Annali di Matematica (2006) 185(4): 627–648 DOI 10.1007/s10231-005-0175-3

¨ · Reinhard Racke · Songmu Zheng Jan Pruss

Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions

Received: 4 May 2004 / Revised: 24 February 2005 / Published online: 3 January 2006
C Springer-Verlag 2005

Abstract This paper deals with the Cahn–Hilliard equation ψt = µ,

µ = −ψ − ψ + ψ 3 ,

(t, x) ∈ J × ,

subject to the boundary conditions 1 ψt = σs || ψ − ∂ν ψ − gs ψ + h, s

∂ν µ = 0,

and the initial condition ψ(0, x) = ψ0 (x) where J = (0, ∞), and  ⊂ Rn is a bounded domain with smooth boundary  = ∂G, n ≤ 3, and s , σs , gs > 0, h are constants. This problem has already been considered in the recent paper of R. Racke and S. Zheng (The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110, 2003), where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal L p -regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor. Keywords Cahn · Hilliard · Maximal regularity · Global attractor Mathematics Subject Classification (2000) 82C26, 35B40, 35B65, 35Q99 J. Pr¨uss Department of Mathematics and Computer Science, Martin-Luther-University, 60120 Halle, Germany E-mail: [email protected]

B

R. Racke ( ) Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany E-mail: [email protected] S. Zheng Institute of Mathematics, Fudan University, 200433 Shanghai, P. R. China E-mail: [email protected]

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1 Introduction Let  ⊂ Rn be a bounded domain with boundary  = ∂ being smooth (cp. Theorem 2.1), and consider the following boundary value problem for the Cahn– Hilliard equation. ψt = µ,

µ = −ψ − ψ + ψ 3 , t > 0, x ∈ , ∂ν µ = 0, t > 0, x ∈ 

1 ∂t ψ − σs || ψ + ∂ν ψ + gs ψ = h, t > 0, x ∈  s ψ = ψ0 t = 0, x ∈ .

(1.1) (1.2) (1.3) (1.4)

Here ν(x) denotes the outer normal of  at x ∈ , || means the Laplace– Beltrami operator on , and s , σs , gs > 0, h are constants. This problem arises from the study of spinodal decomposition of binary mixtures that appears, for examples, in cooling processes of alloys, glasses or polymer mixtures (see [2, 7, 5], and the references cited therein.) Boundary condition (1.3) is usually called the dynamic boundary condition since it also involves the time derivative of ψ. It is derived when the effective interaction between the wall (i.e., the boundary ) and two mixture components are short-ranged (see the references in [5, 10]). This problem was recently studied in [10] and global existence and uniqueness of solution was proved there. Furthermore, it was pointed out that for t > 0 the solution is C ∞ . However, it was not clear whether the solution defines a C0 semigroup in the Sobolev space V = H 1 () ∩ H 1 () introduced in that paper. In the present paper we further investigate this problem. More precisely, we prove the maximal L p -regularity of the solution which implies that the solution defines a C0 -semigroup. Furthermore, we prove the existence of a global attractor. This paper is organized as follows. In Section 2, we first study a linear problem associated with our original problem (1.1)–(1.4), and we establish maximal L p regularity results. Then we consider the corresponding nonlinear problem (1.1)– (1.4), and prove in Sections 3 and 4 first the local well-posedness and then the global well-posedness on a phase manifold M ≡ M p (see Theorem 4.3) which does not coincide with V , which implies that the solution defines a C0 -semigroup in M. In the final section we prove the existence of a global attractor in M2 = H 2 () ∩ H 5/2 () as well as in V .

2 The linear problem In this section we study the following linearized version of (1.1). ∂t v + 2 v = f, ∂ν v = g,

t > 0, x ∈ , t > 0, x ∈ ,

1 ∂t v − σs || v + γ ∂ν v + gs v = h, t > 0, x ∈ , s v = v0 , t = 0, x ∈ .

(2.1)

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Here the functions f , g, h as well as the initial value v0 are given; s , σs , γ > 0, and gs ≥ 0 are given constants (γ = 1 in the original system). Replacing (σs , γ , gs , h) by s (σs , γ , gs , h) we may assume w.l.o.g. that s = 1. Let J = [0, T ] and 1 < p < ∞. We are looking for solutions in the class 
v ∈ H p1 (J ; L p ()) ∩ L p J ; H p4 () , which is the natural class for (2.1) in the L p -setting. Then by well-known trace theorems [1,3,9] the data f , g, v0 necessarily satisfy 
1/4−1/4 p 1−1/ p f ∈ L p (J × ), g ∈ W p (J ; L p ()) ∩ L p J ; W p () , 4−4/ p

v0 ∈ W p

().

As usual, here and in the sequel W ps denote the fractional Sobolev spaces. Furthermore, the traces of v and ∂ν v on  satisfy 
1−1/4 p 4−1/ p v| ∈ W p (J ; L p ()) ∩ L p J ; W p () , and

3/4−1/4 p

∂ν v| ∈ W p


3−1/ p (J ; L p ()) ∩ L p J ; W p () .

This leaves some choice for the setting of the dynamic boundary condition. The 2−1/ p ()). Looking at the possibility of lowest order is the choice h ∈ L p (J ; W p dynamic boundary condition as a heat equation on J × , this will result in 
2−1/ p 4−1/ p v| ∈ H p1 (J ; W p ()) ∩ L p J ; W p () . This regularity implies that the trace of v| at t = 0 necessarily satisfies v0 | ∈ 4−3/ p (). Wp The other extreme possibility consists in taking the regularity of the normal derivative of v as the basic regularity, i.e. we may consider the class 
3/4−1/4 p 3−1/ p h ∈ Wp (J ; L p ()) ∩ L p J ; W p () . This leads to 7/4−1/4 p

v| ∈ W p

 

3−1/ p 5−1/ p (J ; L p ()) ∩ H p1 J ; W p () ∩ L p J ; W p () . 5−3/ p

Then necessarily v0 | ∈ W p

() and the compatibility conditions

∂ν v0 | = g|t=0 ,

for p > 5,

and 3−5/ p

σs || v0 | − γ ∂ν v0 | − gs v0 | + h|t=0 ∈ W p

(),

for p > 5/3,

must hold. More generally, any choice of the space for h of the type 
h ∈ W ps (J ; L p ())∩L p J ; W pr () , 0 ≤ s ≤ 3/4−1/4 p, 2−1/ p ≤ r ≤ 3−1/ p,

