Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2005; 28:709–735Published online 29 November 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mma.590MOS subject classi�cation: 35B 40; 35B 45; 37 L 30

Exponential attractors for the Cahn–Hilliard equation withdynamic boundary conditions

A. Miranville1;∗;† and S. Zelik2

1Laboratoire de Math�ematiques et Applications; Universit�e de Poitiers; UMR 6086—CNRS; SP2MI;Boulevard Marie et Pierre Curie; 86962 Chasseneuil Futuroscope Cedex; France

2Institut f�ur Analysis; Dynamik und Modelierung; Universit�at Stuttgart; Pfa�enwaldring 57;70569 Stuttgart; Germany

Communicated by W. Wendland

SUMMARY

We consider in this article the Cahn–Hilliard equation endowed with dynamic boundary conditions.By interpreting these boundary conditions as a parabolic equation on the boundary and by consideringa regularized problem, we obtain, by the Leray–Schauder principle, the existence and uniqueness ofsolutions. We then construct a robust family of exponential attractors. Copyright ? 2005 John Wiley& Sons, Ltd.

KEY WORDS: Cahn–Hilliard equation; dynamic boundary conditions; Leray–Schauder principle; robustexponential attractors

0. INTRODUCTION

The Cahn–Hilliard equation

@t�+ ��2x� − ��xf(�)=0; �; �¿0

is very important in materials science. This equation describes important qualitative featuresof two-phase systems, in particular, the spinodal decomposition, i.e. a rapid separation ofphases when the material is cooled down su�ciently. We refer the reader to References [1,2]for more details. Here, � is the order parameter, � is related to the surface tension at theinterface, � is the mobility (we shall take �=�≡ 1 in what follows) and f is the derivative

∗Correspondence to: A. Miranville, Laboratoire de Math�ematiques et Applications, Universit�e de Poitiers,UMR 6086—CNRS SP2MI, Boulevard Marie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France.

†E-mail: [email protected]

Contract/grant sponsor: INTAS; contract/grant number: 00-899Contract/grant sponsor: CRDF; contract/grant number: 10545

Copyright ? 2005 John Wiley & Sons, Ltd. Received June 2004

710 A. MIRANVILLE AND S. ZELIK

of a double-well potential (a logarithmic potential; however, such a logarithmic potential isusually approximated by a polynomial of degree four, see Reference [2]).This equation has been much studied and one has now satisfactory existence and uniqueness

results, as well as results on the long-time behaviour of the solutions. We refer the reader tothe review articles [3–5] and the references therein.In all these studies, the Cahn–Hilliard equation is endowed with Neumann or periodic

boundary conditions. Now, physicists have recently considered the study of phase separationin con�ned systems (see References [6–8]). In that case, one has to account for the interactionswith the walls, which leads to additional terms in the free energy and then to the so-calleddynamic boundary conditions (in the sense that the quantity @t� appears in the boundaryconditions).The Cahn–Hilliard equation, endowed with dynamic boundary conditions, has been studied

in References [9–12]. In particular, for a polynomial potential of degree four, the existenceand uniqueness of solutions is proven in Reference [10]. In order to overcome the newmathematical di�culties due to the boundary conditions, the authors introduce an approximateproblem and use results on elliptic operators with highest-order derivatives in the boundaryconditions (which still form elliptic boundary value problems in the sense of H�ormander).We can note that it is not clear, in Reference [10], whether the solutions de�ne a continuoussemigroup in the phase space considered. This problem is overcome in Reference [9], where,based on the maximal Lp-regularity of the solutions, the authors prove that the solutions de�nea continuous semigroup in certain Sobolev spaces and then obtain the existence of the globalattractor.In this article, we consider a third approach in order to handle the problem. More precisely,

we propose to treat the dynamic boundary conditions as a separate parabolic equation on theboundary. We then prove the existence and uniqueness of solutions by the Leray–Schauderprinciple. Actually, following Reference [13], we regularize the problem by considering theviscous Cahn–Hilliard equation (see Reference [14]), i.e. we add the term −�@t�x�, �¿0,to the equation. Compared with the results of [9,10], this approach allows us to considerpotentials with arbitrary growths. Furthermore, we are also able to consider a non-linearterm in the dynamic boundary conditions. Finally, we are able to construct a robust (as�→ 0) family of exponential attractors M� for the problem. As a consequence, we obtain theexistence of the global attractors A� with �nite fractal dimension (we note that the �nitedimensionality of the global attractor was not studied in Reference [9]).We now introduce the problem and give the main assumptions for our study.We consider in this article the following Cahn–Hilliard problem in a bounded smooth

domain �⊂R3:@t�=�x�; @n�|@� =0; �= −�x�+ �@t�+ f(�); �|t=0 =� 0 (1)

where � and � are unknown functions (� is the chemical potential), �x is the Laplacian,�¿0 is a small parameter and f is a given non-linear interaction function.Equation (1) is equipped with the following dynamic boundary conditions:

@t�=�‖� − �� − g(�)− @n�; x∈ @� (2)

where �‖ is the Laplace–Beltrami operator on the boundary @�, g is another given non-linearfunction and � is some given positive constant.

