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The couplings of the Cahn-Hilliard equation with other basic modelling equations have been proposed in various situations to study complicated phenomena in fluid mechanics involving phase transition

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

LARGE TIME BEHAVIOR OF A CAHN-HILLIARD-BOUSSINESQ SYSTEM ON A BOUNDED DOMAIN

KUN ZHAO

Abstract. We study the asymptotic behavior of classical solutions to an initial-boundary value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on bounded domains with large initial data. A sufficient condition is established under which the solutions decay exponentially to constant states as time approaches infinity.

1. Introduction

As one of the fundamental modelling equations, the Cahn-Hilliard equation [6, 7]

plays an important role in the mathematical study of multi-phase flows, and has been studied intensively in the literature both analytically and numerically (see e.g.

[3, 4, 8, 12, 13, 19, 20, 25, 26, 27, 29]). The couplings of the Cahn-Hilliard equation with other basic modelling equations have been proposed in various situations to study complicated phenomena in fluid mechanics involving phase transition. For ex- ample, the coupled Cahn-Hilliard-Navier-Stokes (CHNS) system and its variations, which describe the motion of an incompressible two-phase flow under shear through an order parameter formulation, have been used in order to understand the phe- nomena of phase transition in incompressible fluid flows (c.f. [14, 18, 24]). Recently, a closely related model to the CHNS system has been developed in [10, 11, 15, 16]

to understand the spinodal decomposition of binary fluid in a Hele-Shaw cell, tumor growth, cell sorting, and two phase flows in porous media, which is referred as the Cahn-Hilliard-Hele-Shaw (CHHS) system. In this paper, we consider the following system of equations:

φt+U· ∇φ= ∆µ, x∈Rn, t >0, µ=−α∆φ+F0(φ),

Ut+U · ∇U +∇P =µ∇φ+θen, θt+U· ∇θ=κ∆θ,

∇ ·U = 0,

(1.1)

2000Mathematics Subject Classification. 35Q35, 35B40.

Key words and phrases. Cahn-Hilliard-Boussinesq equations; classical solution;

asymptotic behavior.

c

2011 Texas State University - San Marcos.

Submitted April 11, 2010. Published April 6, 2011.

1

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which is a system of strongly coupled partial differential equations obtained by cou- pling the Cahn-Hilliard equation to the inviscid heat-conductive Boussinesq equa- tions. It describes the motion of an incompressible inviscid two-phase flow subject to convective heat transfer under the influence of gravitational force through or- der parameter formulation. Here, U = (u1, . . . , un),P andθ denotes the velocity, pressure and temperature respectively. φis the order parameter andµis a chem- ical potential derived from a coarse-grained study of the free energy of the fluid (c.f. [17]). The constant κ >0 and α > 0 models heat conduction and diffusion respectively, anden is the n-th unit vector inRn. The functionF usually has a physical-relevant, double-well structure, each of them representing the two phases of the fluid. A typical example ofF takes the form (c.f. [9, 17]): F(z) = 14(z2−1)2. In this paper, we consider a general scenario by imposing appropriate growth con- ditions on F. We remark that, system (1.1) reduces to the CHHS model if the temperature equation and the hydrodynamic effect are dropped. On the other hand, (1.1) becomes the CHNS system if the temperature equation is removed and the viscosity of fluid is added to the velocity equation.

In the real world, flows often move in bounded domains with constraints from boundaries, where the initial-boundary value problems appear. The solutions of the initial-boundary value problems usually exhibit different behaviors and much richer phenomena comparing with the Cauchy problem. In this paper, we consider system (1.1) on a bounded domain in Rn. The system is supplemented by the following initial and boundary conditions:

(φ, µ, U, θ)(x,0) = (φ0, µ0, U0, θ0)(x),

∇φ·n|∂Ω= 0, ∇µ·n|∂Ω= 0, U·n|∂Ω= 0, θ|∂Ω= ¯θ, (1.2) where Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω, n is the unit outward normal to∂Ω and ¯θis a constant.

The initial-boundary value problem (1.1)–(1.2) was first studied in [30], where the global existence and uniqueness of classical solutions are established, for large initial data with finite energy in 2D. However, the large time asymptotic behavior of the solutions is not investigated due to the lack of uniform-in-time estimates of the solutions. We give definite answer to this unsolved issue in current paper for the 2D case, based on new findings of the structure of the system.

Suggested by the conservation of total mass and the boundary conditions, it is expected that the global attractors of φand θ should be ¯φ= |Ω|1 R

φ0(x)dx and θ, respectively, due to diffusion and boundary effects. In this paper, we provide a¯ sufficient condition that guarantees the decay of the solution. We will show that when the diffusion coefficient αpasses a threshold value determined by F and Ω, the functions φandθwill converge exponentially in time to ¯φand ¯θ, respectively, regardless of the magnitude of the initial perturbation. To be precise, we shall assume thatα−F3c0>0, whereF3>0 is a constant such thatF00≥ −F3, andc0

is the constant in Poincar´e inequality on Ω. This condition is crucial in our analysis due to the fact that it produces a positive constant multiple of kφ−φk¯ 2H2 which is one of the major dissipative terms controlling the exponential decay of φ. The condition will trigger a chain reaction leading the energy estimate performed in [30]

to a whole new scenario. As consequences of the convergence ofφ and θ, we will show that the velocity and vorticity are uniformly bounded in time.