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will work. Observe that for such r, s the inequality 2s ≤ r is valid. The correr +2−2/ p (), and if s > 1/ p also sponding trace space for v0 | now becomes W p r (1−1/sp) the time derivative ∂t v| has trace at t = 0, which belongs to W p (). Here is the main result on maximal L p -regularity of (2.1). Theorem 2.1. Let n ∈ N, 1 < p < ∞, fix numbers s ∈ [0, 3/4 − 1/4 p] and r ∈ [2 − 1/ p, 3 − 1/ p], p = 5, sp = 1, and let σs , γ > 0. Suppose  ⊂ Rn is bounded with boundary  = ∂ ∈ C r +2 , and let J = [0, T ]. Then there is a unique solution v of (2.1) such that 
v ∈ Z := H p1 (J ; L p ()) ∩ L p J ; H p4 () , and  

v| ∈ Z  := W ps+1 (J ; L p ()) ∩ H p1 J ; W pr () ∩ L p J ; W pr +2 ()y , if and only if the data are subject to the following conditions. 1/4−1/4 p

(J ; L p ()) ∩ (a) f ∈ X := L p (J × ), g ∈ Y := W p 
1−1/ p () , L p J ; Wp   4−4/ p r +2−2/ p v0 ∈ X p := v ∈ W p () : v| ∈ W p () ; (b) h ∈ X  := W ps (J ; L p ()) ∩ L p (J ; W pr ()); (c) ∂ν v0 | = g|t=0 , if p > 5; r (1−1/ ps) (), if s > 1/ p. (d) v ∗ := σs || v0 | − γ ∂ν v0 | − gs v0 | + h|t=0 ∈ W p Proof. Necessity of (a) ∼ (d) is a consequence of well-known trace theorems and has been already explained. The values p = 5 and sp = 1 have been excluded since the trace theorems leading to (c) and (d) are not true for these values of p. For the sufficiency part, the main idea is to reduce the problem to an equation for ρ¯ := v| on J × . This will be done as follows. Let A := −σs || denote the Laplace–Beltrami operator on W pr (). It is well-known that this operator is the negative generator of an analytic C0 -semigroup which enjoys maximal L p regularity. First, assuming that v| is known, it is convenient to decompose ρ¯ = v| as v| = ρ¯ = ρ + ρ1 , where

ρ1 = e−At v0 | ,

t > 0,

if sp < 1, and α

ρ1 = e−At (v0 | − t Av0 | ) + te−A t v ∗ ,

t > 0,

if sp > 1; here we have set α = r/2s. Then, by (b) and (c) the function ρ1 belongs to Z  , as elementary properties of analytic semigroups and their trace spaces show; cf. e.g. [8], where the necessary arguments are given. Next we employ a recent result due to [3] concerning the solvability of general parabolic initial boundary value problems, which means that for given f , g, v0 1−1/4 p 4−1/ p satisfying (a) and (c), and ρ1 ∈ W p (J ; L p ()) ∩ L p (J ; W p ()) defined

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as above, there is precisely one solution v1 ∈ H p1 (J ; L p ()) ∩ L p (J ; H p4 ()) of the problem ∂t v + 2 v = f, t ∈ J, x ∈ , ∂ν v = g, v| = ρ1 , t ∈ J, x ∈ , v|t=0 = v0 , x ∈ .

(2.2)

Thus the function v1 belongs to Z and is determined only by the data f , g, and v0 , and ∂ν v1 belongs to X  , taking traces. Next we derive the governing equation for the linear problem (2.1), i.e. the equation for ρ. For this purpose we let the solution of ∂t v + 2 v = 0, t ∈ J, x ∈ , ∂ν v = 0, v| = ρ, t ∈ J, x ∈ , v|t=0 = 0, x ∈ ,

(2.3)

be denoted by Rρ. Note that if 1−1/4 p

ρ ∈ 0W p then


4−1/ p (J ; L p ()) ∩ L p J ; W p () ,


Rρ ∈ 0 Z := 0 H 1p (J ; L p ()) ∩ L p J ; H p4 () .

Here and in the sequel the left subscript 0 means that all existing traces at t = 0 vanish. Thus v = v1 + Rρ solves the problem ∂t v + 2 v = f, t ∈ J, x ∈ , ∂ν v = g, v| = ρ + ρ1 , t ∈ J, x ∈ , v|t=0 = v0 , x ∈ . Inserting v into the dynamic boundary condition we obtain the following problem for ρ, ∂t ρ − σs || ρ + γ ∂ν Rρ + gs ρ = h 1 ,

t > 0, x ∈ , ρ(0) = 0,

(2.4)

where h 1 is given by h 1 = h − (∂t − σs || )ρ1 − γ ∂ν v1 − gs ρ1 and belongs to 0X


= 0 W sp (J ; L p ()) ∩ L p J ; W pr () ,

by construction; note that s ≤ 3/4 − 1/4 p and r ≤ 3 − 1/ p. Thus (2.1) is reduced to problem (2.4) for ρ. Introducing the operators L := ∂t − σs || + gs I and Sρ := ∂ν Rρ, we may reformulate the latter problem as an operator equation Lρ + γ Sρ = h 1 . Since A = −σs || has maximal L p -regularity, L : 0 Z  → 0 X  is an isomorphism. Let us consider S in more detail. For obvious reasons, we would call S

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the parabolic Dirichlet-Neumann operator. It has order 1/4 in time and order 1 in space. Therefore, it is subordinate to L. Moreover, it is causal, i.e. S is an integrodifferential operator of Volterra type of inhomogeneous fractional order, and it is of order 1 and nonlocal in the variable x ∈ . Now we are going to prove that L + γ S is an isomorphism for each γ > 0. First we show injectivity. Suppose that Lρ + γ Sρ = 0; let u denote the solution of ∂t u + 2 u = 0, t > 0, x ∈ , ∂ν u = 0, u = ρ, t > 0, x ∈ , u = 0, t = 0, x ∈ . Then  1  ∂t γ |∇u|22 + σs |∇|| ρ|22 + gs |ρ|2 = −γ |∇u|22 − |∂t ρ|22 ≤ 0, 2 which implies u = ρ = 0, i.e. L + γ S is injective. Next we show that the range of L + γ S is closed. If it is not true, then there is a sequence (ρn ) ⊂ Z  with ||ρn || Z  = 1 such that h n = (L + γ S)ρn → 0 in X  . Thus (L −1 Sρn ) ⊂ 0 W p

