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Advances in Differential Equations Volume 8, Number 1, January 2003, Pages 83–110 THE CAHN-HILLIARD EQUATION WITH DYNAMIC BOUNDARY CONDITIONS Reinhard Racke Department of Mathematics and Statistics University of Konstanz, 78457 Konstanz, Germany Songmu Zheng Institute of Mathematics, Fudan University, 200433 Shanghai, P.R. China (Submitted by: Y. Giga) Abstract. This paper is concerned with the following Cahn-Hilliard equation ψt μ, where μ = Δψ ψ + ψ 3 , subject on the boundary Γ to the following dynamic boundary condition σsΔ || ψν ψ+hs gsψ = 1 Γs ψt and ν μ =0, and the initial condition ψ|t=0 = ψ0. This problem was recently proposed by physicists to describe spinodal decomposition of binary mixtures where the effective interaction between the wall (i.e., the boundary Γ) and two mixture components are short-ranged. The global existence and uniqueness of solutions to this initial boundary value problem with highest-order boundary conditions is proved. 1. Introduction It is well known that the following Cahn-Hilliard equation describes spin- odal decomposition of binary mixtures that appears, for example, in cooling processes of alloys, glasses or polymer mixtures (see Cahn and Hilliard [4], Novick-Cohen and Segel [19], Kenzler et al. [16], and the references cited therein): ψ t μ in [0,T ] × Ω, (1.1) μ = Δψ + + 3 , (1.2) where 0 <T ≤∞, Ω is a bounded domain in R n , n =1, 2, 3 with smooth boundary Γ and μ is called the chemical potential with a, b being constants, b> 0,a< 0. Without loss of generality, one can assume that b =1,a = 1. It is clear that the equations (1.1), (1.2) can be written as a single nonlinear parabolic equation for ψ: ψ t = Δ(Δψ + ψ 3 ψ). (1.3) Accepted for publication: May 2002. AMS Subject Classifications: 35K55, 74N20. 83

Transcript of THE CAHN-HILLIARD EQUATION WITH DYNAMIC ... - Project Euclid

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Advances in Differential Equations Volume 8, Number 1, January 2003, Pages 83–110

THE CAHN-HILLIARD EQUATION WITH DYNAMICBOUNDARY CONDITIONS

Reinhard Racke

Department of Mathematics and StatisticsUniversity of Konstanz, 78457 Konstanz, Germany

Songmu Zheng

Institute of Mathematics, Fudan University, 200433 Shanghai, P.R. China

(Submitted by: Y. Giga)

Abstract. This paper is concerned with the following Cahn-Hilliardequation ψt = Δμ, where μ = −Δψ − ψ + ψ3, subject on the boundaryΓ to the following dynamic boundary condition σsΔ||ψ−∂νψ+hs−gsψ =1Γs

ψt and ∂νμ = 0, and the initial condition ψ|t=0 = ψ0. This problem

was recently proposed by physicists to describe spinodal decompositionof binary mixtures where the effective interaction between the wall (i.e.,the boundary Γ) and two mixture components are short-ranged. Theglobal existence and uniqueness of solutions to this initial boundaryvalue problem with highest-order boundary conditions is proved.

1. Introduction

It is well known that the following Cahn-Hilliard equation describes spin-odal decomposition of binary mixtures that appears, for example, in coolingprocesses of alloys, glasses or polymer mixtures (see Cahn and Hilliard [4],Novick-Cohen and Segel [19], Kenzler et al. [16], and the references citedtherein):

ψt = Δμ in [0, T ] × Ω, (1.1)

μ = −Δψ + aψ + bψ3, (1.2)where 0 < T ≤ ∞, Ω is a bounded domain in R

n, n = 1, 2, 3 with smoothboundary Γ and μ is called the chemical potential with a, b being constants,b > 0, a < 0. Without loss of generality, one can assume that b = 1, a = −1.It is clear that the equations (1.1), (1.2) can be written as a single nonlinearparabolic equation for ψ:

ψt = Δ(−Δψ + ψ3 − ψ). (1.3)

Accepted for publication: May 2002.AMS Subject Classifications: 35K55, 74N20.

83

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84 Reinhard Racke and Songmu Zheng

Equations (1.1), (1.2) have to be supplemented by the initial condition

ψ(0, ·) = ψ0 in Ω (1.4)

and two boundary conditions. In the literature, the usual boundary condi-tions considered are the following:

∂νμ|Γ = 0, (1.5)

∂νψ|Γ = 0. (1.6)In the above ∂ν = ν ·∇ denotes the exterior normal derivative at the bound-ary, and ν = ν(x) denotes the exterior normal in x ∈ Γ. The boundarycondition (1.5) has a clear physical meaning: There cannot be any exchangeof the mixture constituents though the boundary Γ; it is easy to see from(1.3) and (1.5) that the total mass

∫Ω ψdx is conserved for all time. The

boundary condition (1.6) is usually called the variational boundary condi-tion, which together with (1.5) results in decreasing of the following bulkfree energy

Fb[ψ] :=∫

Ω(12|∇ψ|2 − 1

2ψ2 +

14ψ4)(x)dx. (1.7)

For the initial boundary value problem for the equations (1.1), (1.2) or forthe equation (1.3) with boundary conditions (1.5), (1.6), the results on globalexistence, uniqueness and large time behavior of solution have been estab-lished in the literature (see, for example, Elliott and Zheng [6], Zheng [26]and Temam [24], see also Kenmochi, Niezgodka and Paw�low [15] for thevariational approach for the problem with constraint).

However, it was proposed by physicists in recent years that, when theeffective interaction between the wall (i.e., the boundary Γ) and both mixturecomponents is short-ranged, the following surface free energy functional

Fs[ψ] =∫

Γ

(σs

2|∇||ψ|2 + fs(ψ)

)dσ (1.8)

withfs(ψ) := −hsψ +

gs

2ψ2 (1.9)

and ∇|| being the tangential gradient operator on Γ, should be added to theabove bulk free energy functional Fb[ψ] to form a total free energy functional

F [ψ] = Fb[ψ] + Fs[ψ]. (1.10)

In the above σs > 0, gs > 0, hs are given constants where gs accounts for amodification of the effective interaction between the components at the wall,and hs �= 0 describes the possible preferential attraction of one of the two

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components by the wall. It turns out that instead of the boundary condition(1.6), the following dynamic boundary condition

1Γs

ψt = σsΔ||ψ − ∂νψ + hs − gsψ on Γ (1.11)

is posed in order that the total free energy functional F [ψ] will decrease withrespect to time, i.e.,

d

dtF [ψ(t, ·)] = −

∫Ω|∇μ|2(t, x)dx − 1

Γs

∫Γ|ψt|2(t, σ)dσ ≤ 0. (1.12)

Here Γs is a positive constant, and Δ|| denotes the tangential Laplace oper-ator on the surface. A similar boundary condition can be derived by takingthe continuum limit of simple lattice models within a direct mean-field ap-proximation or by applying density functional theory. We refer to Fischer,Maass and Dieterich [10], [11], Binder and Frisch [5], Fischer et al. [12] andthe references cited therein for details.

Remark 1.1. In the one-dimensional case (n = 1) it is assumed that theterm with the tangential Laplacian Δ|| simply does not appear in (1.11), or,in other words, the tangential gradient term in (1.8) vanishes.

This paper is concerned with global existence and uniqueness of solutionsto the initial boundary value problem (1.3), (1.5), (1.11), (1.4). While nu-merical experiments were carried out in a very recent paper [16], theoreticalanalysis for this kind of dynamic boundary condition is not available in theliterature.