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Throughout this paper, k · kLp, k · kL and k · kWs,p denote the norms of the usual Lebesgue measurable function spaces Lp (1 ≤ p < ∞), L and the usual Sobolev space Ws,p, respectively. Forp= 2, we denote the norm k · kL2 byk · k andk · kWs,2 byk · kHs, respectively. The function spaces under consideration are C([0, T];Hr(Ω)) andL2([0, T];Hs(Ω)), equipped with norms sup0≤t≤TkΨ(·, t)kHr

and RT

0 kΨ(·, τ)k2Hs1/2

, respectively, where r, s are positive integers. Unless specified,ciwill denote generic constants which are independent ofφ, µ, U, θandt, but may depend onα, κ,Ω and initial data.

For the sake of completeness, we first state the results obtained in [30].

Theorem 1.1. Let Ω⊂R2 be a bounded domain with smooth boundary. Suppose that F(·)satisfies the following conditions:

• F(·)is ofC6 class andF(·)≥0;

• There exist constants F1, F2 > 0 such that |F(n)(φ)| ≤ F1|φ|p−n +F2, n= 1, . . . . ,6, ∀6≤p <∞ andφ∈R;

• There exists a constantF3≥0such that F00≥ −F3.

If the initial data φ0(x) ∈ H5(Ω), µ0(x) ∈ H3(Ω) and (θ0(x), U0(x)) ∈ H3(Ω) are compatible with the boundary conditions, then there exists a unique solution (φ, µ, θ, U)of (1.1)–(1.2)globally in time such that

φ∈C([0, T];H5(Ω))∩L2([0, T];H7(Ω)), µ∈C([0, T];H3(Ω))∩L2([0, T];H5(Ω)),

U ∈C([0, T];H3(Ω))andθ∈C([0, T];H3(Ω))∩L2([0, T];H4(Ω))for0< T <∞.

The next theorem is the main result of this paper regarding the large-time as- ymptotic behavior of the solution obtained in Theorem 1.1.

Theorem 1.2. Suppose that the assumptions in Theorem 1.1 are in force and as- sume that the constantα−F3c0>0, wherec0 is the constant in Poincar´e inequality onΩ. Then the solution to (1.1)–(1.2)satisfies

kφ(·, t)−φk¯ H5+kµ(·, t)−F0( ¯φ)kH3+kθ(·, t)−θk¯ H3 ≤γe−βt; kU(·, t)kW1,p≤γ(p), ∀1≤p <∞; kω(·, t)kL ≤¯γ, ∀t≥0, for some constantsγ, β, γ(p),γ >¯ 0independent oft, whereφ¯= |Ω|1 R

φ0(x)dxand ω=vx−uy is the 2D vorticity.

Remark 1.3. It should be pointed out that, in the theorems obtained above, no smallness restriction is put upon the initial data.

Remark 1.4. We observe that, by assuming small initial perturbation around the equilibrium state and by exploring the structure of the functionF, one can show the exponential decay of the solution. However, the asymptotic result obtained in Theorem 1.2 has an obvious advantage over the case for small perturbation. Indeed, Theorem 1.2 provides a convenient criterion for determining whether the solution collapses to a constant state as time evolves. Based on the result, one only needs to measure the volume of the domain, instead of measuring the “smallness” of the initial perturbation which is usually laborious to perform, to determine whether the solution decays or not when other system parameters are fixed.

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Remark 1.5. It is well-known that the Cahn-Hilliard equation is an effective model in the study of sharp interfaces in two-phase fluid flows. However, based on our results, the order parameterφtends to a uniform constant ¯φinstead of ±1. This suggests that under the conditions of Theorem 1.1 and Theorem 1.2, the modelling equations (1.1) indeed fail to model the sharp interfacial phenomenon. Therefore, our results exhibit some bifurcation phenomena on the effectiveness of the modelling equations.

The proof of Theorem 1.2 involves a series of accurate combinations of energy estimates. The estimates are delicate mainly due to the coupling between the equa- tions by convection, gravitational force and boundary effects. Great efforts have been made to simplify the proof. Current proof involves intensive applications of Sobolev embeddings and Ladyzhenskaya type inequalities, see Lemma 2.1. Roughly speaking, because of the lack of the spatial derivatives of the solution at the bound- ary, our energy framework proceeds as follows: We first apply the standard energy estimate on the solution and the temporal derivatives of the solution. We then apply standard results on elliptic equations to recover estimates of the spatial derivatives.

Such a process will be repeated up to third order, and then with the aid of the assumption on α, the carefully coupled estimates will be composed into a desired one leading to the exponential decay of the solution.

The rest of the paper is organized as follows. In Section 2, we give some basic facts that will be used in the proof of Theorem 1.2. In Section 3, we prove some uniform-in-time energy estimates of the solution based on which the combinations of energy estimates will be performed. We then complete the proof of Theorem 1.2 in Section 4.

2. Preliminaries

In this section, we shall collect several facts which will be used in the proof of Theorem 1.2. First, we recall some inequalities of Sobolev and Ladyzhenskaya type (c.f. [1, 21]).

Lemma 2.1. LetΩ⊂R2be any bounded domain with smooth boundary ∂Ω. Then (i) kfkL ≤c1kfkH2;

(ii) kfkL ≤c2kfkW1,p for all p >2;

(iii) kfkLp≤c3kfkH1 for all1≤p <∞;

(iv) kfk2L4 ≤c4 kfkk∇fk+kfk2

; (v) kfk2L8 ≤c5 kfkk∇fkL4+kfk2

, for some constants ci=ci(p,Ω),i= 1, . . . ,5.

