φ2 .

(2.7)

Combining (2.7) with Young inequality, we find d Δφ22 + Δ2 φ22 dt ≤ C(u2H 1 (Ω) + φ2H 1 (Ω) Δφ22 + φ4H 1 (Ω) + 1)Δφ22 + Cφ6H 1 (Ω) + Cφ2H 1 (Ω) u2H 1 (Ω) + Cφ4H 1 (Ω) ∇Δφ22 .

(2.8)

Thanks to Δφ2 ≤ μ2 + φ36 + φ2 ≤ μ − mμ2 + Cφ33 + φ36 + φ2 + C|I| ≤ C(∇μ2 + φ3H 1 (Ω) + 1) and ∇Δφ2 ≤ ∇μ2 + Cφ26 ∇φ6 + ∇φ2 ≤ ∇μ2 + Cφ3H 1 (Ω) + Cφ2H 1 (Ω) Δφ2 + φH 1 (Ω) , we infer from the uniform Gronwall inequality and (2.5)–(2.6), (2.8) that Δφ(t)22 ≤ ρ3 , t+1 Δ2 φ(s)22 ds ≤ ρ4 t

for any t ≥ T + 1.

(2.9) (2.10)

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2.2.3. H 3 (Ω) ∩ VI estimates of φ Multiplying (1.1) by μt and integrating over Ω, we find 1 d ∇μ22 + ∇φt 22 + 3 2 dt

|φ|2 |φt |2 dx − φt 22 Ω

 = −



∇ · (uφ)(−Δφt + 3φ2 φt − φt )dx Ω

≤ C∇u3 ∇φ6 ∇φt 2 + Cu3 Δφ6 ∇φt 2 + Cu6 ∇φ6 φ6 φφt 2 + u3 φ6 ∇φt 2 .

(2.11)

Multiplying (1.3) by u and integrating over Ω, we obtain u2H 1 (Ω) = ∇u22 + u22  = − (uφ) · ∇μ dx Ω

≤ u3 φ6 ∇μ2 ≤

1 u2H 1 (Ω) + Cφ2H 1 (Ω) ∇μ22 , 2

which implies u2H 1 (Ω) ≤ Cφ2H 1 (Ω) ∇μ22 .

(2.12)

Taking the inner product of (1.3) with −Δu in (L2 (Ω))3 and combining (1.5), we get  Δu22

+

∇u22

(Δuφ) · ∇μ dx

= Ω

≤ Δu2 φL∞ (Ω) ∇μ2 , which implies Δu22 + ∇u22 ≤ Cφ2L∞ (Ω) ∇μ22 ≤ CφH 1 (Ω) φH 2 (Ω) ∇μ22 .

(2.13)

It follows from (2.11)–(2.13) and Young inequality that d ∇μ22 + ∇φt 22 + 3 dt

 |φ|2 |φt |2 dx Ω

≤ CφH 1 (Ω) φ3H 2 (Ω) ∇μ22 + CΔφ26 φ2H 1 (Ω) ∇μ22 + Cφt 22 + Cφ4H 1 (Ω) φ2H 2 (Ω) ∇μ22 + Cφ4H 1 (Ω) ∇μ22 . Denote

(2.14)

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φt 2 ≤ Δμ2 + u3 ∇φ6 ≤ Δ2 φ2 + 6φ6 ∇φ26 + 3φ26 Δφ6 + Δφ2 + u3 ∇φ6 and Δφ6 ≤ ΔφH 1 (Ω) ≤ Δφ2 + ∇Δφ2 , we infer from (2.5)–(2.6), (2.9)–(2.10), (2.12), (2.14) and uniform Gronwall inequality that ∇Δφ(t)22 ≤ ρ5 , t+1 φt (s)2H 1 (Ω) ≤ ρ6

(2.15) (2.16)

t

for any t ≥ T + 2. Moreover, we infer from (2.12)–(2.13) and (2.15) that u(t)2H 2 (Ω) ≤ ρ7

(2.17)

for any t ≥ T + 2. 2.2.4. VI estimates of φt Denote v = ut , q = pt and ψ = φt . It is clear that v, θ respectively satisfy the following equations obtained by differentiating the equations (1.1)–(1.6) with respect to t: ∂ψ + v · ∇φ + u · ∇ψ = Δμt ∂t = −Δ2 ψ + 6ψ|∇φ|2 + 12φ∇ψ · ∇φ + 6φψΔφ + 3φ2 Δψ − Δψ,

(2.18)

−Δv + v = −∇q − ψ∇μ − φ∇μt ,

(2.19)

∇ · v = 0.