7/4−1/4 p

3−1/ p

(J ; L p ())∩ 0 H 1p (J ; W p

5−1/ p

())∩ L p (J ; W p

())

is bounded. Hence, by compact embedding it is relatively compact in 1−1/4 p 4−1/ p (J ; L p ()) ∩ L p (J ; W p ()), which implies that (Sρn ) has a con0W p vergent subsequence in X  . Thus, this yields a convergent subsequence ρn k → ρ0 in Z  because L is an isomorphism. Since S is bounded, we conclude that (L + γ S)ρ0 = 0, i.e. ρ0 = 0 by injectivity of L + γ S, which is a contradiction to ||ρ0 || Z  = 1. Therefore, L + γ S is an injective Fredholm operator of index 0, which means that it is an isomorphism for each γ > 0. This completes the proof of Theorem 2.1.
Remark 2.2. There is an analytic C0 -semigroup hidden in Theorem 2.1. To see this consider the space X := L p () × W pr () and define an operator A by means of A(v, ρ) = (2 v, γ ∂ν v − σs || ρ + gs ρ), for (v, ρ) ∈ D(A), where D(A) = {(v, ρ) ∈ X : v ∈ H p4 (), ρ ∈ W pr +2 (), v| = ρ, ∂ν v| = 0}. Here as before r ∈ [2 − 1/ p, 3 − 1/ p]. This operator A is closed, linear and densely defined, and (2.1) with g = 0 is equivalent to z˙ + Az = F,

z(0) = z 0 ,

(2.5)

where z = (v, ρ), F = ( f, h) and z 0 = (v0 , v0 | ). It is easy to see that (2.5) has maximal L p -regularity if and only if for each f ∈ L p (J × ) and h ∈ L p (J ; W pr ()) there is a unique solution z of (2.5) such that v ∈ Z , ρ = v ∈ Z  with s = 0. Therefore, Theorem 2.1 implies that (2.5) has maximal L p -regularity, and then by a result in [4],(see also [1, 9]), −A generates an analytic C0 -semigroup in X.

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3 Local well-posedness In this section we want to solve the nonlinear problem, i.e. ∂t ψ + 2 ψ + ψ = ψ 3 + f,

t ∈ J, x ∈ ,

∂ν ψ + ∂ν ψ = ∂ν ψ + g, t ∈ J, x ∈  1 ∂t ψ − σs || ψ + γ ∂ν ψ + gs ψ = h, t ∈ J, x ∈  s ψ = ψ0 t = 0, x ∈ . 3

(3.1)

This problem has been solved in a recent paper [10]. In this section and the next section we would like to reconsider this problem in different spaces based on the L p -regularity obtained in the previous section. We first want to employ the contraction mapping principle to solve this problem on a possibly smaller time interval Ja = [0, a] ⊂ J . For this purpose, let ψ0 ∈ X p , f ∈ X , g ∈ Y and h ∈ X  be given such that the compatibility conditions
 ∂ν ψ0 | = ∂ν − ψ0 + ψ03 | + g|t=0 , p > 5, (3.2) r (1−1/ ps)

σs || ψ0 | − γ ∂ν ψ0 | − gs ψ0 | + h|t=0 ∈ W p

(),

s > 1/ p

are satisfied. Here we have used the notation of Section 2. In the sequel we denote the relevant spaces by X (a), Y (a) to indicate the time interval under consideration, e.g. Ja , and so on. The main result of this section reads as follows. Theorem 3.1. Let 1 < p < ∞, p ≥ (n + 4)/5, s ∈ [0, 3/4 − 1/4 p], r ∈ [2 − 1/ p, 3 − 1/ p], and let the data ψ0 ∈ X p , f ∈ X (T ), g ∈ Y (T ), and h ∈ X  (T ) be given such that the compatibility conditions (3.2) are satisfied. Then there is an a ∈ (0, T ] and a unique solution ψ of (3.1) in the class Z (a). Furthermore, ψ| ∈ Z  (a). ψ depends continuously on the data, and for f, g, h independent of t, the map ψ0 → ψ(t) defines a local semiflow in the natural phase manifold M defined by X p and the compatibility conditions (3.2). Proof. Thanks to Theorem 2.1, we may define a function u ∗ ∈ Z (T ) with u ∗ | ∈ Z  (T ) as the solution of the special problem ∂t u ∗ + 2 u ∗ = f, ∗

t ∈ J, x ∈ ,

−A2 t

g0 , t ∈ J, x ∈ , ∂ν u = g − e ∗ ∗ ∂t u − σs || u + γ ∂ν u ∗ + gs u ∗ = h, t ∈ J, x ∈ , u ∗ |t=0 = ψ0 , x ∈ ,

(3.3)

where as before A = −σs || (w.l.o.g. s = 1 again). If p ≤ 5, we set g0 = 0, and g0 = g|t=0 − ∂ν ψ0 | if p > 5. For a given a ∈ (0, T ] to be fixed later, we define E1 := {u ∈ Z (a) : u| ∈ Z  (a)},

and 0 E1 := {u ∈ E1 : u|t=0 = 0},

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with canonical norm || · ||1 and E0 := X (a)×Y (a)× X  (a),

and 0 E0 := {( f, g, h) ∈ E0 : g|t=0 = h|t=0 = 0}

with norm || · ||0 . Define the linear operator L : E1 → E0 by means of ⎡ ⎤ ∂t v + 2 v ⎦. Lv = ⎣ ∂ν v ∂t v| − σs || v| + γ ∂ν v| + gs v|

(3.4)

By Theorem 2.1, L : 0 E1 → 0 E0 is linear, bounded and bijective, hence an isomorphism. Next we define the nonlinear mapping F : E1 × 0 E1 → 0 E0 by ⎡ ⎤ (−(u ∗ + v) + (u ∗ + v)3 ) F(u ∗ , v) = ⎣ ∂ν (−(u ∗ + v) + (u ∗ + v)3 ) − g1 ⎦ , (3.5) 0 where g1 (t) = e−A t [∂ν (−u ∗ + (u ∗ )3 )|t=0 ] ∈ Y . Then u = u ∗ + v is a solution of (3.1) if and only if 2

Lv = F(u ∗ , v)

i.e. v = L−1 F(u ∗ , v).

(3.6)

it is sufficient to show that the To show that F : E1 × 0 E1 → 0 E0 is of class 1/2 3 cubic form u → u is well defined and bounded from E1 to H p (J ; L p ()) ∩ L p (J ; H p2 ()). In fact, if the latter is valid, then u → u 3 is bounded from E1 to X , and after taking traces, the boundedness of u → ∂ν u 3 from E1 to Y follows as well. These properties then show that F is even real analytic in (u ∗ , v). The following lemma takes care of the cubic form. C 1,

Lemma 3.2. Let p ≥ (n + 4)/5. Then the trilinear form b(u, v, w) := uvw 1/2 is bounded from E1 to H p (J ; L p ()) ∩ L p (J ; H p2 ()), moreover there is a constant C > 0 such that the estimate ||uvw|| H 1/2 (L p

p )∩L p




H p2

 ≤ C||u|| Z ||v|| Z ||w|| Z ,

u, v, w ∈ Z ,

is valid. Proof. To prove the boundedness in L p (J ; H p2 ()), by symmetry we have to estimate the two typical terms like (∂i ∂ j u)vw and (∂i u)(∂ j v)w. The first one is estimated via H¨older’s inequality as ||(∂i ∂ j u)vw|| p ≤ ||(∂i ∂ j u)||5 p/3 ||v||5 p ||w||5 p . The mixed derivative theorem and the Sobolev embedding theorem yields 
Z → H p1−θ J ; H p4θ () → L 5 p (J × ), provided 1 − θ − 1/ p ≥ −1/5 p and 4θ − n/ p ≥ −n/5 p, i.e. provided n/5 p ≤ θ ≤ 1 − 4/5 p which is possible since p ≥ (n + 4)/5. On the other hand,  