The new features of the present problem are: the boundary condition(1.11) involves ψt, which is called the dynamic boundary condition in litera-ture, and the term with the tangential Laplacian Δ||. Before going into thedetail of our results and proofs, let us first recall some related papers. Asfar as parabolic equations with dynamic boundary condition is concerned,we refer to Escher [8], [9], Hintermann [13] in which they establish existenceand uniqueness of maximal solutions for quasilinear parabolic equations withdynamic boundary conditions in the framework of Wm,p. It should be men-tioned that in their study the term with the tangential Laplacian Δ|| doesnot appear. It is not clear at all whether and how their approach can beextended to deal with the problem with boundary condition also involvingthe term with the tangential Laplacian Δ||. Besides, it seems to us that forthe problem we studied, it would be very difficult to get uniform estimatesin Wm,p for p �= 2 in order to obtain global existence, if we directly usetheir setting. It is also worthwhile noticing that in [13] the Agmon, Douglis,Nirenberg framework is used, as we shall do below.

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86 Reinhard Racke and Songmu Zheng

We also refer to Sato and Aiki [22] for the results on global existenceand uniqueness of weak solution for the phase filed equations. Again theterm with the tangential Laplacian Δ|| does not appear in their boundaryconditions. Besides, what they established is the existence and uniquenessof weak solution by variational approach. However, what we are concernedin this paper is existence and uniqueness of strong solution. The number ofa priori estimates obtained in weak solution setting is much less than whatneeded for global existence and uniqueness of strong solution of the originalnonlinear problem.

On the other hand, in literature there are some other papers dealing withproblems for partial differential equations involving second partial deriva-tives on the boundary, see, e.g., [23]. But to our knowledge, there are nopapers in literature dealing with problems with both dynamic boundarycondition and the term with the tangential Laplacian Δ|| on the boundary .

To overcome the mathematical difficulties due to the presence of the dy-namic boundary condition as well as the term with the tangential LaplacianΔ|| on the boundary, our strategy is the following. We shall first introducean approximate problem (Pε) depending on a small positive parameter ε;the equations of this problem will, for fixed ε, formally be the phase-fieldequations of Caginalp type (see [3, 27]), but now with a non-homogeneousdynamic boundary condition also involving the highest-order derivative in xwith the tangential Laplacian. To solve this approximate problem, we studythe corresponding linearized problem again with the same kind of boundaryconditions:

εφt − Δφ = f, (1.13)

(σsΔ||φ − ∂νφ − gsφ − 1Γs

φt)|Γ = 0, (1.14)

φ(0, ·) = φ0. (1.15)

After completion of the first version of our paper, a very recent paper byPruss [21] attracts our attention. He discusses (1.13)–(1.15) as an exampleof general abstract parabolic systems and proves optimal regularity results.Since we also need a priori estimates in spaces that are suitable for the orig-inal nonlinear system, we give an alternative approach in different functionspaces (refer to the remark in Section 2 for the detailed discussions).

Namely, in our approach, it will be necessary to use the results on ellipticoperators with highest-order derivatives in the boundary condition whichstill form an elliptic boundary value problem in the sense of Hormander [14](or is elliptic with the complementary condition according to Agmon, Douglis

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and Nirenberg [1, 2], or is a normal elliptic system in the sense of Peetre [20]).In particular we use a result of Visik [25], and the corresponding parabolictheory, respectively, compare Lions and Magenes [17, 18] or Temam [24].

Once the solvability of the linearized problem is proved, we proceed to usethe contraction mapping theorem to prove the local existence and unique-ness of solutions to the approximate problem. The possibility of treatingthe nonlinear problem now naturally depends on the fact that the specialnonlinearity in the differential equation is of semilinear power-type. Then,based on the uniform a priori estimates, we draw the conclusions on theglobal existence and uniqueness of the approximate problem. Furthermore,we show that some uniform a priori estimates we get for the approximateproblems do not depend on ε. Therefore, they allow us to pass to the limit,as ε → 0, to get the global existence of strong solutions for the originalproblem (1.3), (1.5), (1.11), (1.4). The uniqueness of solutions in the sameclass of functions can be proved by the standard energy method.

Remark 1.2. In the one-dimensional case, because the tangential Laplacianin the boundary condition does not appear, some minor modifications in theproof are needed.

This paper is organized as follows: In Section 2, we introduce the ap-proximate problem (Pε), and the solvability of the auxiliary linear problemwith highest-order boundary conditions is extensively studied. In Section3 we shall prove a local existence and uniqueness result for problem (Pε).Uniform a priori estimates will be obtained in Section 4 to prove the globalexistence for the approximate problem. In Section 5 we shall show thatsufficiently strong a priori estimates obtained in Section 4, which do notdepend on ε, will allow us to pass to the limit, as ε → 0, in certain Sobolevspaces as n = 2, 3. The proof of the uniqueness for the original problem(1.3), (1.5), (1.11), (1.4) is also given in that section. In Section 6 the casen = 1 is treated. Finally, we shortly comment on the limiting case Γs = ∞in Section 7.

We use standard notation for Sobolev spaces, e.g. for Hs ≡ Hs(Ω) or forLr(Hs) ≡ Lr([0, T ];Hs), cp. e.g. [17]. ‖ · ‖Y and ‖ · ‖ will denote the normsin the Banach space Y and in L2 ≡ L2(Ω), respectively.

2. The approximate problem (Pε)

In order to find a solution ψ to the original problem (1.3), (1.5), (1.11),(1.4), we now introduce an approximate problem (Pε) for some fixed, but

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88 Reinhard Racke and Songmu Zheng

arbitrary parameter ε, 0 < ε ≤ 1. Observe that the transformation

ψ := ψ − hs

gs(2.1)

turns the Cahn-Hilliard equation into one with the new chemical potential

μ := −Δψ − ψ + ψ3 + 3hs

gsψ2 +

h2s

g2s

ψ +h3

s

g3s

+hs

gs≡ −Δψ + N(ψ), (2.2)

where

N(ψ) = −ψ + ψ3 + 3hs

gsψ2 +

h2s

g2s

ψ +h3

s

g3s

+hs

gs, (2.3)

and it has the effect that the second boundary condition reads1Γs

ψt = σsΔ||ψ − ∂νψ − gsψ on Γ,

having thus removed the nonhomogeneous term hs.The approximate problem (Pε) now consists in finding a solution (φ, u)

satisfyingεut − Δu = −φt, (2.4)

(∂νu)|Γ = 0, (2.5)u(0, ·) = μ0 + εμ1 =: u0 (2.6)

andεφt − Δφ = u − N(φ), (2.7)

(σsΔ||φ − ∂νφ − gsφ − 1Γs

φt)|Γ = 0, (2.8)

φ(0, ·) = ψ0 −hs

gs=: φ0, (2.9)

where μ0 := −Δφ0+N(φ0), and μ1 is an element in H1(Ω) specified later (see(3.1), (3.2)). The system of differential equations in (2.4), (2.7) corresponds,up to scaling, to the phase-field equation of Caginalp type, as proposedand studied in [3], (see also [7], [27]). However, our boundary conditions,especially (2.8), are much more complicated than those studied there.

We now discuss the following auxiliary linear problem

εφt − Δφ = f, (2.10)

(σsΔ||φ − ∂νφ − gsφ − 1Γs

φt)|Γ = 0, (2.11)

φ(0, ·) = φ0. (2.12)This linear problem (2.10)–(2.12) already reflects many of the inherent dif-ficulties of our original problem such as the dynamic boundary condition as

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well as the term with the tangential Laplacian Δ|| on the boundary. Thetools that we use to solve it are variational methods combined with varia-tions of Duhamel’s formula from semigroup theory, and variational evolutionequations techniques.