Next, we recall some classical results on elliptic equations (c.f. [2, 22, 23]), which are useful in the estimation ofθ.

Lemma 2.2. Let Ω⊂R2 be any bounded domain with smooth boundary∂Ω. Con- sider the Dirichlet problem

κ∆Θ =f inΩ, Θ = 0 on∂Ω.

Iff ∈Wm,p, thenΘ∈Wm+2,pand there exists a constantc6=c6(p, κ, m,Ω)such that

kΘkWm+2,p≤c6kfkWm,p

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for any p∈(1,∞)and the integer m≥ −1.

The next lemma is useful for the estimation of the velocity field (see [5]).

Lemma 2.3. Let Ω⊂R2 be any bounded domain with smooth boundary∂Ω, and letF ∈Ws,p(Ω) be a vector-valued function satisfyingF·n|∂Ω= 0, wherenis the unit outward normal to∂Ω. Then there exists a constantc7=c7(s, p,Ω)such that

kFkWs,p ≤c7(k∇ ×FkWs−1,p+k∇ ·FkWs−1,p+kFkLp) for any s≥1 andp∈(1,∞).

Finally, we recall some Poincar´e type inequalities, which will be used in the estimation ofφ, whose proof is straightforward.

Lemma 2.4. LetΩ⊂Rn be any bounded domain with smooth boundary∂Ω. Then, for any functionHs(Ω)3f : Ω→R, there exists a constantc8=c8(s,Ω)>0such that

(1) kf−f¯kH2s ≤c8k∆sfkandkf−f¯kH2s+1≤c8k∇∆sfk,s≥1, if∇f·n|∂Ω= 0;

(2) kfkH2s≤c8k∆sfk andkfkH2s+1≤c8k∇∆sfk,s≥1, iff|∂Ω= 0, wheref¯= |Ω|1 R

f dx.

3. Uniform energy estimates

In this section, we establish some uniform-in-time energy estimates of the solu- tion under the conditionα−F3c0>0, based on which the exponential decay rate of the solution will be proved. The results are stated as a sequence of lemmas and the proofs are carried out by carefully exploring the condition α−F3c0 > 0 and delicate applications of Cauchy-Schwarz and Gronwall inequalities.

To study the asymptotic behavior, we first reformulate the original problem to get the one for the perturbations. For this purpose, let Φ =φ−φ¯and Θ =θ−θ.¯ After plugging Φ and Θ into (1.1) we obtain

Φt+U· ∇Φ = ∆µ, µ=−α∆Φ +F0(φ), Ut+U· ∇U+∇P˜ =µ∇Φ + Θe2,

Θt+U· ∇Θ =κ∆Θ,

∇ ·U = 0,

(3.1)

which is equivalent to (1.1) for sufficiently smooth solutions, where ˜P =P −θy,¯ and the initial and boundary conditions become

(Φ, µ, U,Θ)(x,0) = (Φ0, µ0, U00)(x)≡(φ0−φ, µ¯ 0, U0, θ0−θ)(x),¯

∇Φ·n|∂Ω= 0, ∇µ·n|∂Ω= 0, U ·n|∂Ω= 0, Θ|∂Ω= 0. (3.2) We begin with the uniform estimate ofkΦkL2.

Lemma 3.1. Under the assumptions of Theorem 1.2, it holds that

kΦ(·, t)k2+ Z t

0

kΦ(·, τ)k2H2dτ ≤c9, ∀t≥0. (3.3)

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Proof. TakingL2inner product of (3.1)1with Φ, after integrating by parts we have 1

2 d

dtkΦk2= Z

Φ∆µx=− Z

∇µ· ∇Φdx=−αk∆Φk2− Z

F00(φ)|∇Φ|2dx, (3.4) which gives

1 2

d

dtkΦk2+αk∆Φk2≤F3k∇Φk2, (3.5) where we have used the condition onF. Since Φ =φ−φ¯and ¯φis a constant, using the boundary conditions, Cauchy-Schwarz and Poincar´e inequalities we have

k∇Φk2=− Z

Φ∆Φdx≤ 1 2c0

kΦk2+c0

2k∆Φk2≤ 1

2k∇Φk2+c0

2k∆Φk2, (3.6) which implies

k∇Φk2≤c0k∆Φk2, (3.7)

where c0 is the constant in Poincar´e inequality on Ω. Let α1 ≡ α−F3c0 > 0.

Substituting (3.7) in (3.5) we have 1 2

d

dtkΦk21k∆Φk2≤0. (3.8) Upon integrating (3.8) in time over [0, t] and using Lemma 2.4 we obtain (3.3).

This completes the proof.

Remark 3.2. The estimate (3.8) already implies the decay of kΦk2. However, our ultimate goal is to show the decay rate of the higher order derivatives of the solution. Hence, for the sake of completeness, we leave the proof of the decay rate in the next section.

Next, we prove uniform estimates of Θ, which will be used to settle down the uniform bound ofkUkH1.

Lemma 3.3. Under the assumptions of Theorem 1.2, there exists a constant β0 independent oft such that for anyt≥0, it holds that

kΘ(·, t)k2≤ kΘ0k2e−2β0t, Z t

0

k∇Θ(·, τ)k2eβ0τdτ ≤ 1

κkΘ0k2. (3.9) Proof. Taking theL2inner product of (3.1)4 with Θ we have

1 2

d

dtkΘk2+κk∇Θk2= 0. (3.10)

Since Θ|∂Ω= 0, Poincar´e’s inequality implies d

dtkΘk2+2κ

c0kΘk2≤0, (3.11)

which yields immediately

kΘ(·, t)k2≤ kΘ0k2e−2β0t, (3.12) whereβ0=κ/c0. This proves the first part of (3.9).