(2.20)

Equations (2.18)–(2.20) are subject to the following boundary conditions v(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,

(2.21)

∂Δψ ∂ψ = = 0, (x, t) ∈ ∂Ω × R+ . ∂n ∂n

(2.22)

Multiplying (2.18) by ψ and integrating by parts, we find 1 d ψ22 + Δψ22 2 dt ≤ v3 φ6 ∇ψ2 + 6ψ6 ∇φ26 ψ2 + 12φ6 ∇ψ2 ∇φ6 ψ6 + 6φ6 ψ26 Δφ2 + 3φ26 Δψ2 ψ6 + ∇ψ22 . Taking the L2 (Ω) inner product of (2.19) with v, we obtain

(2.23)

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v2H 1 (Ω) ≤ v3 ψ6 ∇μ2 + v6 ∇φ3 μt 2 ≤

  1 v2H 1 (Ω) + Cψ26 ∇μ22 + C∇φ23 Δψ22 + φ46 ψ26 + ψ22 , 2

which implies v2H 1 (Ω) = ∇v22 + v22

  ≤ Cψ26 ∇μ22 + C∇φ23 Δψ22 + φ46 ψ26 + ψ22 .

(2.24)

We deduce from (2.16), (2.23)–(2.24), Young inequality and uniform Gronwall inequality that φt (t)22 ≤ ρ7 , t+1 Δφt (s)22 ≤ ρ8

(2.25) (2.26)

t

for any t ≥ T + 3. Multiplying (2.18) by −Δψ and integrating by parts, we get 1 d ∇ψ22 + ∇Δψ22 2 dt ≤ v3 φ6 ∇Δψ2 + u3 ψ6 ∇Δψ2 + 6ψ6 ∇φ26 Δψ2 + Cφ6 ∇ψ2 ∇φ6 ∇Δψ2 + Cφ6 ψ6 Δφ2 ∇Δψ2 + Cφ26 Δψ2 ∇Δψ2 + Δψ22 .

(2.27)

We deduce from (2.16), (2.24)–(2.27), Young inequality and uniform Gronwall inequality that ∇φt (t)22 ≤ ρ9

(2.28)

for any t ≥ T + 4. 2.2.5. H 4 (Ω) ∩ VI estimates of φ Taking the inner product of (1.1) with Δμ in L2 (Ω) and using Young inequality, we find Δμ2 ≤ φt 2 Δμ2 + u3 ∇φ6 Δμ2   1 ≤ Δμ2 + C φt 22 + u23 ∇φ26 , 2 which implies   Δμ2 ≤ C φt 22 + u23 ∇φ26 .

(2.29)

Thanks to Δ2 φ2 ≤ Δμ2 + 6φ6 ∇φ26 + 3φ26 Δφ6 + Δφ2 ≤ Δμ2 + 6φ6 ∇φ26 + Cφ26 ∇Δφ2 + Δφ2 ,

(2.30)

it follows from (2.5), (2.9), (2.15), (2.17) and (2.25) that Δ2 φ22 ≤ ρ10 for any t ≥ T + 4.

(2.31)