Z → H p1−θ J ; H p4θ () → L 5 p/3 J ; H52p/3 () ,

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provided 1 − θ − 1/ p ≥ −3/5 p and 4θ − n/ p ≥ 2 − 3n/5 p, i.e. provided (1 + n/5 p)/2 ≤ θ ≤ 1 − 2/5 p which again is possible for p ≥ (n + 4)/5. In a similar way the second type of terms (∂i u)(∂ j v)w can be treated, obtaining ||(∂i u)(∂ j v)w|| p ≤ ||∂i u||5 p/2 ||∂ j v||5 p/2 ||w||5 p . 1/2

To estimate uvw in H p (J ; L p ()), we employ the embedding theorem 
1/2 Z → H p1−θ J ; H p4θ () → H5 p/3 (J ; L 5 p/3 ()), provided 1 − θ − 1/ p ≥ 1/2 − 3/5 p and 4θ − n/ p ≥ −3n/5 p, i.e. provided n/10 p ≤ θ ≤ 1/2 − 2/5 p which is again possible since p ≥ (n + 4)/5.
We note further that we have also shown the estimate ||u|| H 1/2

2 5 p/3 (L 5 p/3 )∩L 5 p/3 (H5 p/3 )

≤ C||u|| Z ,

where C > 0 is independent of the interval under consideration. Hence, ||u|| X (a) ≤ Ca 2 p/5 ||u|| Z (a) ,

||∂ν u||Y (a) ≤ Ca 2 p/5 ||u|| Z (a) ,

(3.7)

on Ja = [0, a]. Consider a ball Br (0) ⊂ 0 E1 , where r > 0 will be fixed later, and define T : Br (0) → 0 E1 by means of T v := L−1 F(u ∗ , v). To show that T is a contraction, we infer from Lemma 2.2 and (3.7) that ||T v − T w||1 ≤ ||L−1 || ||F(u ∗ , v) − F(u ∗ , w)||0   ≤ C ||v − w|| H 1/2 (L )∩L (H 2 ) + ||v − w||1 (||v||1 + ||w||1 )2 p p p p   ≤ C a 2 p/5 + 4r 2 ||v − w||1 1 ≤ ||v − w||1 , v, w ∈ Br (0), 2 provided C[a 2 p/5 + 4r 2 ] ≤ 1/2, i.e. if r > 0 and a > 0 are sufficiently small. To show that T Br (0) ⊂ Br (0), in a similar way we obtain that ||T v||1 ≤ ||L−1 || ||F(u ∗ , v)||0  ≤ C ||u ∗ + v|| H 1/2 (L )∩L p

∗

≤ C[||u ||1 + a

2 p/5

p

2 p (H p )

+ ||u ∗ + v||31 + ||g0 ||Y (a)



r + (||u ∗ ||1 + r )3 + ||g0 ||Y (a) ≤ r,

provided a > 0 and r > 0 are small enough. Note that ||u ∗ ||1 → 0 and also ||g0 ||Y (a) → 0 as a → 0, since u ∗ is a fixed function. Therefore the contraction mapping principle yields a unique fixed point v ∈ Br (0) ⊂ 0 E1 which in turn gives the unique solution u = v + u ∗ of (3.1). It is clear that u ∗ depends continuously on the data and v is continuous in u ∗ , hence u depends continuously on the data as well. If f, g, h are constant in t, problem (3.1) is autonomous, hence translation is invariant, which by uniqueness of solution

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shows that the map u 0 → u(t) is a local semiflow in the phase space X p . This completes the proof.
A successive application of Theorem 3.1 yields a solution u on a maximal time interval Jmax = [0, amax ). This time interval is characterized by the two equivalent conditions lim ψ(t)

t→tmax

does not exist in X p ,

and ||ψ|| Z (amax ) + ||ψ| || Z  (amax ) = ∞. Moreover, the function F defined above is real analytic, and Dv F(u ∗ , v) is of lower order, compared to L. Therefore, by arguments as in the proof of Theorem 3.1 one can show that L − Dv F(u ∗ , v) : 0 E1 → 0 E0 is an isomorphism for each u ∗ and v. Therefore the solution ψ depends on the data even analytically.

4 Global well-posedness In this section we want to solve the nonlinear problem ∂t ψ + 2 ψ + ψ = ψ 3 + f,

t ∈ J,

x ∈ ,

∂ν ψ + ∂ν ψ = ∂ν ψ + g, t ∈ J, x ∈  1 ∂t ψ − σs || ψ + ∂ν ψ + gs ψ = h, t ∈ J, x ∈  s ψ = ψ0 t = 0, x ∈ . 3

(4.1)

globally in time in the setting established in previous sections. In particular, we would like to show that the solution defines a semiflow in a suitable phase space M (see Theorem 4.3). Here we finally have to restrict our considerations to dimensions n < 4. For this purpose let data ψ0 ∈ X p , f ∈ X (T ), g ∈ Y (T ) and h ∈ X  (T ) be given such that the compatibility conditions (3.2) are satisfied. By Theorem 3.1 there is a unique solution on some maximal time interval Jmax = [0, amax ). We fix some arbitrary 0 < a < amax . To prove the global existence, it is crucial to use the following well-known energy functional (cf. [5] or [10]):

  1 4 1 2 2 [σs |∇|| ψ|2 + gs |ψ|2 ] ds, |∇ψ| − |ψ| + |ψ| dx + 2 2   (4.2) where ds means the surface measure on  and ∇|| is the surface gradient. Computing the time derivative of F(u(t)), we obtain by integration by parts, employing 1 F[ψ] := 2

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the boundary condition and using µ = −ψ − ψ + ψ 3  d F[ψ(t)] = [∂t ∇ψ · ∇ψ − ψ∂t ψ + ψ 3 ∂t ψ] dx dt   + [σs ∂t ∇|| ψ∇|| ψ + gs ψ∂t ψ] ds    ∂t ψ[−ψ − ψ + ψ 3 ]dx + ∂t ψ[∂ν ψ − σs || ψ + gs ψ] ds =      1 = (µ + f )µdx + ∂t ψ h − ∂t ψ ds s     1 [|∂t ψ|2 − ∂t ψh − µg] ds, = − [|∇µ|2 − µf ]dx −    s   3 2 ≤− |∇µ| dx − |∂t ψ|2 ds + s |h|2 ds 4s    

 + µf dx + µg ds . 