Next, we multiply (3.10) byeβ0tand use (3.12) to obtain d

dt eβ0tkΘk2

+ 2κeβ0tk∇Θk2≤β0e−β0t0k2. (3.13)

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For anyt >0, upon integrating (3.13) in time we obtain eβ0tkΘ(·, t)k2− kΘ0k2+ 2κ

Z t

0

eβ0τk∇Θ(·, τ)k2dτ ≤ 1−e−β0t

0k2, (3.14) which implies the second part of (3.9) immediately. This completes the proof.

With the help of Lemma 3.3 we now prove the uniform estimates ofkUkH1 and kΦkH1.

Lemma 3.4. Under the assumptions of Theorem 1.2, for allt≥0, kU(·, t)k2H1+kΦ(·, t)k2H1+

Z t

0

k∇µ(·, τ)k2+kΦ(·, τ)k2H3

dτ ≤c10. (3.15) Proof. Step 1. Note that due to Lemma 2.3 and the boundary condition onU, it suffices to estimate kUk2 and kωk2, in order estimate kU(·, t)k2H1. Taking the L2 inner product of (3.1)3 withU we have

1 2

d

dtkUk2= Z

µ(∇Φ·U)dx+ Z

Θe2·U dx. (3.16) TakingL2 inner product of (3.1)1withµwe have

d dt

α

2k∇Φk2+ Z

F(φ)dx

+k∇µk2=− Z

µ(∇Φ·U)dx. (3.17) Adding (3.16) and (3.17), we obtain

d dt

1

2kUk2

2k∇Φk2+ Z

F(φ)dx

+k∇µk2= Z

Θe2·U dx. (3.18) Applying Cauchy-Schwarz inequality to the right-hand side of (3.18) and using (3.12), we obtain

d dt

1

2kUk2

2k∇Φk2+ Z

F(φ)dx

+k∇µk2≤e−β0tkUk2+e−β0t0k2. (3.19) After droppingk∇µk2from the left hand side (LHS) of (3.19), we have

d dt

1

2kUk2

2k∇Φk2+ Z

F(φ)dx

≤e−β0tkUk2+e−β0t0k2. (3.20) SinceF ≥0, Gronwall’s inequality then gives

1

2kUk2

2k∇Φk2+ Z

F(φ)dx≤c11, ∀t≥0. (3.21) Applying (3.21) to (3.19) and integrating with respect tot we obtain

kUk2+kΦk2H1+ Z

F(φ)dx+ Z t

0

k∇µ(·, τ)k2dτ ≤c12, ∀t≥0. (3.22) Step 2. By the definition of Φ, Lemma 2.4 and (3.1)2, we observe that

kΦk2H3 ≤c8k∇(∆Φ)k2≤c13 k∇µk2+kF00(φ)∇Φk2

. (3.23)

Using the condition onF, H¨older inequality, Lemma 2.1 (iii) and (3.22) we have kF00(φ)∇Φk2≤c14 kφk2(p−2)L4(p−2)k∇Φk2L4+k∇Φk2

≤c15 (kΦk2(p−2)H1 +kφk¯ 2(p−2)H1 )k∇Φk2H1+k∇Φk2

≤c16kΦk2H2.

(3.24)

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substituting (3.24) in (3.23) we have

kΦk2H3≤c17 k∇µk2+kΦk2H2

, (3.25)

which, together with (3.3) and (3.22), implies Z t

0

kΦ(·, τ)k2H3dτ ≤c18, ∀t≥0. (3.26) Step 3. By taking the curl of the velocity equation, we obtain

ωt+U· ∇ω=µxΦy−µyΦx+ Θx, (3.27) where ω =vx−uy is the 2D vorticity. Taking L2 inner product of (3.27) withω and applying H¨older inequality we have

d

dtkωk ≤2k∇µkk∇ΦkL+k∇Θk. (3.28) Upon integrating (3.28) in time using H¨older and Sobolev inequalities we have

kω(·, t)k ≤ Z t

0

2k∇µkk∇ΦkL+k∇Θk

dτ +kω0k

≤c19

Z t

0

k∇µk21/2Z t 0

kΦk2H31/2

+Z t 0

eβ0τ /2k∇Θk21/2Z t 0

e−β0τ /21/2

+kω0k.

(3.29)

Since the right hand side of (3.29) is uniformly bounded in time by virtue of previous estimates, we have

kω(·, t)k ≤c20, ∀t≥0. (3.30) Thus, (3.15) follows from (3.22), (3.26) and (3.30). This completes the proof.

With the aid of Lemma 3.4 we are now able to improve the estimates of Φ andµ.

Due to the lack of spatial derivatives of the solution on∂Ω, we shall alternatively work on the temporal derivatives and use an iteration program to recover the spatial derivatives.