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From (2.5), (2.9), (2.15) and (2.31), we obtain the following result. Theorem 2.3. Let {SI (t)}t≥0 be a semigroup generated by (1.1)–(1.6). Then there exists a (VI , H 4 (Ω) VI )absorbing set. That is, there exists a positive constant R satisfying for any bounded subset B ⊂ VI , there is a positive time T = T (B) depending on the VI -norm of B such that u(t)2H 1 (Ω) + φt (t)2H 1 (Ω) + φ(t)2H 4 (Ω) ≤ R2 for any t ≥ T . 3. The existence of global attractors In this section, we prove the existence of a global attractor in H 4 (Ω) ∩VI for the Cahn–Hilliard–Brinkman system. In [3], the authors proved the existence of a global attractor in VI for the Cahn–Hilliard–Brinkman system, we first recall it as follows. Theorem 3.1. The semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) has a global attractor A1 in VI . Next, for any fixed s ∈ (1, 4), from Theorem 2.3, we deduce that the semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) is norm-to-weak continuous in H s (Ω) ∩ VI . Furthermore, from the compactness of H 4 (Ω) ∩ VI ⊂ H s (Ω) ∩ VI , we immediately obtain the existence of a global attractor in H s (Ω) ∩ VI from the theory of global attractor in [20]. Theorem 3.2. Assume that s ∈ (1, 4). Then the semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) has a global attractor As in H s (Ω) ∩ VI . Finally, we prove the asymptotical compactness in H 4 (Ω) ∩ VI of the semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) to obtain the existence of a global attractor in H 4 (Ω) ∩ VI . Theorem 3.3. The semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) is asymptotically compact in H 4 (Ω) ∩ VI . Proof. Let B0 be an absorbing set in H 4 (Ω) ∩ VI obtained in Theorem 2.3, then we need to show that for 4 any tn → +∞ and φ0n ∈ B0 , {φn (tn )}∞ n=0 is pre-compact in H (Ω) ∩ VI , where φn (tn ) = SI (tn )φ0n . 1,3 In fact, from Theorem 2.3, we know that {φn (tn )}∞ (Ω) ∩ VI and H 3 (Ω) ∩ VI , n=0 is pre-compact in W 3 ∞ 2 {un (tn )}∞ n=0 is pre-compact in L (Ω) and {(φn )t (tn )}n=0 is pre-compact in L (Ω). Without loss of generality, ∞ 1,3 we assume that {φn (tn )}n=0 is a Cauchy sequence in W (Ω) ∩ VI , {un (tn )}∞ n=0 is a Cauchy sequence in 2 L3 (Ω) and {(φn )t (tn )}∞ is a Cauchy sequence in L (Ω). n=0 2 In the following, we prove that {Δμn (tn )}∞ n=0 is a Cauchy sequence in L (Ω), where μn (tn ) = −Δφn (tn ) + 3 φn (tn ) − φn (tn ). Then, we obtain

Δμnk (tnk ) − Δμnj (tnj )22

 d d ≤ − φnk (tnk ) + φnj (tnj ), Δμnk (tnk ) − Δμnj (tnj ) dt dt   + −unk (tnk ) · ∇φnk (tnk ) + unj (tnj ) · ∇φnj (tnj ), Δμnk (tnk ) − Δμnj (tnj ) . From the simple calculations and Hölder inequality, we deduce that

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Δμnk (tnk ) − Δμnj (tnj )22 ≤

d d φn (tn ) − φnj (tnj )2 Δμnk (tnk ) − Δμnj (tnj )2 dt k k dt

+ unk (tnk ) − unj (tnj )3 ∇φnj (tnj )6 Δμnk (tnk ) − Δμnj (tnj )2 + unk (tnk )6 ∇φnk (tnk ) − ∇φnj (tnj )3 Δμnk (tnk ) − Δμnj (tnj )2 . Combining Theorem 2.3, Sobolev embedding compactness Theorem with Young inequality, yields 2 {Δμn (tn )}∞ n=0 is a Cauchy sequence in L (Ω). Thanks to μnk (tnk ) − μnj (tnj )     = −Δ φnk (tnk ) − φnj (tnj ) − φnk (tnk ) − φnj (tnj )

  + φnk (tnk ) − φnj (tnj ) φ2nk (tnk ) + φnk (tnk )φnj (tnj ) + φ2nj (tnj ) and Δ2 φnk (tnk ) − Δ2 φnj (tnj )2 = Δμnk (tnk ) − Δμnj (tnj )2 + Δφnk (tnk ) − Δφnj (tnj )2 + CR2 φnk (tnk ) − φnj (tnj )H 2 (Ω) , 2 it implies {Δ2 φn (tn )}∞ n=0 is a Cauchy sequence in L (Ω), where R is specified in Theorem 2.3. Therefore, ∞ 4 {φn (tn )}n=0 is a Cauchy sequence in H (Ω) ∩ VI . 2