In the last step we have used Young’s inequality. The embedding H21 () → L 2 (), and Poincar´e’s second inequality   2     2 2 |v| dy ≤ C |∇v| dy +  vdy  





imply further          µf dx  ≤ C| f |2 |∇µ|2 +  µdx  ,               µg  ≤ C|g|2, |∇µ|2 +  µdx  ,     



where | · | p and | · | p, denote the L p -norms on  and on. , respectively. Since    3 µdx = (ψ − ψ)dx + (−h + ∂t ψ + gs ψ)ds, 

we obtain





       µdx  ≤ C |ψ|3/4 + |h|2, + |∂t ψ|2, + |ψ|2, . 4   

Combining these estimates, we deduce from Young’s inequality that  d 1 F[ψ(t)] ≤ − |∇µ|22 + |∂t ψ|22, dt 2   + C |ψ|44 + |ψ|22, + | f |42 + |g|42, + |h|22, . By Gronwall’s inequality, this implies the following a priori estimates.

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Lemma 4.1. Let p ≥ 2 and suppose ψ is a solution of (4.1) on the maximal interval Jmax . Then there is a constant C > 0 depending only on T > 0, , and on the parameters gs and σs such that   sup |ψ(t)| H 1 () + |ψ(t)| L 4 () + |ψ(t)| H 1 () 2

t∈Jmax



+ 

tmax



tmax



0

  |µ(t)|2H 1 () + |∂t ψ|2L 2 () dt ≤ C |ψ0 | H 1 ()∩L 4 ()∩H 1 () 2

2

0

+

2

| f (t)|4L 2 () + |g(t)|4L 2 () + |h(t)|2L 2 ()



2

 dt .

Actually, in this section we will use only ψ ∈ L ∞ (Jmax ; H21 ()). The following lemma is an analogue of Lemma 3.2 and will lead to global existence. Lemma 4.2. Suppose p ∈ [2, ∞), n ≤ 4, and let ψ ∈ Z (a). Then there is a constant C > 0 independent of a > 0 such that ||ψ 3 || H 1/2 (L p

2 p )∩L p (H p )

≤ C||ψ||δZ (a) sup |ψ(t)|3−δ , 1 H2 ()

t∈Ja

where δ = 1 for n = 4 and δ < 1 in case n ≤ 3. Proof. The proof is very similar to that of Lemma 3.2. We estimate as follows, using the Gagliardo-Nirenberg inequality, |∂i ∂ j ψψ 2 | p ≤ |∂i ∂ j ψ|3 p |ψ|23 p ≤ C|∇ 4 ψ|α+2β |ψ| p

3−α−2β H21 ()

≤ C|ψ|δH 4 () sup |ψ(t)|3−δ , 1 p

H2 ()

t∈Ja

(4.3)

where δ = α + 2β and α, β ∈ [0, 1] are such that α(3 + n(1/2 − 1/ p)) ≥ 1 + n(1/2 − 1/3 p), β(3 + n(1/2 − 1/ p)) ≥ −1 + n(1/2 − 1/3 p). Similarly, |∂i ψ∂ j ψψ| p ≤ |∂i ψ|3 p |∂ j ψ|3 p |ψ|3 p ≤ C|∇ 4 ψ|2α+β |ψ| p

3−2α−β H21 ()

≤ C|ψ|δH 4 () sup |ψ(t)|3−δ , (4.4) 1 p

H2 ()

t∈Ja

where δ = 2α + β and α, β ∈ [0, 1] are such that α(3+n(1/2−1/ p)) ≥ n(1/2−1/3 p), β(3+n(1/2−1/ p)) ≥ −1+n(1/2−1/3 p). Integrating with respect to t and using H¨older’s inequality, the second part follows. 1/2 The estimate of ψ 3 in H p (L p ) is more involved. We begin with the estimate ||ψ 3 || H 1/2 (L p

p ))

≤ C||ψ|| H 1/2 (L pσ

pρ ))

||ψ||2L

pσ  (L pρ  )

,

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which follows from the characterization of H ps via differences and from H¨older’s inequality; here 1 1 2 2 +  = +  = 1. σ σ ρ ρ Next we use the Gagliardo-Nirenberg inequality to obtain β

||ψ|| L pσ  (L pρ  ) ≤ C||ψ|| Z ||ψ||

1−β 1/2

L ∞ (H2 )

,

where β(3 + n(1/2 − 1/ p)) ≥ n/2 − n/ pρ  − 1 and 0 ≤ β ≤ 1/σ  . We choose ρ  = 2n/ p if p ≤ n and ρ  = ∞ otherwise (then ρ  ∈ [1, ∞]), and set β = 1/σ  = [n − 2n/ pρ  − 2]+ /2(3 + n(1/2 − 1/ p)) to meet these requirements. Next with s = 1 − n/2 pρ  we have 
s Z → H p1−θ J ; H p4θ () → H pσ (J ; L pρ ()), hence complex interpolation yields with α = 1/2s ||ψ|| H 1/2 (L pσ

provided

pρ ))

≤ C||ψ||αZ ||ψ||1−α L pτ (L pρ) ,

1/σ ≥ α + (1 − α)/τ.

(4.5)

Note that s ∈ {3/4, 1}, hence α ∈ {2/3, 1/2}. Employing the Gagliardo-Nirenberg inequality one more time, we get ||ψ|| L pτ (L pρ ) ≤ C||ψ||κZ ||ψ||1−κ

1/2

L ∞ (H2 )

,

with κ(3 + n(1/2 − 1/ p)) ≥ n/2 − n/ pρ − 1, 0 ≤ κ ≤ 1/τ . We choose 1/τ = κ = [n/2 − n/ pρ − 1]+ /(3 + n(1/2 − 1/ p)), and verify that for these choices the inequality (4.5) is valid if and only if n ≤ 4, and the strict inequality holds if and only if n ≤ 3, namely: For p ≤ n we have ρ  = 2n/ p, s = 3/4, α = 2/3, hence (4.5) is equivalent to 1−

[n − 3]+ 2 n(1/2 − 1/ p) ≥ + 3 + n(1/2 − 1/ p) 3 3(3 + n(1/2 − 1/ p))

which is equivalent to For p > n we have

ρ

[n − 3]+ ≤ 1. = ∞, s = 1, α = 1/2, hence (4.5) is equivalent to

3 + n(1/2 − 1/ p) − 2[n − 2]+ ≥ [n(1/2 − 1/ p) − 1]+ and the claim follows easily distinguishing the two cases of n(1/2 − 1/ p) being larger than or less than 1, respectively. Combining these estimates, we obtain ||ψ 3 || H 1/2 (L p

p ))

≤ C||ψ||δZ ||ψ||3−δ

L ∞ (H21 )

,

(4.6)

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with

δ = 2β + α + (1 − α)κ ≤ 2/σ  + 1/σ = 1,

in fact δ < 1 if strict inequality holds in (4.5) which is the case if n ≤ 3. Integrating (4.3) and (4.4) with respect to t and using H¨older’s inequality, and combining it with (4.6) proves the lemma.
It is now easy to prove global existence for p ≥ 2 and n ≤ 3. For this purpose let ψ be a solution as above. Then by maximal L p -regularity, there is a constant C > 0 such that ||ψ|| Z (a) ≤ C + C||ψ 3 || H 1/2 (L p

2 p )∩L p (H p )

.