Lemma 3.5. Under the assumptions of Theorem 1.2, it holds that

kΦ(·, t)k2H2+kµ(·, t)k2+ Z t

0

tk2+k∇µk2H1

dτ ≤c21, ∀t≥0. (3.31) Proof. Step 1. By takingL2 inner product of (3.1)1 with Φt we have

tk2+ Z

Φt(U · ∇Φ)dx= Z

Φt∆µdx. (3.32)

Using the boundary conditions we calculate the RHS of (3.32) as:

Z

Φt∆µdx=−d dt

α

2k∆Φk2+1 2

Z

F00(φ)|∇Φ|2dx +1

2 Z

F000(φ)Φt|∇Φ|2dx.

(3.33) Substituting (3.33) in (3.32) we obtain

d dt

α

2k∆Φk2+1 2 Z

F00(φ)|∇Φ|2dx

+kΦtk2

= 1 2 Z

F000(φ)Φt|∇Φ|2dx− Z

Φt(U· ∇Φ)dx.

(3.34)

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Using Cauchy-Schwarz inequality and Lemma 3.4 we estimate the first term on the RHS of (3.34) as

1 2

Z

F000(φ)Φt|∇Φ|2dx ≤1

4kΦtk2+1 4

Z

|F000(φ)|2|∇Φ|4dx

≤1

4kΦtk2+c22kφk2(p−3)L4(p−3)k∇Φk4L8+c22k∇Φk4L4

≤1

4kΦtk2+c23 k∇Φk4L8+k∇Φk4L4

.

(3.35)

Lemma 2.1 (iii)–(v) and Lemma 3.4 then give

k∇Φk4L4+k∇Φk4L8≤c24 k∇Φk2k∇2Φk2+k∇Φk4+k∇Φk2k∇2Φk2L4+k∇Φk4

≤c25 k∇2Φk2+k∇Φk2+k∇2Φk2H1

≤c26kΦk2H3.

(3.36) So we update (3.35) as

1 2

Z

F000(φ)Φt|∇Φ|2dx ≤1

4kΦtk2+c27kΦk2H3. (3.37) The second term on the RHS of (3.34) is estimated as

Z

Φt(U · ∇Φ)dx ≤ 1

4kΦtk2+c28kUk2H1k∇Φk2H1

≤ 1

4kΦtk2+c29kΦk2H2,

(3.38)

where we have used Lemma 3.4. Combining (3.34), (3.37) and (3.38) we have d

dt α

2k∆Φk2+1 2

Z

F00(φ)|∇Φ|2dx +1

2kΦtk2≤c30kΦk2H3. (3.39) Upon integrating (3.39) in time and using (3.26) we have

α

2k∆Φk2+1 2

Z

F00(φ)|∇Φ|2dx+1 2

Z t

0

tk2dτ ≤c31. (3.40) SinceF00≥ −F3, we have

Z

F00(φ)|∇Φ|2dx≥ −F3k∇Φk2≥ −F3c0k∆Φk2, (3.41) where we have used (3.6). Substituting (3.41) in (3.40) we have

α1

2 k∆Φk2+1 2

Z t

0

tk2dτ ≤c31, which, together with Lemma 2.4, implies

kΦk2H2+ Z t

0

tk2dτ ≤c32. (3.42) Step 2. We derive some consequences of (3.42). From (3.1)2 and Lemma 2.1 (i) we see that

kµ(·, t)k2≤c33 k∆Φk2+kF0(φ)k2

≤c34 k∆Φk2+kφk2(p−1)L + 1

≤c35 k∆Φk2+kΦk2(p−1)H2 +kφk¯ 2(p−1)L + 1 .

(3.43)

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Therefore, using (3.42) we have

kµ(·, t)k2≤c36, ∀t≥0. (3.44) Since∇µ·n|∂Ω= 0, by Lemma 2.4 and Lemma 3.4 we have

k∇µk2H1 ≤c37k∆µk2≤c38tk2+kU · ∇Φk2

≤c39tk2+kUk2H1k∇Φk2H1

≤c40tk2+kΦk2H2

,

(3.45)

which, together with (3.15) and (3.42), implies that Z t

0

k∇µk2H1dτ ≤c41. (3.46)

Therefore, (3.31) follows from (3.42), (3.44) and (3.46). This completes the proof.

The next lemma gives the uniform estimate of kUt(·, t)k2 whose proof requires more careful examination of the energy estimate for the temperature.

Lemma 3.6. Under the assumptions of Theorem 1.2, there exists a constantβ1>0 independent oft such that

eβ1tkΘ(·, t)k2H1+ Z t

0

eβ1τ /2kΘ(·, τ)k2H2dτ ≤c42, ∀t≥0. (3.47) Proof. Step 1. TakingL2 inner product of (3.1)4 with Θt we have

κ 2

d

dtk∇Θk2+kΘtk2≤ kU· ∇Θk2+1

4kΘtk2. (3.48) Using Lemma 3.4 we have

kU · ∇Θk2≤c43kUk2H1k∇Θk2L4 ≤c44k∇Θk2L4. (3.49) So we update (3.48) as

κ 2

d

dtk∇Θk2+3

4kΘtk2≤c44k∇Θk2L4. (3.50) The estimate of the RHS of (3.50) is tricky. First, applying Lemma 2.1 (iv) to

∇Θ to obtain

c44k∇Θk2L4 ≤c45 k∇ΘkkD2Θk+k∇Θk2

≤c46(δ)k∇Θk2+δkD2Θk2, (3.51) where δ is a number to be determined. Since Θ|∂Ω = 0, by the elliptic estimate (c.f. Lemma 2.2), we have

kΘk2H2 ≤c47tk2+kU· ∇Θk2

. (3.52)