For any fixed I ∈ R, from Theorem 2.3, we deduce that the semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) is norm-to-weak continuous in H 4 (Ω) ∩VI . Furthermore, from Theorem 3.3, we immediately get the existence of a global attractor in H 4 (Ω) ∩ VI . Theorem 3.4. The semigroup {SI (t)}t≥0 generated by (1.1)–(1.6) has a global attractor A in H 4 (Ω) ∩ VI . 4. Estimates of the fractal dimension of the global attractor In this section, we will give the estimate of the fractal dimension of the global attractor for the Cahn– Hilliard–Brinkman system (1.1)–(1.6). To start with, we recall the definition of fractal dimension. Definition 4.1. (See [15].) Let H be a separable real Hilbert space. For any non-empty compact K ⊂ H, the fractal dimension of K is the number df (K) = lim sup →0+

log(N (K)) , log( 1 )

where N (K) denotes the minimum number of open balls in H with radii  > 0 that are necessary to cover K. Let A be the stokes operator. Equation (1.1)–(1.4) can be formulated in the form 

dφ dt

= g(φ, t),

φ(0) = φ0 ,

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11

where ⎧ ⎪ ⎪ ⎨g(φ, t) = Δμ − u · ∇φ, μ = −Δφ + f (φ), ⎪ ⎪ ⎩Au + u = −φ∇μ. Then, g(·, t) is Gateaux differentiable in H 3 (Ω) ∩ VI for any t ∈ R with ⎧ ⎪ ⎪g  (φ, t)ϕ = Δν − u · ∇ϕ − v · ∇φ, ⎨ ν = −Δϕ + 3φ2 ϕ − ϕ, ⎪ ⎪ ⎩Av + v = −φ∇ν − ϕ∇μ,

for any φ, ϕ ∈ H 3 (Ω) ∩ VI and the mapping g  : φ ∈ H 3 (Ω) ∩ VI × R → g  (φ, t) ∈ L(H 3 (Ω) ∩ VI , VI ) is continuous. Obviously, there exists a unique solution ϕ(t) = ϕ(t; ϕ0 ) ∈ C([0, T ]; VI )∩L2 (0, T ; H 3 (Ω)) for any 0 ≤ t ≤ T of the following equation: 

dϕ dt

= g  (S(t)φ0 , t)ϕ,

(4.1)

ϕ(0) = ϕ0 .

Theorem 4.2. Assume that φi,0 ∈ VI for i = 1, 2. Then, there exists a bounded linear operator Λ(t; φ2,0 ) : VI → VI such that S(t)φ1,0 − S(t)φ2,0 − Λ(t; φ2,0 )(φ1,0 − φ2,0 )VI ≤ γ(t, φ1,0 − φ2,0 VI )φ1,0 − φ2,0 VI , where γ(t, ζ) → 0 as ζ → 0+ for all t ≥ 0 and ϕ(t) = Λ(t; φ2,0 )ϕ0 is the solution of the linearized equation (4.1) with φ0 = φ2,0 . Proof. Let φi (t) = S(t)φi,0 , μi = −Δφi + f (φi ), Aui + ui = −φi ∇μi for i = 1, 2 and let ϕ(t) be a solution of the linearized equation (4.1) with φ0 = φ2,0 and initial data ϕ0 = φ1,0 − φ2,0 ∈ V0 . Denote φ = φ1 − φ2 , μ = μ1 − μ2 and u = u1 − u2 for any t ≥ 0. Obviously, φ(t) ∈ C([0, T ]; VI ) ∩ L2 (0, T ; H 3 (Ω)) satisfies φt = Δμ − u · ∇φ1 − u2 · ∇φ,

(4.2)

μ = −Δφ + f (φ1 ) − f (φ2 ),

(4.3)

Au + u = −φ1 ∇μ − φ∇μ2 .