Using Lemmas 4.1 and 4.2 this yields ||ψ|| Z (a) ≤ C + C||ψ||δZ (a) , with a different constant C. Now, if n < 4, we have δ < 1, which yields a bound on ||ψ|| Z (a) independent of a < amax . This in turn proves also boundedness of ||ψ|| Z  (a) , i.e. tmax = T , hence the global existence follows. We summarize these considerations for the case r = 2 − 1/ p, s = 0 in Theorem 4.3. Let p ∈ [2, ∞), n ≤ 3, T > 0, and let the data 4−4/ p

ψ0 ∈ W p

4−3/ p

() ∩ W p

(),

f ∈ L p (J × ) ∩ L 4 (J ; L 2 ()),  1/4−1/4 p 1−1/ p g ∈ Wp (J ; L p ()) ∩ L p J ; W p () ∩ L 4 (J ; L 2 ()), 
2−1/ p () , h ∈ L p J ; Wp


be given such that the compatibility condition ∂ν ψ0 | + ∂ν ψ0 | = ∂ν ψ03 | + g|t=0 is valid. Then there is a unique global solution ψ of (4.1) on J = [0, T ] which satisfies 
ψ ∈ H p1 (J ; L p ()) ∩ L p J ; H p4 () , and

 

2−1/ p 4−1/ p ψ| ∈ H p1 J ; W p () ∩ L p J ; W p () .

If f, g, h are constant in t, the map ψ0 → ψ(t) defines a global semiflow on the nonlinear phase manifold   4−4/ p 4−3/ p M := ψ ∈ W p () : ψ| ∈ W p (), ∂ν [ψ + ψ − ψ 3 ]| = g . 4−4/ p

Note that for p < 5 the phase manifold is the linear space W p () ∩ 4−3/ p (), since for p < 5 no compatibility conditions are involved. In parWp 5/2 ticular, for p = 2 the phase space is H22 () ∩ H2 (). Observe also that by 4−4/ p Sobolev embedding for p > (n + 4)/4 W p () is a subset of the bounded continuous functions on , in particular for p = 2 and n ≤ 3.

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5 Existence of global attractors In this section we shall now prove the existence of global attractor for the semiflow in M2 := H 2 () ∩ H 5/2 () given by Theorem 4.3; moreover the existence of a semiflow in V := H 1 () ∩ H 1 () together with the existence of global attractor in V will be obtained simultaneously. Notice that the basic space V was introduced in the work [10]. We should mention that for the Cahn–Hilliard equation subject to the usual linear boundary conditions, i.e. ∂ν ψ| = ∂ν ψ| = 0, the problem of existence of global attractors has been extensively considered in the literature, e.g., in the book [11]. However, in the present paper we consider the problem of the global attractor for the Cahn–Hilliard equation subject to the new nonlinear and dynamic boundary condition (1.3). First we consider the existence of a global attractor in M2 for the problem (1.1). Based on the results in the previous section, the solution to the problem (1.1) defines a semiflow in M2 . Therefore, for the existence of a global attractor, we infer from Theorem 1.1 in Chapter 1 in the book [11], that it remains to verify : I. the existence of an absorbing set, II. the uniform compactness of the orbits. Since we work in the L 2 -setting, we use in this section the notation  ·  for the L 2 -norm, and H k for the Sobolev spaces H2k , with norm  ·  H k . Step I—part (a) Existence of an absorbing set Let ψ be the solution to the Cahn–Hilliard system ψt = µ

∂ν µ| = 0,

(5.1)

µ = −ψ + ψ 3 − ψ

(5.2)

1 ψt | = σs || ψ − ∂ν ψ − gs ψ + h s

(5.3)

ψ|t=0 = ψ0

(5.4)

with ψ0 ∈ M2 . In the above s , σs , gs and h are given constants. By the boundary condition ∂ν µ = 0 and (5.1) we have   ψ(t, ·)dx = ψ0 dx, t ≥ 0. 



Due to this mass conservation, in analogy with the treatment of the Cahn–Hilliard equation subject to the usual linear boundary conditions, we have to restrict ourselves to the metric spaces M2,γ defined as follows. For any given γ > 0, let      1    Vγ := ψ ∈ V  ψ dx| ≤ γ , M2,γ := M2 ∩ Vγ ||  

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  1 1 ψ(t, ·)dx = ψ0 dx. ||  ||  If − N denotes the usual realization of the negative Laplace operator in L 2 () according to Neumann boundary conditions, a multiplication of (5.1) in L 2 () by ¯ yields (− N )−1 (ψ − ψ)   ¯ ¯ ψt (− N )−1 (ψ − ψ)dx + (−ψ + ψ 3 − ψ)(ψ − ψ)dx = 0. and let

ψ¯ :=



H 1 ()|





:= {v ∈  v dx = 0} with inner product u, v∗ := Introducing ∇u ∇v dx, the norm  ·  given by −1    −1 ((− N ) h hdx)1/2 h−1 := H∗1



is just the norm in the dual space H −1 := (H∗1 )∗ . We then obtain, using (5.3),   1 d ¯ 2−1 + (|∇ψ|2 + ψ 4 − ψ 2 )dx + (σs |∇|| ψ|2 + gs ψ 2 − hψ)ds ψ − ψ 2 dt      1 3 ¯ ¯ + (gs ψ − h)ψ¯ ds. =− ψt (ψ − ψ)ds + (ψ − ψ)ψdx (5.5) s    Recalling from [5] or [10] the relevant energy functional (cp. (4.2))

    1 1 gs 2 1 σs F[ψ] := |∇ψ|2 + ψ 4 − ψ 2 dx + |∇|| ψ|2 + ψ − hψ ds, 4 2 2  2  2 (5.6) we have d 1 F[ψ] + ∇µ2 + dt s

 

|ψt |2 = 0,

(5.7)

where, as mentioned before,  ·  will denote the L 2 ()-norm in this section. For small η > 0 to be specified later, we multiply (5.5) by η and add it to (5.7), and obtain   d  1 η ¯ 2−1 + ∇µ2 + ψ 2 ds F[ψ] + ψ − ψ dt 2 s  t   2 4 2 + η [|∇ψ| + ψ − ψ ]dx + η [σs |∇|| ψ|2 + gs ψ 2 − hψ] dx      η 3 ¯ t ds + η (ψ − ψ)ψ¯ dx + η (gs ψ − h)ψ¯ ds. (5.8) (ψ − ψ)ψ =− s    Using Young’s inequality we have    η2 η 1 ¯ t ds ≤ ¯ 2 ds, − (ψ − ψ)ψ ψt2 ds + |ψ − ψ| s  2s   2s   1 3 ¯ (ψ − ψ)ψ dx ≤ ψ 4 dx + Cγ , 4  

(5.9) (5.10)