As already mentioned, problem (1.13)–(1.15) has also been studied re-cently by Pruss [21]. He discusses it as an example of general abstractparabolic systems and proves the Lp-maximal regularity results. More pre-cisely, what he proves for the case p = 2 is the following (see [21], p.2):problem (1.13)–(1.15) admits a unique solution φ ∈ H1([0, T ];L2(Ω)) ∩L2([0, T ];H2(Ω)) such that φ|Γ ∈ H5/4([0, T ];L2(Γ)) ∩ L2([0, T ];H5/2(Γ))if and only if f ∈ L2([0, T ] × Ω), φ0 ∈ H3/2(Ω), and the compatibility con-dition φ0|Γ ∈ H3/2(Γ) is satisfied. As one can see from Theorems 2.2–2.4in Section 2 that we use a quite different approach and obtain existence,uniqueness and a priori estimate results in different function space settingsuch as V, H spaces, as introduced below in Subsection 2.1, to fit them intoour original nonlinear problem. Roughly speaking, in comparison with [21],p. 2, in our results less requirements on compatibility of initial data andmore assumptions on the right-hand term f are needed. Besides, the con-stants appearing in our a priori estimates do not depend on t, T . This isimportant for the study of our original nonlinear problem as well as for thefurther study of the asymptotic behavior of solution to the original nonlinearproblem as time tends to infinity.

2.1. The homogenous problem (2.10)–(2.12): f = 0. We first look atthe homogeneous problem, i.e., we assume in this subsection that f = 0. Tosolve the homogeneous problem

εφt − Δφ = 0, (2.13)

(σsΔ||φ − ∂νφ − gsφ − 1Γs

φt)|Γ = 0, (2.14)

φ(0, ·) = φ0, (2.15)

we formally use the method of separation of variables. The ansatz

φ(t, x) ≡ Φ(x)h(t) (2.16)

leads toεΦht − (ΔΦ)h = 0, (2.17)

((σsΔ||Φ − ∂νΦ − gsΦ)h − 1Γs

Φht)|Γ = 0. (2.18)

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90 Reinhard Racke and Songmu Zheng

Therefore,ΔΦΦ

=εht

h≡ −λ ∈ R,

and then

(σsΔ||Φ − ∂νΦ − gsΦ

Φ)|Γ =

ht

Γsh=

−λ

εΓs.

Thus, h should satisfy

h(t) = h(0)e−λεt (2.19)

and the pair (Φ, λ) should satisfy the eigenvalue problem

−ΔΦ = λΦ in Ω, (2.20)

(σsΔ||Φ − ∂νΦ − gsΦ)|Γ =−λ

εΓsΦ|Γ. (2.21)

Multiplying (2.20) by a smooth function Ψ and using (2.21) we obtain that∫Ω∇Φ∇Ψdx+σs

∫Γ∇||Φ∇||Ψdσ+gs

∫Γ

ΦΨdσ = λ( ∫

ΩΦΨdx+

1εΓs

∫Γ

ΦΨdσ).

(2.22)The observations above suggest to establish the following abstract framework(we refer, e.g., to [24] for the basic knowledge on spectral analysis on Hilbertspaces and linear evolution equations): Let V be a Hilbert space with theinner product defined by the left-hand side of equation (2.22), i.e., for anyΨ,Φ ∈ V ,

(Ψ,Φ)V =∫

Ω∇Φ∇Ψdx + σs

∫Γ∇||Φ∇||Ψdσ + gs

∫Γ

ΦΨdσ.

In other words,

V := completion of C1(Ω) under ‖ · ‖V .

Clearly [17, Thm 9.4], we have H3/2(Ω) ⊂ V ⊂ H1(Ω). Let H be anotherHilbert space with the inner product defined by the right-hand side of equa-tion (2.22), i.e., for any Ψ,Φ ∈ H,

(Ψ,Φ)H =( ∫

ΩΦΨdx +

1εΓs

∫Γ

ΦΨdσ).

In other words,

H := completion of C0(Ω) under ‖ · ‖H .

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cahn-hilliard equation 91

Clearly, H is a subset of L2(Ω), H1/2+α(Ω) ⊂ H, for α > 0, and the injectionfrom V into H is continuous and compact. Then it is well-known (e.g., see[24]) that the bilinear form

a(u, v) := (u, v)V (2.23)

defines a strictly positive self-adjoint unbounded operator A from

D(A) = {u ∈ V, Au ∈ H} (2.24)

into H, and for any v ∈ V ,

(Au, v)H = (u, v)V . (2.25)

Moreover, the standard spectral theory allows us to define the power As

of A for s ∈ R, and we infer that there exists a complete orthonormalfamily of H, {wj}, j ∈ IN , with wj ∈ D(As) for s ∈ R, and a sequence λj ,0 < λ1 ≤ λ2, · · · , λj → ∞ as j → ∞ such that

Awj = λjwj . (2.26)

Remark 2.1. We have from (2.23) that V = D(A1/2). Furthermore, itfollows from (2.25) and the definition of the inner product in V and H thatfor any u ∈ D(A), v ∈ V ,∫

ΩAuvdx +

1εΓs

∫Γ

Auvdσ =∫

Ω∇u∇vdx + σs

∫Γ∇||u∇||vdσ + gs

∫Γ

uvdσ.

(2.27)Taking v ∈ C∞

0 (Ω) in (2.27) and taking integration by parts yields thatAu = −Δu ∈ H. Furthermore, taking integration by parts in (2.27) yieldsthat for v ∈ V ,

−∫

Γ∂νuvdσ − 1

εΓs

∫Γ

Δuvdσ = σs

∫Γ∇||u∇||vdσ + gs

∫Γ

uvdσ. (2.28)

Thus, the following boundary condition on Γ holds in H−1(Γ):

(σsΔ||u − ∂νu − gsu − 1εΓs

Δu)|Γ = 0. (2.29)

Similarly, it follows from (2.26) that wj satisfies

−Δwj = λjwj in Ω, (2.30)

(σsΔ||wj − ∂νwj − gswj −1

εΓsΔwj)|Γ = 0. (2.31)

Having established this framework, we now state and prove the followingtheorem on solvability of the problem (2.13)–(2.15):

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92 Reinhard Racke and Songmu Zheng

Theorem 2.2. For any φ0 ∈ H, there exists a unique solution

φ ∈ C0([0,∞), H) ∩ C∞((0,∞), C∞(Ω))

in the following sense that, for any t > 0, φ satisfies the equation (2.13)and the boundary condition (2.14) in the classical sense; as t tends to zero,‖φ(t, ·) − φ0‖H → 0. Moreover, the solution φ can be explicitly expressed by

φ(t, x) =∞∑

k=1

ake−λk

εtwk(x), (2.32)

whereak = (φ0, wk)H . (2.33)

The family of operators {S(t)}t≥0 on H given by (2.32),

φ = S(t)φ0,

define a C0-semigroup on H.

Proof. We first prove that wk belongs to C∞(Ω). Indeed, taking the innerproduct of (2.26) with any element v ∈ V in H yields (wj , v)V = λj(wj , v)H ,i.e.,∫

Ω

∇wj∇vdx+σs

∫Γ

∇||wj∇||vdσ + gs

∫Γ

wjvdσ = λj

( ∫Ω

wjvdx+1

εΓs

∫Γ

wjvdσ).