(4.4)

Multiplying (4.2) and (4.4) by −Δφ and u, respectively, and integrating by parts, we obtain 1 d ∇φ22 + ∇u22 + u22 2 dt   = − (Δμ − u · ∇φ1 − u2 · ∇φ)Δφ dx + u · ∇φμ2 dx Ω

Ω



u · ∇φ1 (−Δφ + f (φ1 ) − f (φ2 )) dx

+ Ω

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AID:19801 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.159; Prn:23/09/2015; 9:14] P.12 (1-18)

F. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

12

 = −

 u2 · ∇φΔφ dx +

ΔμΔφ dx + Ω



Ω



u · ∇φμ2 dx Ω

u · ∇φ1 (f (φ1 ) − f (φ2 )) dx.

+

(4.5)

Ω

Using Hölder inequality, Sobolev inequality and interpolation inequality, we estimate each term of the right hand side of (4.5) as follows.  −

 ΔμΔφ dx = −

Ω

Δ(−Δφ + f (φ1 ) − f (φ2 ))Δφ dx Ω

 ∇(f (φ1 ) − f (φ2 )) · ∇Δφ dx

= − ∇Δφ22 + Ω

≤ −

∇Δφ22

+ ∇(f (φ1 ) − f (φ2 ))2 ∇Δφ2 ,

and 

 u2 · ∇φΔφ dx + Ω

 u · ∇φμ2 dx +

Ω

u · ∇φ1 (f (φ1 ) − f (φ2 )) dx Ω

≤ u2 3 φ6 ∇Δφ2 + u3 ∇μ2 2 φ6 + u3 φ1 6 ∇(f (φ1 ) − f (φ2 ))2 ≤ C(u2 H 1 (Ω) ∇φ2 ∇Δφ2 + ∇u2 μ2 H 1 (Ω) ∇φ2 + ∇u2 φ1 H 1 (Ω) ∇(f (φ1 ) − f (φ2 ))2 ). Thanks to ∇(f (φ1 ) − f (φ2 ))2 ≤ f  (φ1 )L∞ (Ω) ∇φ2 + 3φ1 + φ2 6 φ6 ∇φ2 6 ≤ f  (φ1 )L∞ (Ω) ∇φ2 + Cφ1 + φ2 6 ∇φ2 φ2 H 2 (Ω) , we infer from Young inequality that d ∇φ22 + ∇Δφ22 + ∇u22 + u22 dt ≤ C(1 + φ1 2H 1 (Ω) )(f  (φ1 )2L∞ (Ω) + φ1 + φ2 26 φ2 2H 2 (Ω) )∇φ22 + C(μ2 2H 1 (Ω) + u2 2H 1 (Ω) )∇φ22 . From the classical Gronwall inequality, we deduce t ∇φ22

(∇Δφ22 + ∇u22 + u22 ) ds

+ 0

≤ ∇φ0 22 exp where

t 0

H1 (s) ds

,

(4.6)

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AID:19801 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.159; Prn:23/09/2015; 9:14] P.13 (1-18)

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13

H1 (t) = C(1 + φ1 2H 1 (Ω) )(f  (φ1 )2L∞ (Ω) + φ1 + φ2 26 φ2 2H 2 (Ω) ) + Cμ2 2H 1 (Ω) + Cu2 2H 1 (Ω) . Now, we denote ψ(t) = φ1 (t) − φ2 (t) − ϕ(t), ϑ = μ1 − μ2 − ν and w = u1 − u2 − v for any t ≥ 0. Obviously, ψ(t) ∈ C([0, T ]; V0 ) ∩ L2 (0, T ; H 3 (Ω)) satisfies ψt = −u · ∇φ − w · ∇φ2 − u2 · ∇ψ + Δϑ, ϑ = −Δψ − ψ + φ3 + 3φ2 φ2 + 3φ22 ψ, Aw + w = −φ∇μ − ψ∇μ2 − φ2 ∇ϑ,

(4.7) (4.8) (4.9)

ψ(0) = 0.

(4.10)

Multiplying (4.7) and (4.9) by −Δψ and w, respectively, and integrating by parts, we obtain 1 d ∇ψ22 + ∇w22 + w22 2 dt  = − (Δϑ − u · ∇φ − w · ∇φ2 − u2 ∇ψ)Δψ dx Ω



(−φ∇μ − ψ∇μ2 − φ2 ∇ϑ) · w dx

+ Ω



= −

 (Δϑ − u · ∇φ)Δψ dx +

Ω

 u2 ∇ψΔψ dx +

Ω



(w · ∇φμ + w · ∇ψμ2 ) dx Ω

w · ∇φ2 (−ψ + φ3 + 3φ2 φ2 + 3φ22 ψ) dx.