Cahn–Hilliard equation

643

where Cγ denotes here and in the sequel a positive constant depending only on γ and . Similarly, we have   gs 2 (gs ψ − h)ψ¯ ds ≤ (5.11) ψ ds + Cγ .   4 Let η :=

s gs . 4

Then we deduce from (5.8) that   η η d  2 ¯ ψ 4 dx ≤ Cγ . F[ψ] + ψ − ψ−1 + ηF[ψ] + dt 2 4 

(5.12)

Notice that ¯ 2−1 ≤ Cψ − ψ ¯ 2≤ ψ − ψ

1 2η

 

ψ 4 dx + Cγ ,

(5.13)

where C denotes here and in the sequel a positive constant depending only on . Thus,    η d  η ¯ 2−1 + η F[ψ] + ψ − ψ ¯ 2−1 ≤ Cγ , F[ψ] + ψ − ψ dt 2 2

(5.14)

which implies   C η η ¯ 2−1 + γ . ¯ 2−1 ≤ e−ηt F[ψ0 ] + ψ0 − ψ F[ψ] + ψ − ψ 2 2 η

(5.15)

Since 1 (5.16) ψ2V − C, 2 we conclude that there is a ball B0 in Vγ with radius depending only on γ ,  such that for any initial data in M2,γ which are in a V -bounded subset B ⊂ Vγ there is t1 = t1 (B) > 0 such that for all t ≥ t1 the orbits enter into B0 . Before continuing with the proof of the existence of an absorbing set in M2,γ , we turn to the question of uniform compactness. F[ψ] ≥

Step II: Uniform compactness of the orbits. It follows from (5.7) that  t  t 1 2 F[ψ] + ∇µ dτ + ψt2 ds dτ = F[ψ0 ].  s 0  0

(5.17)

For initial data varying in a given bounded set of Vγ we have F[ψ0 ] ≤ C,

(5.18)

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where C > 0 depends only on ψ0 V . Thus  t ∇µ2 dτ ≤ C, 0

 t 0



ψt2 ds dτ ≤ C,

(5.19)

ψ(t)V ≤ C.

F[ψ(t)] ≤ C,

(5.20)

Differentiating (5.2) with respect to t, we obtain µt = −ψt + 3ψ 2 ψt − ψt .

(5.21)

Multiplying (5.21) by ψt and integrating over  yields   1 d 1 d 3ψ 2 ψt2 dx + ψt2 ds ∇µ2 + ∇ψt 2 + 2 dt 2 dt s         2 2 2 ψt dx = µ ψt dx = − ∇µ∇ψt dx σs |∇|| ψt | + gs ψt ds = + 







1 1 ≤ ∇ψt 2 + ∇µ2 . 2 2

(5.22)

Multiplying (5.22) by t and integrating with respect to t, we obtain    t    1 t ∇µ2 + ψt2 ds + τ ∇ψt 2 + ψ 2 ψt2 dx s   0     2 2 + σs |∇|| ψt | + gs ψt ds dτ   t

  t  2 2 2 ≤C τ ∇µ dτ + ∇µ + ψt ds dτ  0 0   t

  t  2 2 2 ≤C t ∇µ + ∇µ + ψt ds dτ . 0



0

(5.23)

Thus, by (5.19), for t > 0, 1 ∇µ(t) + s



2



ψt2 ds

  1 ≤C 1+ , t

(5.24)

where C in this Step II always denotes a positive constant depending at most on ψ0 V . On the other hand, integration of (5.2) over  yields that

   1 µ dx = [ψ 3 − ψ]dx + ψt + gs ψ − h ds. (5.25)    s Using Young’s inequality, we conclude from (5.20), (5.24), (5.25) that for t ≥ δ > 0,      µ dx  ≤ Cδ , (5.26)  

Cahn–Hilliard equation

645

where Cδ denotes here and in the sequel a positive constant depending only on ψ0 V and on δ. It also follows from (5.23) that for t > 0,        1 t 1 τ ∇ψt 2 + ψ 2 ψt2 dx + [σs |∇ ψt |2 + gs ψt2 ]ds dτ ≤ C 1 + . t 0 t   (5.27) Differentiating (5.2) with respect to t and multiplying by ψtt in L 2 () yields     1 d 1 d 1 d 2 2 3 d 2 2 ψ ψt dx + ∇ψt  − ψt  + σs |∇|| ψt |2 + gs ψt2 ds 2 dt 2 dt 2 dt  2 dt    1 ψ 2 ds = 3 ψψt3 dx. (5.28) + ∇µt 2 + s  tt  Since n ≤ 3, we obtain by (5.20)     3   ψψt dx  ≤ C ψt  L 18 .  

5

Using the Gagliardo-Nirenberg inequality, we get
 ψt  L 18 ≤ C ψt 2H 1 ψt  + ψt 3 .

(5.29)

5

Multiplying (5.28) by t 2 and integrating with respect to t, we obtain      1 3 1 2 1 ψ 2 ψt2 dx + ∇ψt 2 − ψt 2 + t σs |∇|| ψt |2 + gs ψt2 ds 2 2 2  2 

  t  t   1  + ψ 2 ds dτ + τ ψt 2 dτ τ 2 ∇µt 2 + s  tt 0 0   t      = τ ∇ψt 2 + 3 ψ 2 ψt2 dx + σs |∇|| ψt |2 + gs ψt2 ds dτ 0    t  +3 τ2 ψψt3 dx dτ, (5.30) 0



i.e., by (5.29),      3 1 2 1 t ψ 2 ψt2 dx + ∇ψt 2 + σs |∇|| ψt |2 + gs ψt2 ds 2 2  2    t  t    1 + τ 2 ∇µt 2 + ψtt2 ds dτ + τ ψt 2 dτ  s  0 0   t      t2 2 2 2 2 2 2 σs |∇|| ψt | + gs ψt ds dτ ≤ ψt  + τ ∇ψt  +3 ψ ψt dx + 2   0  t   +C τ 2 ψt 2H 1 ψt  + ψt 3 dτ. (5.31) 0

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Using again  ψt 2 =





ψt µ dx = −



∇ψt ∇µ dx ≤ ∇ψt  ∇µ,

(5.32)

we have by (5.24)   t2 t2 1 2 2 2 ψt  ≤ ∇ψt  + Ct 1 + . 2 4 t

(5.33)

Moreover,  0

t

τ 2 ψt 2H 1 ψt dτ ≤ sup (τ ψt (τ ) H 1 ) 0≤τ ≤t



t

× 0



t

 τ ψt  dτ ≤

0

2

t

0



t

×

∇µ dτ

0

2

τ ψt 2H 1 dτ

 12 

t

τ ψt  dτ

≤ Ct

1 2

 12

,

(5.34)

0

τ ∇ψt  ∇µdτ ≤ t  12

2



1 2



t

0 t

τ ∇ψt  dτ 2

τ ∇ψt  dτ

 12

2

,

 12

(5.35)