(2.34)In other words, wj is a weak solution in V for the boundary value problem(2.20), (2.21). Notice that the following boundary value problem

−Δu = f, (2.35)

(σsΔ||u − ∂νu − gsu)|Γ = g (2.36)

is an elliptic boundary value problem in the sense of Hormander [14], or inthe sense of Agmon, Douglis and Nirenberg [1, 2], or in the sense of Peetre[20], as mentioned before. For any f ∈ Hm(Ω), g ∈ Hm−1/2(Γ), m ∈ IN , theproblem (2.35), (2.36) admits a unique solution u ∈ Hm+2(Ω). Moreover,the following estimate holds:

‖u‖Hm+2(Ω) ≤ C(‖f‖Hm(Ω) + ‖g‖Hm−1/2(Γ)). (2.37)

Therefore, it follows that wj is a solution to the problem (2.35), (2.36) withf = λjwj , g = λjwj . Then a bootstrap argument yields that wj belongs toHm+2(Ω) for any m ∈ IN , thus belongs to C∞(Ω). Moreover, it follows from(2.37) that

‖wj‖H3(Ω) ≤ C(‖λjwj‖H1(Ω) + ‖λjwj‖H1/2(Γ)) ≤ C(λj + λ2j ). (2.38)

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cahn-hilliard equation 93

By induction, it easily follows that for any m ∈ IN ,

‖wj‖H2m+1(Ω) ≤ Cm(λmj + λm+1

j ) (2.39)

with Cm being a positive constant depending on m. Then it follows thatφ ∈ C0([0,∞), H)∩C∞((0,∞), C∞(Ω)) and, for t > 0, φ explicitly expressedby (2.32) satisfies (2.13), (2.14), and the initial condition (2.15) is satisfied inthe sense that as t tends to zero, ‖φ(t, ·)−φ0‖H → 0. The uniqueness can beeasily seen from the following identity which is derived by multiplying (2.13)by φ, then integrating with respect to x, and using the boundary condition(2.14):

ε

2d

dt‖φ‖2 +

12Γs

d

dt

∫Γ

φ2dσ + ‖∇φ‖2 +∫

Γ(σs|∇||φ|2 + gsφ

2)dσ = 0. (2.40)

It is a routine procedure to verify that the operators S(t) from φ0 to φdefined by (2.32) form a C0-semigroup on H, and we can omit the detailshere. The proof of the theorem is completed. �

2.2. The problem (2.10)–(2.12) with general f . Now we return to thenonhomogeneous auxiliary problem (2.10)–(2.12). Since ε is a fixed positiveconstant, without loss of generality, we can assume in this section that ε = 1.

Notice that, because the boundary condition (2.11) involves φt, we cannotdirectly apply the Duhamel principle. For the time being, we assume thatφ0 ∈ V, f ∈ H2([0, T ];L2) with some arbitrary, but fixed T > 0. We nowintroduce z as a solution to

−Δz = f,Bz := σsΔ||z − ∂νz − gsz = 0 on Γ.

}(2.41)

It follows from the results on general elliptic equations as given in Hormander[14], in particular a result of Visik [25] that applies to (2.41), cp. [14, page264], that we may conclude that the problem (2.41) admits a unique solutionz with z ∈ H2([0, T ];H2).

Now, define w := φ − z. Then w satisfies

wt − Δw = φt − Δφ − zt + Δz = −zt =: f ,

and on Γ, we have

Bw − 1Γs

Δw = Bφ − 1Γs

Δφ − Bz +1Γs

Δz

= Bφ − 1Γs

(φt − f) − 1Γs

f = Bφ − 1Γs

φt = 0.

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94 Reinhard Racke and Songmu Zheng

That is, w satisfies

wt − Δw = f ∈ H1([0, T ];V ),

(σsΔ||w − ∂νw − gsw − 1Γs

Δw)|Γ = 0,

w(0, ·) = φ0 − z(0, ·) ∈ V = D(A12 ).

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(2.42)

But this last problem can be solved since the corresponding homogeneousproblem (f = 0) is equivalent to the problem (2.13)–(2.15), not involvingtime derivatives on the boundary in this formulation. In other words, it canbe written as an initial value problem for an abstract evolution equation:

dw

dt+ Aw = f , (2.43)

w(0) = φ0 − z(0). (2.44)Hence the usual Duhamel principle applies to (2.42), and we can solve forw according to Subsection 2.1. Then the desired solution to (2.10)–(2.12) isof course given by φ := w + z. A representation of the solution describing amodified Duhamel’s principle can be given as follows. Let {S(t)}t≥0 be thesemigroup defined in Subsection 2.1, and let Δ−1

B f(t, ·) denote the solutionz(t, ·) to (2.41). Then the solution φ to (2.10)–(2.12) is given by

φ(t, x) = S(t)(φ0 − Δ−1

B f(0, ·))−

∫ t

0S(t− τ)∂τΔ−1

B (f(τ, ·))dτ + Δ−1B f(t, ·).

(2.45)Now we have the following result on solvability of the nonhomogeneous prob-lem (2.10)–(2.12).

Theorem 2.3. Let φ0 ∈ V, f ∈ C0([0, T ];H1), ft ∈ H1([0, T ];L2). Thenthere is a unique solution φ ∈ C0([0, T ];V ) ∩ C1((0, T ];V ) ∩ C0((0, T ];H3)to the problem (2.10)–(2.12) such that φ = w + z with z ∈ C0([0, T ];H3),zt ∈ H1([0, T ];H2) being the unique solution to the problem (2.41) and wbeing the unique solution to the problem (2.42) with w ∈ C0([0, T ];V ) ∩C1((0, T ];V ) ∩ C0((0, T ];D(A

32 )).

Proof. It is clear from the previous constructions of z and w that the sumφ = w + z is a solution to the problem (2.10)–(2.12). The uniqueness of thesolutions in the indicated class can be easily seen from the following identitywhich can be derived by multiplying (2.10) by φ, and integrating over Ω:

d

dt

(12‖φ‖2 +

12Γs

+∫

Γφ2dσ

)+ ‖∇φ‖2 +

∫Γ(σs|∇||φ|2 + gsφ

2)dσ =∫

Ωfφdx.

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cahn-hilliard equation 95

The proof is completed. �The following regularity result will be needed in the next section:

Theorem 2.4. Let φ0 ∈ H3, f ∈ C0([0, T ];H1), ft ∈ L2([0, T ];L2), and

w(0) = w0 := φ0 − z(0) ∈ D(A3/2) (2.46)

which implies thatφ1 := Δφ0 + f(0) ∈ V. (2.47)

Then the problem (2.10)–(2.12) admits a unique solution φ such that φ ∈C0([0, T ];H3), φt ∈ C0([0, T ];V ) ∩ L2([0, T ];H2), φtt ∈ L2([0, T ];H). More-over, for t ∈ [0, T ] the following estimates hold:

‖φ(t)‖2V +

∫ t

0‖φt‖2

Hdτ ≤ C(‖φ0‖2

V +∫ t

0‖f‖2dτ

), (2.48)

‖φt(t)‖2V +

∫ t

0‖φtt‖2

Hdτ ≤ C(‖φ1‖2

V +∫ t

0‖ft‖2dτ

), (2.49)

‖φ(t)‖2H3 ≤ C

(‖φ1‖2

V + ‖f(t)‖2H1 +

∫ t

0‖ft‖2dτ

), (2.50)

where C is a positive constant independent of the solution φ, t, φ0 and f .