+

(4.11)

Ω

Using Hölder inequality, Sobolev inequality and interpolation inequality, we estimate each term of the right (4.11) as follows.  − (Δϑ − u · ∇φ)Δψ dx Ω





Δ(−Δψ − ψ + φ + 3φ φ2 +

= −

3

2

3φ22 ψ)Δψ dx

Ω

u · ∇φΔψ dx

+ Ω

≤ − ∇Δψ22 + Δψ22 + ∇(φ3 + 3φ2 φ2 + 3φ22 ψ)2 ∇Δψ2 + u3 φ6 ∇Δψ2 ≤ − ∇Δψ22 + C∇ψ2 ∇Δψ2 + ∇(φ3 + 3φ2 φ2 + 3φ22 ψ)2 ∇Δψ2 + CuH 1 (Ω) ∇φ2 ∇Δψ2 ,  w · ∇φμdx Ω



w · ∇φ(−Δφ + f (φ1 ) − f (φ2 )) dx

= Ω

≤ w3 φ6 ∇Δφ2 + w3 φ6 ∇(f (φ1 ) − f (φ2 ))2 ≤ C∇w2 ∇φ2 ∇Δφ2 + C∇w2 ∇φ2 ∇(f (φ1 ) − f (φ2 ))2

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AID:19801 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.159; Prn:23/09/2015; 9:14] P.14 (1-18)

F. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

14

and  u2 · ∇ψΔψ + w · ∇ψμ2 + w · ∇φ2 (−ψ + φ3 + 3φ2 φ2 + 3φ22 ψ) dx Ω

≤ u2 3 ψ6 ∇Δψ2 + w3 ψ6 ∇μ2 2 + w3 φ2 6 ∇ψ2 + w3 φ2 6 ∇(φ3 + 3φ2 φ2 + 3φ22 ψ)2 ≤ Cu2 H 1 (Ω) ∇ψ2 ∇Δψ2 + C∇w2 ∇ψ2 μ2 H 1 (Ω) + C∇w2 φ2 H 1 (Ω) ∇ψ2 + C∇w2 φ2 H 1 (Ω) ∇(φ3 + 3φ2 φ2 + 3φ22 ψ)2 . Thanks to ∇(φ3 + 3φ2 φ2 + 3φ22 ψ)2 ≤ φ26 ∇φ6 + φ2 6 φ6 ∇φ6 + ∇φ2 6 φ26 + φ2 6 ∇φ2 6 ψ6 + φ2 2L∞ (Ω) ∇ψ2 ≤ C∇φ22 Δφ2 + Cφ2 H 1 (Ω) ∇φ2 Δφ2 + Cφ2 H 2 (Ω) ∇φ22 + Cφ2 2L∞ (Ω) ∇ψ2 + Cφ2 H 1 (Ω) φ2 H 2 (Ω) ∇ψ2 , we infer from Young inequality that d ∇ψ22 + ∇Δψ22 + ∇w22 + w22 dt ≤ H2 (t)∇ψ22 + H3 (t) where H2 (t) = C(1 + u2 2H 1 (Ω) + ∇μ2 22 + φ2 2H 1 (Ω) + φ2 2H 1 (Ω) ∇φ2 2H 1 (Ω) + φ2 4L∞ (Ω) + φ2 4H 1 (Ω) ∇φ2 2H 1 (Ω) + φ2 2H 1 (Ω) φ2 4L∞ (Ω) ), H3 (t) = C(u2H 1 (Ω) ∇φ22 + ∇φ22 ∇Δφ22 + ∇φ22 ∇(f (φ1 ) − f (φ2 ))22 4 6 2 4 4 10 + ∇φ10 2 + φ2 H 1 (Ω) ∇φ2 + ∇φ2 H 1 (Ω) ∇φ2 + φ2 H 1 (Ω) ∇φ2

+ φ2 8H 1 (Ω) ∇φ62 + φ2 2H 1 (Ω) ∇φ2 2H 1 (Ω) ∇φ42 ). From the classical Gronwall inequality, we deduce t ∇ψ22

(∇Δψ22 + ∇w22 + w22 ) ds

+ 0

⎞ ⎛ t  t ≤ ⎝ H3 (s) ds⎠ exp 0 H2 (s) ds .