For the second term on the RHS of (3.52), we use (3.49) and (3.51) to get kU· ∇Θk2≤c48 k∇ΘkkD2Θk+k∇Θk2

. (3.53)

Then, using Cauchy-Schwarz inequality we update (3.52) as kΘk2H2 ≤c49tk2+k∇ΘkkD2Θk+k∇Θk2

≤c50tk2+k∇Θk2 +1

2kΘk2H2, (3.54) which implies

kΘk2H2≤c51tk2+k∇Θk2

. (3.55)

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By choosingδ= 1/(4c51) in (3.51), and coupling the result with (3.55) we have c44k∇Θk2L4 ≤c52k∇Θk2+1

4kΘtk2. (3.56) Combining (3.50) and (3.56) we obtain

κ 2

d

dtk∇Θk2+1

2kΘtk2≤c52k∇Θk2. (3.57) Step 2. We multiply (3.10) by 2c52/κand add the result to (3.57) to obtain

d dt

c52

κ kΘk2

2k∇Θk2

+c52k∇Θk2+1

2kΘtk2≤0. (3.58) It is clear that, by Poincar´e inequality, there exists a constantc53>0 such that

c53

c52

κ kΘk2

2k∇Θk2

≤c52k∇Θk2. (3.59) Substituting (3.59) in (3.58) we have

d dt

c52

κ kΘk2

2k∇Θk2 +c53

c52

κ kΘk2

2k∇Θk2 +1

2kΘtk2≤0, (3.60) which implies (by dropping 12tk2from the LHS) that

c52

κ kΘ(·, t)k2

2k∇Θ(·, t)k2

≤c52

κ kΘ0k2

2k∇Θ0k2

e−c53t. (3.61) Therefore, fort≥0,

kΘ(·, t)k2H1

min{c52/κ, κ/2}−1c52

κ kΘ0k2

2k∇Θ0k2

e−c53t. (3.62) Using (3.58) and (3.61), by repeating the same procedure as in Lemma 3.3, we have

ec53t/2c52

κ kΘ(·, t)k2

2k∇Θ(·, t)k2 +

Z t

0

ec53τ /2

c52k∇Θ(·, τ)k2+1

2kΘt(·, τ)k2

≤2c52

κ kΘ0k2

2k∇Θ0k2 ,

(3.63)

which yields Z t

0

ec53τ /2

c52k∇Θ(·, τ)k2+1

2kΘt(·, τ)k2

dτ ≤2c52

κ kΘ0k2

2k∇Θ0k2 . (3.64) In view of (3.55) we see that

Z t

0

ec53τ /2kΘ(·, τ)k2H2dτ ≤c54. (3.65) Therefore, (3.47) follows from (3.62) and (3.65). This completes the proof.

With the help of Lemma 3.6, we have the following result.

Lemma 3.7. Under the assumptions of Theorem 1.2, for allt≥0,

kµ(·, t)k2H1+kΦt(·, t)k2+kU(·, t)k2W1,4+kU(·, t)k2L+kUt(·, t)k2≤c55. (3.66)

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Proof. Step 1. Taking L2 inner product of (3.27) with |ω|2ω and using H¨older inequality we have

d

dtkωkL4 ≤2k∇µkL8k∇ΦkL8+k∇ΘkL4≤c56(k∇µkH1kΦkH2+kΘkH2). (3.67) Integrating (3.67) in time and using the previous lemmas, we have

kω(·, t)kL4≤c56

Z t

0

k∇µk2H11/2Z t 0

kΦk2H21/2

+c56

Z t

0

e−β1τ /21/2Z t 0

eβ1τ /2kΘk2H21/2

+kω0kL4

≤c57,

(3.68)

which, together with Lemma 2.3 and Sobolev embedding, implies

kU(·, t)kW1,4+kU(·, t)kL ≤c58, ∀t≥0. (3.69) For the estimate ofkUtk2, taking L2 inner product of (3.1)3withUtwe have kUtk2≤c59 kUk2Lk∇Uk2+kµk2H1kΦk2H2+kΘ0k2

≤c60 k∇µk2+ 1

, (3.70) where we have used (3.42), (3.44) and (3.69).

Step 2. We now deal withµand Φt. TakingL2 inner product of (3.1)1withµt

we have 1 2

d

dtk∇µk2+αk∇Φtk2=− Z

F00(φ)Φ2tt(U· ∇Φ)

dx. (3.71) From (3.42) and Sobolev embedding we know that

kΦkL≤c61, (3.72)

which according to the condition onF implies kF(p−n)(φ)kL ≤F1c(p, n) kΦkp−nL +kφk¯ p−nL

+F2≤c62, n= 1, . . . . ,6. (3.73) Using (3.73) we estimate the RHS of (3.71) as follows:

Z

F00(φ)Φ2tt(U · ∇Φ) dx

≤ kF00(φ)kLtk2+ 2αkUk2Lk∇Φk2+ 1 8αkµtk2

≤c62tk2+c63k∇Φk2+ 1

8α 2α2k∆Φtk2+ 2kF00(φ)k2Ltk2

≤c64tk2+c65k∆Φk2

4k∆Φtk2,

where we have used (3.69) and Lemma 2.4. We update (3.71) as 1

2 d

dtk∇µk2+αk∇Φtk2≤c64tk2+c65k∆Φk2

4k∆Φtk2. (3.74) Differentiating (3.1)1 with respect totwe have

Φtt+Ut· ∇Φ +U· ∇Φt= ∆µt. (3.75) TakingL2 inner product of (3.75) with Φtwe have

1 2

d

dtkΦtk2+αk∆Φtk2= Z

F00(φ)Φt∆Φtdx+ Z

Φ(Ut· ∇Φt)dx. (3.76)