Proof. We use a density argument. For the time being, we assume that f ∈C0([0, T ];H1), ft ∈ H1([0, T ];L2). As proved previously, the unique solutionφ can be decomposed into φ = w + z. By the elliptic theory previouslymentioned, we now have z ∈ C0([0, T ];H3), zt ∈ H1([0, T ];H2). Therefore,f = −zt ∈ C0([0, T ];V ), ft ∈ L2([0, T ];V ). Since w is the unique solution tothe initial value problem for the abstract evolution equation (2.43), (2.44),and now w(0) = φ0 − z(0) ∈ D(A3/2), i.e., Δφ0 + f(0) ∈ V , and we inferfrom semigroup theory that w ∈ C0([0, T ];D(A3/2)), wt ∈ C0([0, T ];V ), wt ∈L2([0, T ];D(A)), wtt ∈ L2([0, T ];H). On the other hand, it follows from(2.10), (2.11) that

−Δφ = f − φt ∈ C0([0, T ];H1), (2.51)

and on the boundary Γ

σsΔ||φ − ∂νφ − gsφ =1Γs

φt ∈ H1(Γ). (2.52)

Therefore, by the elliptic theory, we have

‖φ(t)‖H3 ≤ C(‖f(t) − φt(t)‖H1 + ‖φt(t)‖H1/2(Γ)

), (2.53)

hence,‖φ(t)‖H3 ≤ C(‖f(t)‖H1 + ‖φt(t)‖H1). (2.54)

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96 Reinhard Racke and Songmu Zheng

Thus, the solution φ belongs to the desired function class as stated. Toobtain the estimate (2.48), we multiply the equation (2.10) by φ and φt,respectively, then integrate over Ω and use the boundary condition (2.11) toget12

d

dt

(‖φ(t)‖2 +

1Γs

∫Γ

φ2dσ)

+‖∇φ(t)‖2 +∫

Γ(σs|∇||φ|2 +gsφ

2)dσ =∫

Ωfφdx,

12

d

dt

( ∫Γ(σs|∇||φ|2 + gsφ

2)dσ +‖∇φ(t)‖2)

+‖φt‖2 +1Γs

∫Γ

φ2t dσ =

∫Ω

fφtdx.

Thus we easily deduce the estimate (2.48) by integrating with respect to tand using the Holder inequality.

To obtain the estimate (2.49), we use the density argument again. Forthe time being, we assume that the solution φ is more regular with moreregular f and initial data. Similarly, we differentiate the equation (2.10)with respect to t, then multiply it by φtt, integrate with respect to x and t,and use the boundary condition (2.11) to get

12

d

dt

( ∫Γ(σs|∇||φt|2+gsφ

2t )dσ+‖∇φt(t)‖2

)+‖φtt‖2+

1Γs

∫Γ

φ2ttdσ=

∫Ω

ftφttdx.

(2.55)We then easily deduce that

‖φt(t)‖2V +

∫ t

0‖φtt‖2

Hdτ ≤ C(‖φ1‖2

V +∫ t

0‖ft‖2dτ

). (2.56)

Thus, the estimate (2.49) follows. For the initial data and f as regular asstated in the present theorem, we use a sequence of more regular initial dataand f to approximate, then pass to the limit. Combining (2.53) with (2.49)yields (2.50). Notice that in the estimates (2.48)–(2.50), it does not involveftt. Therefore, we can conclude the proof by the density argument. �

Lemma 2.5. Suppose that the following compatibility condition

(σsΔ||φ0 − ∂νφ0 − gsφ0 −1

εΓsΔφ0 −

1εΓs

f(0, ·))|Γ = 0 (2.57)

is satisfied andΔφ0 + f(0, ·) ∈ V, (2.58)

then the condition (2.46) in the statement of Theorem 2.4 holds.

Proof. It follows from (2.44) and (2.41) that

σsΔ||w(0) − ∂νw(0) − gsw(0) − 1εΓs

Δw(0))|Γ = 0, (2.59)

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cahn-hilliard equation 97

i.e., w(0) ∈ D(A), and Aw(0) = −Δw(0) = −Δφ0 − f(0, ·) ∈ V . Thus, theproof is completed. �

3. Local existence for (Pε) for n = 2, 3

Having solved the auxiliary problem (2.10)–(2.12), we are now able to solvethe approximate problem (Pε) given by (2.4)–(2.6), (2.7)–(2.9) by using thecontraction mapping principle in an appropriate Banach space.

From now on we always assume that the following conditions (3.1), (3.2)on the initial data φ0 = ψ0 − hs

gs, u0 = μ0 + εμ1 = N(φ0) − Δφ0 + εμ1 are

satisfied:φ0 ∈ H3, Bφ0 ∈ H1(Γ), μ1 ∈ V, (3.1)

which implies that u0 ∈ H1,(Bφ0 −

1εΓs

(Δφ0 + u0 − N(φ0)))|Γ =

(Bφ0 −

1Γs

μ1

)|Γ = 0. (3.2)

It turns out that

φ1 :=Δφ0 + (u0 − N(φ0))

ε= μ1 ∈ V. (3.3)

Then we we have the following result on local existence and uniqueness.

Theorem 3.1. Suppose that the initial data φ0, u0 satisfy the conditionsabove. Then there is a positive constant δ, which may depend on the initialdata and on ε such that the problem (Pε) admits a unique local solutionφ, u such that φ ∈ C0([0, δ];H3), φt ∈ C0([0, δ];V ), φt ∈ L2([0, δ];H2), φtt ∈L2([0, δ];H), u ∈ C0([0, δ];H1)∩ L2([0, δ];H2), ut ∈ L2([0, δ];L2).

Proof. Let

Y1 := C0([0, δ], H3) ∩ C1([0, δ], V ) ∩ H2([0, δ], L2),

Y2 := C0([0, δ], H1) ∩ H1([0, δ];L2) ∩ L2([0, δ];H2),and Xδ = Y1 × Y2. For (χ, v) ∈ Xδ, we define (φ, u) to be the solution to

εut − Δu = −χt =: g,(∂νu)|Γ = 0,u(0, ·) = u0,

⎫⎬⎭ (3.4)

andεφt − Δφ = v − N(χ) =: f,

(σsΔ||φ − ∂νφ − gsφ − 1Γs

φt)|Γ = 0,

φ(0, ·) = φ0.

⎫⎬⎭ (3.5)

For (χ, v) ∈ Xδ, thus g ∈ H1([0, δ];L2), by the well-known results for theheat equation, we have a unique solution u ∈ Y2, u ∈ C0((0, δ];H3), ut ∈

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98 Reinhard Racke and Songmu Zheng

C0((0, T ];H1) to the problem (3.4). On the other hand, by the Sobolevimbedding theorem the right-hand side f in (3.5) satisfies f ∈ C0([0, δ];H1),ft ∈ L2([0, δ];L2). Moreover, it is easy to see from (3.1)–(3.3) that the con-ditions on initial data in Theorem 2.4 are satisfied. Therefore, by Theorem2.4, the problem (3.5) admits a unique solution φ ∈ Y1. Thus, the mappingS : (χ, v) �→ (φ, u) is well defined as a mapping from Xδ into itself. Letφ1 := 1

ε (Δφ0 − N(φ0) + u0) and

Zδ :={

(φ, u) ∈ Xδ : max0≤t≤δ

‖u(t)‖2H1 +

∫ δ

0‖ut(τ)‖2 + ‖u(τ)‖2

H2dτ ≤ M1,

max0≤t≤δ

‖φ(t)‖2V ≤ M2, max

0≤t≤δ(‖φt(t)‖2

V + ‖φ(t)‖2H3) +

∫ δ

0‖φtt(τ)‖2dτ ≤ 2M3,

φ(0, ·) = φ0, φt(0, ·) = φ1, u(0, ·) = u0

},

where the positive constants M1, M2, M3 will be determined below. Weshall prove that S maps Zδ into itself as a contraction in a suitable norm,if 0 < δ ≤ 1 is sufficiently small. For this purpose let (φ, u) = S((χ, v))again, now for (χ, v) ∈ Zδ. Then we conclude as usual from (3.4) that for0 ≤ t ≤ δ,

‖u(t)‖2H1 +

∫ t

0‖ut(τ)‖2 + ‖u(τ)‖2

H2dτ ≤ C1(‖u0‖2H1 +

∫ t

0‖g(τ)‖2dτ)

≤ C1(‖u0‖2H1 + δM3), (3.6)

where C1 (and in the sequel C2, . . . ) denotes a positive constant dependingat most on ε. We choose

M1 := 2C1‖u0‖2H1 (3.7)

and δ such thatδM3 ≤ ‖u0‖2

H1 , (3.8)then we get from (3.7)