(4.12)

0

Combining (4.6), (4.12) with the regularity of the solution, we deduce ∇ψ22 + ψ22 ≤ γ(t, φ1,0 − φ2,0 H 1 (Ω) )φ1,0 − φ2,0 2H 1 (Ω) , where γ(t, ζ) satisfies

(4.13)

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AID:19801 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.159; Prn:23/09/2015; 9:14] P.15 (1-18)

F. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

15

lim γ(t, ζ) = 0.

ζ→0+

This completes the proof of Theorem 4.2.

2

Theorem 4.3. Assume that the semigroup {S(t)}t≥0 generated by the Cahn–Hilliard–Brinkman system (1.1)–(1.6) possesses a global attractor A2 in H 2 (Ω) ∩ VI . Then,

df (A2 ) ≤

C(R4 + 1) λ1

 32 .

Proof. Let L(t, φ)ζ = g  (φ, t)ζ be given by the following equations  g  (φ, t)ζ = −Δ2 ζ − Δζ + 3Δ(φ2 ζ) − u · ∇ζ − v · ∇φ, Av + v = −φ∇(−Δζ + ζ 3 − ζ) − ζ∇μ, then L∗ (t, φ) satisfies the following equations ⎧ ∗ ⎪ ⎪ ⎨L (t, φ)η, η = ⎪ ⎪ ⎩Aω + ω = φ∇η.

−Δ2 η + 3φ2 Δη − Δη + u · ∇η − Δ(ω · ∇φ) + (3φ2 − 1)ω · ∇φ  dx, − ω · ∇μ, η + ∂Ω η ∂(ω·∇φ) d n

Therefore, Lc (t, φ) = L(t, φ) + L∗ (t, φ) satisfies the following equations ⎧ ⎪ Lc (t, φ)η, η = −2Δ2 η + 3φ2 Δη + 3Δ(φ2 η) − 2Δη − v · ∇φ − ω · ∇μ ⎪ ⎪ ⎪ ⎨ + (3φ2 − 1)ω · ∇φ − Δ(ω · ∇φ), η +  η ∂(ω·∇φ) dx, d n ∂Ω ⎪ ⎪Aω + ω = φ∇η, ⎪ ⎪ ⎩ Av + v = −φ∇(Δη + η 3 − η) − η∇μ. In the following, we consider the quadratic form Lc (t, φ)η, η =

−2Δη22  +

+



2∇η22

+

 2  3φ ηΔη + 3φ2 ηΔη − vη · ∇φ − (ω · ∇φ)Δη dx

Ω



 −φ3 ω · ∇η + φω · ∇η − (Δφ − φ3 + φ)ω · ∇η dx

Ω

≤ −2Δη22 + 2∇η22 + 6φ26 η6 Δη2 + v3 φ6 ∇η2 + ω3 φH 2 Δη2 + ω6 φH 2 ∇η3 ≤ −2Δη22 + 2∇η22 + CvH 1 φH 1 ∇η2 + C(φ2H 1 ∇η2 + φH 2 ωH 1 )Δη2 . Thanks to ∇v22 + v22 = −φ∇ν + η∇μ, v = v · ∇φ, ν + v · ∇η, μ

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[m3L; v1.159; Prn:23/09/2015; 9:14] P.16 (1-18)

F. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

= v · ∇φ, −Δη + 3φ2 η − η + v · ∇η, −Δφ + φ3 − φ = v · ∇φ, −Δη + v · ∇η, Δφ ≤ v6 ∇φ3 Δη2 + Δφ2 v6 ∇η3 ≤ CφH 2 ∇v2 Δη2 ≤

1 ∇v22 + Cφ2H 2 Δη22 , 2

and ∇w22 + w22 = (φ · ∇η, w) ≤ φ6 ∇η2 w3 ≤ CφH 1 ∇η2 ∇w2 ≤

1 ∇w22 + Cφ2H 1 ∇η22 , 2

we obtain vH 1 ≤ CφH 2 Δη2 ,

(4.14)

wH 1 ≤ CφH 1 ∇η2 .