Copyright ? 2005 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2005; 28:709–735

EXPONENTIAL ATTRACTORS FOR THE CAHN–HILLIARD EQUATION 711

We assume that the non-linearities f and g belong to C2(R;R) and satisfy the followingstandard dissipativity assumptions:

lim inf|v|→∞

f′(v)¿0; lim inf|v|→∞

g′(v)¿0 (3)

We also recall that system (1)–(2) possesses a natural conservation law:

〈�(t)〉 ≡ 〈� 0〉=M0 (4)

where 〈:〉 denotes the average over �. Moreover, if the value of �(t) is known for somet=T , then the value of �(T ) can be found by solving the following linear elliptic problem:

�(T )− ��x�(T )= −�x�(T ) + f(�(T )); @n�(T )|@� =0 (4′)

Thus, we only need to �nd the function �(t). It is however more convenient to introduce onemore unknown function, namely, (t) :=�(t)|@�, de�ned on the boundary @�, and to rewritethe above system as follows:

@t�=�x�; @n�|@� =0�= −�x�+ �@t�+ f(�); �|t=0 =� 0

@t =�‖ − � − g( )− @n�; x∈ @�; |t=0 = 0

�|@� =

(5)

where the boundary condition (2) is now interpreted as an additional second-order parabolicequation on the boundary @�.Following Reference [13], we introduce the following ‘natural’ phase space for

problem (5):

D� := {(�; )∈H 2(�)× H 2(@�); �∈H 1(�);

�1=2�∈H 2(�); �|@� = ; @n�|@� =0} (6)

(where � is computed from � via (4′)), with the obvious norm

‖(�; )‖2D�:= ‖�‖2H 2(�) + ‖�‖2H 1(�) + �‖�‖2H 2(�) + ‖ ‖2H 2(�) (7)

A solution of problem (5) is a pair of functions (�(t); (t)) belonging to the spaceL∞([0; T ];D�) with @t�∈L2([0; T ]; H 1(�)) and @t ∈L2([0; T ]; H 1(@�)) and which satis�esthe equations in the sense of equalities in the spaces L2([0; T ]; L2(�)) and L2([0; T ]; L2(@�)).

Remark 0.1We recall that, due to the embedding H 2 ⊂C, we have f(�)∈C([0; T ] × �) and g( )∈C([0; T ]; L2(@�)) and the non-linearities in (5) are well-de�ned. Moreover, since@t�∈L2([0; T ]; H 1(�)), then �∈L2([0; T ]; H 3(�)) and thus the boundary conditions are alsowell-de�ned.



We also need to introduce the weak energy spaces L� for problem (5) de�ned by thefollowing norms:

‖(�; )‖2L� := �‖�‖2L2(�) + ‖�‖2H−1(�) + ‖ ‖2L2(@�) (8)

where H−1(�) := [H 1(�)]′ endowed with the norm

‖v‖2H−1(�) = ‖(−�x)−1=2N (v − 〈v〉)‖2L2(�) + 〈v〉2

(−�x)−1N : L20(�)→L20(�) being the inverse Laplacian with Neumann boundary conditionsde�ned on the subspace of functions with null average. We note that, for � �=0, the space L�coincides with L2(�)× L2(@�) and, for �=0, we have L0 =H−1(�)× L2(@�).

1. A PRIORI ESTIMATES

The main task of this section is to derive uniform (with respect to �→ 0) a priori estimatesfor the solutions of (5) in the phase space D�. The existence of a solution will be veri�ed inthe next section.We derive the required estimates in three steps. In a �rst step, we obtain a divergent

estimate in D�. In a second step, we derive dissipative estimates in the weak energy space L�and, in a third step, we prove some kind of a smoothing property which allows to eliminatethis divergence and obtain dissipative estimates in D�.

Theorem 1.1Let the non-linearities f and g satisfy assumptions (3) and let a pair of functions (�(t); (t))be a solution of problem (5). Then, the following estimates hold:

‖(�(t); (t))‖2D�+∫ t

0(‖@t�(s)‖2H 1(�) + ‖@t (s)‖2H 1(�)) ds

6Q(‖(�(0); (0))‖2D�eKt) (9)

where the positive constant K and the monotonic function Q depend on M0 (introduced in(4)), but are independent of �.