‖u(t)‖2H1 +

∫ t

0‖ut(τ)‖2 + ‖u(τ)‖2

H2dτ ≤ M1. (3.9)

The estimate (2.48) from Theorem 2.4 yields that for 0 ≤ t ≤ δ,

‖φ(t)‖2V ≤ C2

(‖φ0‖2

V + δM1 + δ(1 + M2 + M32 )

). (3.10)

ChoosingM2 := 2C2‖φ0‖2

V (3.11)and δ such that

2C2δ(M1 + 1 + M2 + M32 ) ≤ M2, (3.12)

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cahn-hilliard equation 99

we conclude from (3.10)‖φ(t)‖2

V ≤ M2. (3.13)The estimate (2.49) from Theorem 2.4 yields that for 0 ≤ t ≤ δ,

‖φt(t)‖2V +

∫ t

0‖φtt(τ)‖2

H ≤ C3

(‖φ1‖2

V + M1 + δ(M3 + M33 )

). (3.14)

ChoosingM3 ≤ 2C3(‖φ1‖2

V + M1) (3.15)and δ such that

2C3δ(M3 + M33 ) ≤ M3, (3.16)

we obtain from (3.14)

‖φt(t)‖2V +

∫ t

0‖φtt(τ)‖2

H ≤ M3. (3.17)

We use (2.50) from Theorem 2.4 to estimate ‖φ(t)‖2H3 :

‖φ(t)‖2H3 ≤ C4(‖φ1‖2

V +M1 +δ(M3 +M33 )+1+M2 +M3

2 +‖∇χ3‖2). (3.18)

The last term can be estimated as follows:

‖∇χ3‖2 = ‖∇(φ30 + 3

∫ t

0(χ2χt)(τ)dτ)‖2 ≤ C5(‖φ0‖6

H2 + δ2M33 ). (3.19)

Combining (3.18) and (3.19), we obtain

‖φ(t)‖2H3 ≤ C6(‖φ1‖2

V +M1 +δ(M3 +M33 )+1+M2 +M3

2 +‖φ0‖6H2 +δ2M3

3 ).(3.20)

Choosing

M3 ≥ 2C6(‖φ1‖2V + M1 + 1 + M2 + M3

2 + ‖φ0‖6H2). (3.21)

and δ such that2C6(δ(M3 + M3

3 ) + δ2M33 ) ≤ M3 (3.22)

we obtain from (3.20)‖φ(t)‖2

H3 ≤ M3. (3.23)Therefore, choosing M1, M2, M3 and δ this way, we conclude from (3.9),(3.13), (3.17) and (3.23) that (u, φ) ∈ Zδ, that is, S maps Zδ into itself. Wenotice that M1, M2, M3 and δ depend at most on (‖u0‖H1 , ‖φ1‖V , ‖φ0‖H3)and ε.

Finally, we show that S : Zδ −→ Zδ is a contraction mapping with respectto the following norm

‖(φ, u)‖Zδ:=

(max0≤t≤δ

‖u(t)‖2H1 +

∫ δ

0‖ut(τ)‖2 + ‖u(τ)‖2

H2dτ

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100 Reinhard Racke and Songmu Zheng

+ max0≤t≤δ

‖φ(t)‖2V + K1( max

0≤t≤δ(‖φt(t)‖2

V +∫ δ

0‖φtt(τ)‖2dτ) + K2‖φ(t)‖2

H3

),

where the positive constants K1, K2 will be defined below. Let (χj , vj) ∈ Zδ,j = 1, 2, and (φj , uj) := S((χj , vj). Then φ := φ1 − φ2, u := u1 − u2,χ := χ1 − χ2 and v := v1 − v2 satisfy

εut − Δu = −χt =: g,

(∂νu)|Γ = 0, u(0, ·) = 0,

}(3.24)

andεφt − Δφ = v + N(χ2) − N(χ1) =: f,

(σsΔ||φ − ∂νφ − gsφ − 1Γs

φt)|Γ = 0, φ(0, ·) = 0.

⎫⎬⎭ (3.25)

As in (3.6) we now obtain

‖u(t)‖2H1 +

∫ t

0‖ut(τ)‖2 + ‖u(τ)‖2

H2dτ ≤ C7δ‖(χ, v)‖2Zδ

≤ 124

‖(χ, v)‖2Zδ

,

(3.26)if

δ ≤ 124C7

. (3.27)

In analogy to (3.10), we get

‖φ(t)‖2V ≤ C8δ(2 + M2

2 )‖(χ, v)‖2Zδ

≤ 124

‖(χ, v)‖2Zδ

, (3.28)

ifδ ≤ 1

24C8(2 + M22 )

. (3.29)

As in (3.14) we now conclude

‖φt(t)‖2V +

∫ t

0‖φtt(τ)‖2

H ≤ C9

(‖(χ, v)‖2

Zδ+ δ(1 + M2

3 )‖(χ, v)‖2Zδ

). (3.30)

With K1 := 124C9

we obtain from (3.30)

K1

(‖φt(t)‖2

V +∫ t

0‖φtt(τ)‖2

H

)≤ 2

24‖(χ, v)‖2

Zδ(3.31)

ifδ ≤ 1

24C9(1 + M23 )

. (3.32)

The estimate (3.20) now carries over to

‖φ(t)‖2H3 ≤ C10(1 + M2

2 )(‖(χ, v)‖2Zδ

+ δ(1 + M23 + δM2

3 )‖(χ, v)‖2Zδ

. (3.33)

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cahn-hilliard equation 101

Defining K2 := 124C10(1+M2

2 ), we get from (3.33)

K2‖φ(t)‖2H3 ≤ 2

24‖(χ, v)‖2

Zδ(3.34)

if

δ ≤ 124C10(1 + 2M2

3 ). (3.35)

Combining (3.26), (3.28) and (3.34), we conclude

‖(φ, u)‖Zδ≤ 1

2‖(χ, v)‖Zδ

.

Thus, S is a contraction mapping and the proof of the local existence anduniqueness theorem is completed. �

Remark 3.2. As the proof demonstrated, the parameter δ in Theorem 3.1depends at most on (‖u0‖H1 , ‖φ1‖V , ‖φ0‖H3) and ε.

4. Global existence for (Pε) for n = 2, 3

Denoting by (φε, uε) the local solution of (Pε) given by Theorem 3.1 weshall now prove further a priori estimates to justify the global existence.To do so, we reverse the transformation (2.1) and observe that (φ, u) thensatisfies (dropping the index ε in most places)

εut − Δu = −φt,

(∂νu)|Γ = 0, u(0, ·) = u0,

}(4.1)

andεφt − Δφ − φ + φ3 = u,

(σsΔ||φ − ∂νφ)|Γ = (1Γs

φt − hs + gsφ)|Γ,

φ(0, ·) = ψ0.

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(4.2)

Theorem 4.1. Suppose that the initial data φ0, u0 satisfy the conditions(3.1)–(3.3) in Section 3. Then problem (4.1), (4.2) admits a unique globalsolution such that for any T > 0, φ ∈ C0([0, T ];H3), φt ∈ C([0, T ];V ), φt ∈L2([0, T ];H2), φtt ∈ L2([0, T ];H), u ∈ C0([0, T ];H1) ∩ L2([0, T ];H2), ut ∈L2([0, T ];L2).

Proof. By the local existence and uniqueness result in the previous section itsuffices to obtain uniform a priori estimates on ‖φ(t)‖H3 , ‖u(t)‖H1 , ‖φt(t)‖V .