(4.15)

We conclude from (4.14)–(4.15) and Young inequality that Lc (t, φ)η, η ≤ −Δη22 + C(φ2H 1 φ2H 2 + φ4H 1 + 1)∇η22 .

(4.16)

Consider an initial orthogonal set of infinitesimal displacements η1,0 , · · · , ηn,0 for some n ≥ 1. The volume of the parallelepiped they span is given by Vn (0) = |η1,0 ∧ · · · ∧ ηn,0 |. We deduce that the evolution of such displacements satisfies the evolution equation 

∂ηi ∂t

= L(φ; t)ηi ,

ηi (0) = ηi,0 , ∀ i = 1, 2, · · · , n and the volume elements Vn (t) = |η1 (t) ∧ · · · ∧ ηn (t)| satisfies Vn (t) = Vn (0) exp

t 0

T r(Pn (s)L(φ;s)) ds

,

where the orthogonal projection Pn (s) is onto the linear span of {η1 (s), · · · , ηn (s)} in H 2 (Ω) ∩ V0 , and T r(Pn (s)L(φ; s)) =

n 

L(φ; s)ϕj (s), ϕj (s)

j=1

for n ≥ 1 and {ϕ1 (s), · · · , ϕn (s)} is an orthonormal set spanning Pn (s)(H 2 ∩ V0 ). We obtain

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AID:19801 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.159; Prn:23/09/2015; 9:14] P.17 (1-18)

F. Li et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

T r(Pn (s)L(φ; s)) ≤ −

n 

Δϕj 22 + C(φ2H 1 φ2H 2 + φ4H 1 + 1)

j=1

17 n 

∇ϕj 22

j=1

≤ −n + C(φ2H 1 φ2H 2 + φ4H 1 + 1)

n  1 , λ j=1 j

where λj is the eigenvalue of −Δ with homogeneous Neumann boundary condition on Pj (s)(H 2 ∩ V0 ). Set ⎞ ⎛ T ⎝1 q˜n = lim sup sup sup T r(Pn (s)L(φ; s)) ds⎠ . T i 2 T →∞ φ0 ∈A φ ∈(H ∩V0 ), 0

0

φi0 2 ≤1, 1≤j≤n 2

Therefore, from the fact that φ ∈ A2 and λj ≥ cλ1 j 3 for any j ≥ 1, we obtain q˜n ≤ −n + C(2R4 + 1) ≤ −n +

n  1 λ j=1 j

C(R4 + 1) 1 n3 λ1

which implies that

df (A) ≤

C(R4 + 1) λ1

 32 .

2

Acknowledgments The authors of this paper would like to express their sincere thanks to the reviewer for valuable comments and suggestions. This work was supported by the National Science Foundation of China Grant (11401459, 11031003). References [1] H. Abels, On a diffusive interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal. 194 (2009) 463–506. [2] H. Abels, Longtime behavior of solutions of a Navier–Stokes/Cahn–Hilliard system, in: Proceedings of the Conference “Nonlocal and Abstract Parabolic Equations and Their Applications”, Bedlewo, in: Banach Center Publications, vol. 86, 2009, pp. 9–19. [3] S. Bosia, M. Conti, M. Grasselli, On the Cahn–Hilliard–Brinkman system, arXiv:1402.6195v2. [4] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal. 20 (1999) 175–212. [5] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., A 1 (1947) 27–36. [6] C.S. Cao, C.G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity 25 (2012) 3211–3234. [7] C. Collins, J. Shen, S.M. Wise, An efficient, energy stable scheme for the Cahn–Hilliard–Brinkman system, Commun. Comput. Phys. 13 (2013) 929–957. [8] A. Diegel, X. Feng, S. Wise, Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system, SIAM J. Numer. Anal. 53 (2015) 127–152. [9] C.G. Gal, M. Grasselli, Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) 401–436. [10] C.G. Gal, M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B 31 (2010) 655–678. [11] C.G. Gal, M. Grasselli, Instability of two-phase flows: a lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system, Phys. D, Nonlinear Phenom. 240 (2011) 629–635.

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