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Using (3.73) and (3.70) we estimate the RHS of (3.76) as follows:

Z

F00(φ)Φt∆Φtdx+ Z

Φ(Ut· ∇Φt)dx

≤ α

4k∆Φtk2+1

αkF00(φ)k2Ltk2+ 1

2αkΦk2LkUtk2

2k∇Φtk2

≤ α

4k∆Φtk2+c66tk2+c67kΦk2H2(k∇µk2+ 1) +α

2k∇Φtk2

≤ α

4k∆Φtk2+c66tk2+c68 k∇µk2+k∆Φk2

2k∇Φtk2.

(3.77)

So we update (3.76) as 1

2 d

dtkΦtk2+3α

4 k∆Φtk2≤c66tk2+c68 k∇µk2+k∆Φk2

2k∇Φtk2. (3.78) Combining (3.74) and (3.78) we obtain

d dt

k∇µk2+kΦtk2

k∇Φtk2+k∆Φtk2

≤c69tk2+k∇µk2+k∆Φk2 . (3.79) After integrating (3.79) in time and using (3.3) and (3.31) we have

k∇µ(·, t)k2+kΦt(·, t)k2+ Z t

0

k∇Φtk2+k∆Φtk2

dτ ≤c70, ∀t≥0. (3.80) Substituting (3.80) in (3.70) we have kUt(·, t)k2 ≤c71. This completes the proof.

As consequences of previous lemmas, we have the following result.

Lemma 3.8. Under the assumptions of Theorem 1.2, it holds

kΦ(·, t)k2H4+kµ(·, t)k2H2 ≤c72, ∀t≥0. (3.81) Proof. First, by (3.45) we have

kµ(·, t)k2H2 ≤c40t(·, t)k2+kΦ(·, t)k2H2

+kµ(·, t)k2. (3.82) Then the uniform estimate ofkµ(·, t)k2H2 follows from Lemma 3.5 and Lemma 3.7.

Second, by Lemma 2.4 and (3.1)2we have

kΦ(·, t)k2H4≤c73 kµ(·, t)k2H2+kF0(φ)(·, t)k2H2

. (3.83)

Using the second condition (H2) on F, (3.31) and (3.73), it is straightforward to show that kF0(φ)(·, t)k2H2 ≤ c74, which together with the uniform bound of kµ(·, t)k2H2 imply thatkΦ(·, t)k2H4 ≤c75. This completes the proof.

4. Large time asymptotic behavior

In this section we prove Theorem 1.2, based on a sequence of accurate combina- tions of energy estimates. For the convenience of the reader, we first collect some uniform-in-time estimates. From (3.31), (3.66) and (3.81) we have, for anyt≥0:

kΦk2H4+kµk2H2+kΦtk2+kF(p−n)(φ)k2L+kUk2W1,4+kUk2L+kUtk2

(t)≤c76. (4.1)

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4.1. Decay of (Φ, µ). Step 1. First, by (3.1)2we have

kµ−F0( ¯φ)k2H3 =k −α∆Φ +F0(φ)−F0( ¯φ)k2+k∇µk2H2

≤2α2k∆Φk2+ 2kF0(φ)−F0( ¯φ)k2+k∇µk2H2. (4.2) Using (4.1) and Lemma 2.3 we estimate the last two terms on the RHS of (4.2) as follows:

2kF0(φ)−F0( ¯φ)k2+k∇µk2H2

≤2kF00(ζ)k2LkΦk2+c77k∇∆µk2

≤c78kΦk2+c79 k∇Φtk2+k∇(U· ∇Φ)k2

≤c78kΦk2+c80 k∇Φtk2+k∇Uk2k∇Φk2+kUk2k∇2Φk2

≤c81 k∇Φtk2+kΦk2H3

.

(4.3)

Combining (4.2) and (4.3) we have

kµ−F0( ¯φ)k2H3 ≤c82 k∇Φtk2+kΦk2H3

. (4.4)

Combining (3.25) and (4.4) we then have

kµ−F0( ¯φ)k2H3≤c83tk2H1+kΦk2H2+k∇µk2

. (4.5)

Second, by Lemma 2.4 and (3.1)2we have

kΦk2H5 ≤c8k∇∆2Φk2≤c84k∇∆µk2+c85k∇∆F0(φ)k2. (4.6) By direct calculations and Sobolev embeddings we can show that

k∇F0(φ)k2H2 ≤c86kF000(φ)k2L k∇Φk4H1+ 2k∇Φk2H1k∇Φk2H2

+c87kF0000(φ)k2Lk∇Φk4H2k∇Φk2. (4.7) Using (4.1) we obtain from (4.7) thatk∇F0(φ)k2H2 ≤c88k∇Φk2H2, which, together with (4.6), implies that

kΦk2H5≤c89 k∇Φtk2+kΦk2H3

. (4.8)

Combining (4.5), (4.8) and (3.25) we have

kµ−F0( ¯φ)k2H3+kΦk2H5 ≤c90tk2H1+kΦk2H2+k∇µk2

. (4.9)

Therefore, it suffices to show the decay of RHS of (4.9) in order to prove the decay of Φ andµ.