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102 Reinhard Racke and Songmu Zheng

Multiplying equations (4.1) and (4.2) by u and φt, respectively, and inte-grating with respect to x, we obtain that

d

dt(12ε‖u‖2 + F [φ]) + ‖∇u‖2 + ε‖φt‖2 +

1Γs

∫Γ

φ2t dσ = 0,

where F [φ] denotes the total energy functional defined in (1.10), implyingafter integration with respect to t and using the elementary estimate |a| ≤a2 + 1/4 for a ∈ R,

ε‖u‖2 ≤ C, ‖φ‖V ≤ C,

∫ t

0‖∇u‖2dτ ≤ C,

ε

∫ t

0‖φt‖2dτ ≤ C,

∫ t

0

∫Γ

φ2t dσdτ ≤ C,

⎫⎪⎪⎬⎪⎪⎭

(4.3)

where C denotes — also in the sequel in different places with possibly dif-ferent values — a positive constant that depends at most on φ0, u0, φ1 butis independent of t, δ and ε.

Differentiating (4.2) with respect to t, multiplying the result by φt, inte-grating over Ω, and using (4.1), we get∫

Ω(−εφtt + Δφt − 3φ2φt + φt)φtdx = −

∫Ω

utφtdx = ε

∫Ω

u2t dx −

∫Ω

Δuutdx

which implies that12

d

dt(‖∇u‖2 + ε‖φt‖2 +

1Γs

∫Γ

φ2t dσ) + ‖∇φt‖2 + ε‖ut‖2 + 3

∫Ω

φ2φ2t dx

+∫

Γ(|∇||φt|2 + gsφ

2t )dσ = ‖φt‖2.

Multiplying (4.1) by φt and integrating by parts leads to

‖φt‖2 = −ε

∫Ω

utφtdx −∫

Ω∇u∇φtdx ≤ ε‖ut‖‖φt‖ + ‖∇u‖‖∇φt‖.

By the inequality ab ≤ a2

2 + b2

2 , we deduce that

‖φt‖2 ≤ ε2‖ut‖2 + 2‖∇u‖‖∇φt‖.Hence, using (4.3),∫ t

0‖φt‖2dτ ≤ ε2

∫ t

0‖ut‖2dτ + 2(

∫ t

0‖∇u‖2dτ)1/2(

∫ t

0‖∇φt‖2dτ)1/2

≤ ε2

∫ t

0‖ut‖2dτ + C(

∫ t

0‖∇φt‖2dτ)1/2.

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cahn-hilliard equation 103

Thus, we deduce, observing 0 < ε ≤ 1,

‖∇u‖ ≤ C, ε‖φt‖2 ≤ C,

∫Γ

φ2t dσ ≤ C,

∫ t

0‖∇φt‖2dτ ≤ C,

ε

∫ t

0‖ut‖2dτ ≤ C,

∫ t

0

∫Γ(|∇||φt|2 + φ2

t )dσdτ ≤ C,

∫ t

0‖φt‖2dτ ≤ C.

⎫⎪⎪⎬⎪⎪⎭(4.4)

It follows from equation (4.1) that∫ t

0‖Δu‖2dτ ≤ C. (4.5)

As (2.55) in the proof in Theorem 2.4, we have

12

d

dt

(∫Γ(σs|∇||φt|2 + gsφ

2t )dσ + ‖∇φt(t)‖2

)+ ε‖φtt‖2 +

1Γs

∫Γ

φ2ttdσ

=∫

Ω(ut − 3φ2φt + φt)φttdx, (4.6)

hence,

12

d

dt

(∫Γ(σs|∇||φt|2 + gsφ

2t )dσ + ‖∇φt(t)‖2 − ‖φt‖2 +

∫Ω

32φ2φ2

t dx

)

+ε‖φtt‖2 +1Γs

∫φ2

ttdσ =∫

Ω(utφtt + 3φφ3

t )dx. (4.7)

Since n ≤ 3, by the Young inequality and Sobolev’s imbedding theorem, wehave

|3∫

Ωφφ3

t dx| ≤ C‖φ‖L6‖φt‖3

L185≤ C‖φt‖3

L185

.

By the Gagliardo–Nirenberg interpolation inequality we get

‖φt‖3

L185≤ C1‖∇φt‖

2n3 ‖φt‖

9−2n3 + C2‖φt‖3 ≤ C‖φt‖2

H1‖φt‖.

Therefore, it follows from (4.4) that∫ t

0‖φt‖3

L185≤ C max

0≤τ≤t‖φt(τ)‖ ≤ η max

0≤τ≤t‖φt(τ)‖2 + Cη

with η being a small positive constant. Integrating (4.7) with respect to t,then taking the maximum with respect to time over [0, t], and choosing ηsmall enough yields∫

Γ(σs|∇||φt|2 + gsφ

2t )dσ + ‖∇φt(t)‖2 +

∫ t

0ε‖φtt‖2dτ +

∫ t

0

2Γs

∫Γ

φ2ttdσdτ

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104 Reinhard Racke and Songmu Zheng

≤ C

ε+

C

ε

∫ t

0‖ut‖2dτ ≤ C

ε+

C

ε2. (4.8)

Thus,ε‖φt‖V ≤ C. (4.9)

Integrating (4.2) over Ω yields

−ε

∫Ω

φtdx +∫

Γ∂νφdσ −

∫Ω(φ3 − φ)dx =

∫Ω

udx.

From the boundary conditions in (4.2) we conclude that∫Γ

∂νφdσ = σs

∫Γ

Δ||φdσ − 1Γs

∫Γ

φtdσ +∫

Γ(hs − gsφ)dσ

= − 1Γs

∫Γ

φtdσ +∫

Γ(hs − gsφ)dσ.

On the other hand we deduce from (4.3) and (4.4) that

|∫

Γ(hs−gsφ)dσ| ≤ C, |

∫Ω(φ3−φ)dx| ≤ C, |ε

∫Ω

φtdx| ≤ C, |∫

Γφtdσ| ≤ C.

Therefore, we have

|∫

Ωudx| ≤ C,

and, in combination with (4.4),

‖u‖H1 ≤ C. (4.10)

Finally, we have the elliptic estimate (cp. Section 2 and the references [14,1, 2, 20])

‖φ‖H3 ≤ C(‖f‖H1 + ‖g‖H1/2(Γ)),

where

f := Δφ = −u + εφt + φ3 − φ, g :=1Γs

φt − hs + gsφ.

Since by (4.3)–(4.10)

‖f‖H1 ≤ C, ‖g‖H1/2(Γ) ≤ C/ε,

we get‖φ(t)‖H3 ≤ C/ε. (4.11)

By these estimates, the global existence and uniqueness of strong solutionsfor problem (Pε) with ε > 0 being fixed follows. �

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cahn-hilliard equation 105

5. Global existence for the original Cahn-Hilliard equation

We now turn to the original problem (1.3), (1.5), (1.11), (1.4) for theCahn-Hilliard equation, i.e.,

ψt = Δμ in [0, T ] × Ω, (5.1)

μ = −Δψ − ψ + ψ3, (5.2)

∂νμ|Γ = 0, (5.3)1Γs

ψt = σsΔ||ψ − ∂νψ + hs − gsψ on Γ, (5.4)

ψ(0, ·) = ψ0 in Ω. (5.5)

Then we have the following result.

Theorem 5.1. Suppose that n = 2 or n = 3 and that the initial data ψ0

satisfies ψ0 ∈ H3, Bφ0 ∈ H1(Γ). Then the initial boundary value problem(5.1)–(5.5) admits a unique global solution ψ such that for any T > 0, ψ ∈C0([0, T ];H1) ∩ L2([0, T ];H3), ψt ∈ L2([0, T ];V ),Δψ ∈ L2([0, T ];H3), μ :=−Δψ − ψ + ψ3 ∈ L2([0, T ];H3).