0

where we have used (5.19). It follows from (5.27), (5.34), (5.35) that    t 1 τ 2 ψt 2H 1 ψt dτ ≤ sup (τ ψt (τ ) H 1 )Ct 1 + t 0 0≤τ ≤t   C 1 2 ≤ ε sup τ 2 ψt (τ )2H 1 + t 2 1 + (5.36) ε t 0≤τ ≤t with ε > 0 being a small constant to be specified below. Thus we obtain from (5.31)–(5.36), using (5.27), for t > 0       3 1 1 t2 ψ 2 ψt2 dx + ∇ψt 2 + σs |∇|| ψt |2 + gs ψt2 ds 4 2  2        1 C 1 1 2 ≤ Ct 2 1 + + Ct 1 + + t2 1 + t t ε t
2  2 + ε sup τ ψt (τ ) H 1 . (5.37) 0≤τ ≤t

Taking the supremum from 0 to t and taking ε small enough and fixed now, we deduce from (5.37) that      
2  1 1 1 2 2 2 2 sup τ ψt (τ ) H 1 ≤ Ct 1 + + Ct 1 + + Ct 1 + . (5.38) t t t 0≤τ ≤t

Cahn–Hilliard equation

647

holds. Thus it follows from (5.37) and (5.38) that for t ≥ δ > 0, where δ < 1 without loss of generality,       1 C ∇ψt (t)2 + ψ 2 ψt2 dx + 1+ . (5.39) σs (|∇|| |ψt |2 + gs ψt2 ds ≤ δ δ   By the elliptic estimate (cf. (4.11), [10])
 ψ(t) H 3 ≤ C µ H 1 + ψt  H 1/2 () + h H 1/2 () , we obtain, using (5.24), (5.26), (5.39), for t ≥ δ > 0, ψ(t) H 3 ≤ Cδ ,

(5.40)

with Cδ being a positive constant depending only on δ, γ and the norm of initial data in V . This means that for initial data belonging to M2 and staying in a bounded subset of Vγ , for t ≥ δ > 0 the orbits are uniformly bounded in H 3 (). To prove the uniform compactness in M2,γ = H 2 () ∩ H 5/2 () ∩ Vγ for t ≥ δ > 0, it suffices to prove that ψ(t) H 3 () is also uniformly bounded. From (5.39) we can conclude that for t ≥ δ > 0, ψt (t) H 1 () ≤ Cδ .

(5.41)

Thus, from the boundary condition (5.3) we have σs || ψ =

1 ψt + ∂ν ψ + σs ψ − h ∈ H1 (). s

By elliptic regularity for || , we get for t ≥ δ > 0,
 ψ(t) H 3 () ≤ C ψt (t) H 1 () + ∂ν ψ(t) H 1 () + ψ(t) H 1 () + C (5.42) ≤ Cδ , due to (5.40), (5.41). Thus (5.40) and (5.42) imply the uniform compactness of the orbits in M2,γ for t ≥ δ > 0. Step I—part (b): Existence of an absorbing set in M2,γ For initial data in any given bounded set in M2 , which naturally are in a bounded set B in V , by (5.15), (5.16), there is a ball B0 in V with radius r B0 depending only on γ , and t1 (B) such that for t ≥ t1 , the orbits enter B0 . Now by (5.40), for t ≥ t1 + δ > 0 with δ being fixed, e.g., δ = 1, ψ(t) M2 ≤ Cδ , Cδ depending only on r B0 and δ. Thus, the existence of an absorbing set in M2,γ follows. Combining the existence of an absorbing set in M2,γ and the uniform compactness of orbits in M2,γ proved in Steps I and II, respectively, by Theorem 1.1 in Chapter 1 in the book [11], we have proved

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Theorem 5.1. For any γ > 0 there is a global attractor Aγ in M2,γ for the semiflow associated to the Cahn–Hilliard system (5.1)–(5.4). Finally we want to prove that for initial data in V the Cahn–Hilliard system (5.1)–(5.4) admits a unique global solution which also defines a semiflow in V . This was not proven in the previous work by [10]. This is also the last step now needed to prove the existence of a global attractor. Since M2 is dense in V , for any initial data in V there is a sequence of initial data in M2 converging to the given initial data in V . Therefore, by the results in the previous section, there is a sequence of approximate solutions which belong to M2 for t ≥ 0. By a similar discussion as in the proof of the uniform compactness in M2 above, we can conclude that the approximate solutions will converge in M2 for any t > 0. Thus the limit is a global solution to the Cahn–Hilliard system (5.1)–(5.4). The uniqueness can also be easily proved by the energy method. In a similar way as deriving (5.7), we can easily show that the solution is also continuous in V for t ≥ 0. Thus the solutions defines a continuous semiflow in V . From (5.15), (5.16), (5.40) we can conclude the existence of an absorbing ball in Vγ as well as the uniform compactness of the orbits in Vγ for t ≥ δ > 0. Thus we have proved the following result. Theorem 5.2. The solution for the Cahn–Hilliard system (5.1)–(5.4) defines a continuous semiflow in V . Moreover, for any γ > 0 there is a global attractor Aγ in Vγ . Acknowledgements The author S. Zheng is supported by NSF of China under the grant No. 10371022.

References 1. Amann, H.: Linear and Quasilinear Parabolic Problems, Vol. I. Abstract Linear Theory. Monographs in Mathematics. Basel: Birkh¨auser (1995) 2. Cahn, J.W., Hilliard, E.: Free energy of a nonuniform system, I. Interfacial free energy. J. Chem. Phys. 28, 258–367 (1958) 3. Denk, R., Hieber, M., Pr¨uss, J.: Optimal L p − L q -regularity for vector-valued parabolic problems with inhomogeneous boundary data. Submitted. 4. Hieber, M., Pr¨uss, J.: Heat kernels and maximal L p - L q estimates for parabolic evolution equations. Commun. Part. Different. Eq. 22, 1647–1669 (1997) 5. Kenzler, R., Eurich, F., Maass, P., Rinn, B., Schropp, J., Bohl, E., Dieterich, W.: Phase separation in confined geometries: solving the Cahn–Hilliard equation with generic boundary conditions. Comput. Phys. Commun. 133, 139–157 (2001) 6. Ladyˇzhenskaya, O.A., Solonnikov, V.N., Uralt’seva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Moskow: Izdat: Nauka (1968) 7. Novick-Cohen, A., Segel, L.A.: Nonlinear aspects of the Cahn–Hilliard equation. Physica D 10, 277–298 (1984) 8. Pr¨uss, J.: Maximal regularity for abstract parabolic problems with inhomogeneous boundary data. Math. Bohemica 127, 311–327 (2002) 9. Pr¨uss, J.: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Basel: Birkh¨auser (1993) 10. Racke, R., Zheng, S.: The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110 (2003) 11. Temam, R.: Infinite-dimensional dynamical systems in mechanics anc physics. Appl. Math. Sci. 68 (1988)