Step 2. We recall (3.17), d

dt α

2k∇Φk2+ Z

F(φ)dx

+k∇µk2=− Z

µ(∇Φ·U)dx. (4.10) Due to the structure of the function F(·), there may be a constant term in the integralR

F(φ)dxin general, which is impossible to decay. In order to resolve this issue, we observe, sinceR

(φ−φ)dx= 0, it holds that Z

F(φ)−F(φ)dx= Z

F0(φ)(φ−φ)dx+1 2

Z

F00(ξ)(φ−φ)2dx

=1 2

Z

F00(ξ)Φ2dx, for someξ betweenφand ¯φ.

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Then we update (4.10) as d

dt α

2k∇Φk2+1 2 Z

F00(ξ)Φ2dx

+k∇µk2=− Z

µ(∇Φ·U)dx. (4.11) Using (4.1) and Lemma 2.4 we estimate the RHS of (4.11) as

Z

µ(∇Φ·U)dx =

Z

ΦU· ∇µdx

≤ 1

2k∇µk2+1

2kUk2L4kΦk2L4

≤ 1

2k∇µk2+c91kΦk2H1

≤ 1

2k∇µk2+c92k∆Φk2.

(4.12)

Substituting (4.12) in (4.11), we have d

dt

αk∇Φk2+ Z

F00(ξ)Φ2dx

+k∇µk2≤c93k∆Φk2. (4.13) Step 3. Recalling (3.34) and using (4.1) and Lemma 2.4 we have

d dt

α

2k∆Φk2+1 2

Z

F00(φ)|∇Φ|2dx

+kΦtk2

=1 2

Z

F000(φ)Φt|∇Φ|2dx− Z

Φt(U· ∇Φ)dx

≤1

2kΦtk2+c94(k∇Φk4H1+k∇Φk2H1)

≤1

2kΦtk2+c95k∆Φk2, which yields

d dt

αk∆Φk2+ Z

F00(φ)|∇Φ|2dx

+kΦtk2≤c96k∆Φk2. (4.14) Step 4. From (3.77) and (4.1) we have

1 2

d

dtkΦtk2+αk∆Φtk2≤ α

4k∆Φtk2

2k∇Φtk2+c97(kΦtk2+k∆Φk2). (4.15) Combining (3.74) and (4.15), we have

d

dt k∇µk2+kΦtk2

+α k∇Φtk2+k∆Φtk2

≤c98tk2+k∆Φk2

. (4.16) Step 5. TakingL2 inner product of (3.75) withµtwe have

α 2

d

dtk∇Φtk2+k∇µtk2= Z

(UtΦ +UΦt)· ∇µtdx− Z

F00(φ)ΦtΦttdx. (4.17) For the last term on the RHS of (4.17), we have

− Z

F00(φ)ΦtΦttdx=−1 2

d dt

Z

F00(φ)Φ2tdx+1 2

Z

F000(φ)Φ3tdx.

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So we update (4.17) as d dt

αk∇Φtk2+ Z

F00(φ)Φ2tdx

+ 2k∇µtk2

= 2 Z

(UtΦ +UΦt)· ∇µtdx+ Z

F000(φ)Φ3tdx.

(4.18)

Using (4.1) and Lemma 3.6 we estimate the first two terms on the RHS of (4.18) as

2

Z

(UtΦ +UΦt)· ∇µtdx

≤ k∇µtk2+kUtk2kΦk2L+kUk2Ltk2

≤ k∇µtk2+c99kΦk2H2+c76tk2

≤ k∇µtk2+c100k∆Φk2+c101k∆Φtk2.

(4.19)

Similarly, for the term involving Φ3t, we have

Z

F000(φ)Φ3tdx

≤ kF000(φ)ktk3L3

≤c76tk2tkL1

≤c102k∆Φtk2tk ≤c103k∆Φtk2.

(4.20)

Substituting (4.19) and (4.20) in (4.18), we have d

dt

αk∇Φtk2+ Z

F00(φ)Φ2tdx

+k∇µtk2≤c104 k∆Φtk2+k∆Φk2

. (4.21) Step 6. In this step, we make combinations of energy estimates, which will be used to prove the exponential decay of Φ andµ. First, we collect energy inequalities from Steps 2–5:

(4.13) dtd

αk∇Φk2+R

F00(ξ)Φ2dx

+k∇µk2≤c93k∆Φk2 (4.14) dtd

αk∆Φk2+R

F00(φ)|∇Φ|2dx

+kΦtk2≤c96k∆Φk2 (4.16) dtd k∇µk2+kΦtk2

+α k∇Φtk2+k∆Φtk2

≤c98tk2+k∆Φk2 (4.21) dtd

αk∇Φtk2+R

F00(φ)Φ2tdx

+k∇µtk2≤c104 k∆Φtk2+k∆Φk2 First, multiply (4.16) by 2cα104 and then add (4.21) to obtain

d dt

J0(t)

+K0(t)≤c105tk2+k∆Φk2

, (4.22)

where

J0(t)≡ 2c104

α k∇µk2+kΦtk2

+αk∇Φtk2+ Z

F00(φ)Φ2tdx, K0(t)≡c104 2k∇Φtk2+k∆Φtk2

+k∇µtk2. Second, multiply (4.14) by 2c105then add (4.22) to obtain

d dt

J1(t)

+K1(t)≤c106k∆Φk2, (4.23) where

J1(t)≡J0(t) + 2c105

αk∆Φk2+ Z

F00(φ)|∇Φ|2dx , K1(t)≡c105tk2+K0(t).

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