Proof. As in Section 3, let μ1 be an element in V such that(Bψ0 −

1Γs

μ1

)|Γ = 0. (5.6)

Consider the following (Pε) problem:

εut − Δu = −φt,

(∂νu)|Γ = 0, u(0, ·) = u0,

}(5.7)

andεφt − Δφ − φ + φ3 = u,

(σsΔ||φ − ∂νφ)|Γ = (1Γs

φt − hs + gsφ)|Γ,

φ(0, ·) = ψ0,

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(5.8)

whereu0 = μ0 + εμ1, (5.9)

andμ0 = −Δψ0 − ψ0 + ψ3

0.

Then it is easy to see from (5.6) and (5.9) that the conditions (3.1)–(3.2) inSection 3 are satisfied. Thus, by Theorem 4.1, problem (5.1)–(5.5) admits a

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106 Reinhard Racke and Songmu Zheng

unique global solution φ(ε), u(ε). Moreover, by the estimates (4.3)–(4.11) inthe previous section, for any T > 0, we have

φ(ε) uniformly bounded in C0([0, T ];H1) ∩ L2([0, T ];H3), (5.10)

φ(ε)t uniformly bounded in L2([0, T ];V ), (5.11)

u(ε) uniformly bounded in C0([0, T ];H1), (5.12)√

εu(ε)t uniformly bounded in L2([0, T ];L2), (5.13)

Δu(ε) uniformly bounded in L2([0, T ];L2). (5.14)Therefore, we have a subsequence in ε and φ(ε), u(ε), still denoted by ε andφ(ε), u(ε), and ψ, μ such that, as ε → 0,

φ(ε) → ψ, weak-* in L∞([0, T ];H1), (5.15)

φ(ε) → ψ, weakly in L2([0, T ];H3), (5.16)

φ(ε)t → ψt, weakly in L2([0, T ];V ), (5.17)

u(ε) → μ, weak-* in L∞([0, T ];H1), (5.18)

εu(ε)t → 0 strongly in L2([0, T ];L2), (5.19)

Δu(ε) → Δμ weakly in L2([0, T ];L2). (5.20)It follows from (5.15), (5.17) that ψ also belongs to C0([0, T ];H1) andψ|t=0 = ψ0. By the well-known Aubin compactness theorem, we deducefrom (5.16), (5.17) that

φ(ε) → ψ strongly in L2([0, T ];H3−η) (5.21)

with η being a sufficiently small positive constant. Then it follows from theSobolev imbedding theorem that

(φ(ε))3 → ψ3 strongly in L2([0, T ];L2). (5.22)

Taking the weak limit in both (5.7) and (5.8) yields that (5.1) holds inL2([0, T ];L2), and (5.2) holds in L2([0, T ];H1). The boundary conditions(5.3) and (5.4) are satisfied in L2([0, T ];H−1/2(Γ)) and in L2([0, T ];H1/2(Γ)),respectively. Since ψt ∈ L2([0, T ];V ), by the regularity theorem for theelliptic equation, we deduce that μ ∈ L2([0, T ];H3). Therefore, the proof forthe global existence part of the present theorem is completed.

To prove the uniqueness, let ψ1, ψ2 be two solutions, and let ψ = ψ1 −ψ2.Then ψ and the corresponding μ satisfy

Δμ = ψt, (5.23)

−Δψ + ψ31 − ψ3

2 − ψ = μ, (5.24)

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cahn-hilliard equation 107

∂νμ|Γ = 0, (5.25)

1Γs

ψt = σsΔ||ψ − ∂νψ − gsψ on Γ, (5.26)

ψ(0, ·) = 0 in Ω. (5.27)

Multiplying (5.24) by ψt and integrating over Ω, then using (5.23) and theboundary conditions, we have for n ≤ 3

12

d

dt

( ∫Ω(|∇ψ|2 + (ψ2

1 + ψ1ψ2 + ψ22)ψ

2)dx +∫

Γ(σs|∇||ψ|2 + gsψ

2)dσ)

+ ‖∇μ‖2 +1Γs

∫Γ|ψt|2dσ (5.28)

= −∫

Ω∇μ∇ψdx +

12

∫Ω(ψ1t(2ψ1 + ψ2) + ψ2t(ψ1 + 2ψ2))ψ2dx

≤ 12(‖∇μ‖2 + ‖∇ψ‖2) + 2(‖ψ1t‖ + ‖ψ2t‖)(‖ψ1‖L6 + ‖ψ2‖L6)‖ψ‖2

L6

≤ 12(‖∇μ‖2 + ‖∇ψ‖2) + C(‖ψ1‖H1 + ‖ψ2‖H1)(‖ψ1t‖ + ‖ψ2t‖)‖ψ‖2

H1 .

Let y(t) be the nonnegative function defined as follows:

y(t) =∫

Ω(|∇ψ|2 + (ψ2

1 + ψ1ψ2 + ψ22)ψ

2)dx +∫

Γ

(σs|∇||ψ|2 + gsψ

2)dσ.

Then it follows from (5.28) that

dy

dt≤ α(t)y(t), (5.29)

where

α(t) = C(‖ψ1‖H1 + ‖ψ2‖H1)(‖ψ1t‖ + ‖ψ2t‖). (5.30)

It is easy to see from (5.16), (5.17) that α ∈ L1([0, T ]; R). Thus, we deducefrom (5.29), the Gronwall inequality and (5.27) that ψ(t) = 0 for all t ∈ [0, T ].The proof is completed. �

Remark 5.2. We could also conclude the C∞ smoothness of the solutionas t > 0, since the system is now a nonlinear parabolic one, cp. e.g. [27] forthis general aspect.

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108 Reinhard Racke and Songmu Zheng

6. The case n = 1

In the one-dimensional case the boundary condition (1.11) turns out to be1Γs

ψt = −∂νφ + hs − gsφ on Γ, (6.1)

where ∂ν = ∂x now. Therefore, we should make some changes in the previoussections. For instance, in Section 2, we should replace V by the usual H1 inwhich the inner product is defined as follows

(u, v)H1 =∫

Ω∇u∇vdx + gs

∫Γ

uvdσ.

Accordingly, instead of using the results on elliptic equations with the bound-ary condition involving the operator Δ||− ∂ν − gs, we should use the elliptictheory for the usual third kind of boundary condition, i.e., Robin’s boundarycondition. For this reason some estimates, e.g. (2.37) and (2.53) in Section 2no longer hold the same way. However, for n = 1, we have Δφ = φxx, hence(2.54) still follows, and it finally turns out that Theorem 2.4 still holds.Keeping these minor changes in mind, we now have the corresponding resultfor problem (5.1)–(5.5) for n = 1.

Theorem 6.1. Suppose that n = 1 and that the initial data ψ0 satisfiesψ0 ∈ H3. Then the initial boundary value problem (5.1)–(5.5) (without theoperator Δ||) admits a unique global solution ψ such that for any T > 0,ψ ∈ C0([0, T ];H1)∩L2([0, T ];H5), ψt ∈ L2([0, T ];V ), μ := −Δψ−ψ + ψ3 ∈L2([0, T ];H3).

7. The limiting case Γs = ∞In the limiting case Γs = ∞, the nonlinear boundary condition (1.11)

turns into a boundary condition of lower-order since the highest-order termvanishes now. This simpler case can be dealt with in an analogous way tothe previous sections. For example, the eigenvalue problem (2.20), (2.21)becomes just an ordinary eigenvalue problem for the Laplace operator withboundary condition BΦ = 0, which in turn is reflected in the fact thatH equals L2. Also because the decomposition technique in Section 2 isno longer needed, the auxiliary function z in that section should be thetrivial zero now, and so on. Keeping these remarks in mind, we obtain thecorresponding results to Theorem 5.1, Theorem 6.1 by making minor changesin the proofs in the previous sections.

Acknowledgement: The first author thanks the Institute of Mathematics

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cahn-hilliard equation 109

at Fudan University, Shanghai, for its financial support during his visit inMarch 2001, where the research for this work was done.

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