ProofWe di�erentiate (5) in t and set (u(t); v(t); w(t)) := @t(�(t); �(t); (t)). Then, we have

@tu=�xv; @nv|@� =0v+ �@tu= −�xu+ f′(�)u; u|@� =w

@tw=�‖w − �w − g′( )w − @nu

(10)

We then note that, due to (4), we have 〈u〉=0. Consequently, multiplying the �rst equationof (10) scalarly by (−�x)−1N u, the second equation by u and the third one by and taking



the sum of the equations that we obtain, we have the following identity:

1=2@t(‖u(t)‖2H−1(�) + �‖u(t)‖2L2(�) + ‖w(t)‖2L2(@�))

+�‖w(t)‖2L2(@�) + ‖∇xu(t)‖2L2(�) + ‖∇‖w(t)‖2L2(@�)= (f′(�(t))u(t); u(t)) + (g′( (t))w(t); w(t))@� (11)

where (·; ·) and (·; ·)@� denote the inner products in L2(�) and L2(@�), respectively, and ∇‖is the covariant gradient at @�. We now note that, due to assumptions (3), f′(�)¿−K andg′( )¿− K for some constant K . Consequently, applying Gronwall’s inequality to (11) andusing the interpolation inequality ‖u‖2L2(�)6C‖u‖H−1(�)‖∇xu‖L2(�), we have

‖u(t)‖2H−1(�) + �‖u(t)‖2L2(�) + ‖w(t)‖2L2(@�)

+∫ t

0(‖∇xu(s)‖2L2(�) + ‖∇‖w(s)‖2L2(@�)) ds

6CeK1t(‖u(0)‖2H−1(�) + �‖u(0)‖2L2(�) + ‖w(0)‖2L2(@�)) (12)

where the positive constants C and K1 are independent of �. We now recall that u= @t�=∇x�.Consequently, (12) can be rewritten as follows:

‖∇x�(t)‖2L2(�) + �‖�x�(t)‖2L2(�) + ‖@t (t)‖2L2(@�)6CeK1t(‖∇x�(0)‖2L2(�) + �‖�x�(0)‖2L2(�) + ‖@t (0)‖2L2(@�)) (13)

So, in order to obtain the �-part of estimate (9), we only need to estimate the average of�(t). To this end, averaging the expression for � from the second equation of (5), we have

〈�(t)〉= − 〈�x�(t)〉+ 〈f(�(t))〉=−〈@n�(t)〉@� + 〈f(�(t))〉= 〈� (t) + g( (t))〉@� − 〈@t (t)〉@� + 〈f(�(t))〉 (14)

where we denote by 〈·〉@� the average value on the boundary and have used the third equationof (5) in order to �nd the average of @n�. Thus, since the average of @t is already estimatedin (13), we only need to estimate the averages of , f(�) and g( ). In order to do so,we introduce the functions ��(t) :=�(t) − 〈�(t)〉=�(t) − M0 and � (t) := (t) − M0. Then,multiplying the expression for � by ��(t), we have, after standard transformations,

‖∇x�(t)‖2L2(�) + ‖∇‖ (t)‖2L2(@�) + (f(�(t)); ��(t)) + (g( (t)); � (t))@�+�( (t); (t)− M0)@�

= (�(t); �(t))− (�@t�(t); ��(t))− (@t (t); � (t))L2(@�) (15)



We note that the �rst term on the right-hand side of (15) can be estimated by 1=2‖∇x�‖2L2 +C‖∇x�(t)‖2L2 (since 〈 ��〉=0). Consequently, estimates (12), (13) and (15) give

‖∇x�(t)‖2L2(�) + ‖∇‖ (t)‖2L2(@�)+2(f(�(t)); ��(t)) + 2(g( (t)); � (t))@�6C(‖(�(0); (0))‖2D�

+ 1)eK1t (16)

It remains to note that assumptions (3) imply that

1=2|f(v)|(1 + |v|)6f(v)(v − M0) + Cf;M0(17)

1=2|g(v)|(1 + |v|)6 g(v)(v − M0) + Cg;M0 ∀v∈R

and, consequently, (16) yields

‖ (t)‖L1(@�) + ‖f(�(t))‖L1(�) + ‖g( (t))‖L1(@�)

6C(1 + ‖(�(0); (0))‖2D�)eK1t (18)

Thus, the required average estimates for the terms , f(�) and g( ) are obtained and (14)now implies that

|〈�(t)〉|6C(1 + ‖(�(0); (0))‖D�2)eK1t (19)

Consequently (taking into account (13) and (19)), the �-part of estimate (9) is veri�ed. Inorder to obtain the required estimates for the H 2-norms of � and and thus complete theproof of Theorem 1.1, we rewrite (for every �xed t) problem (5) as a second-order non-linearelliptic boundary value problem:{

�x� − f(�)= h1(t) := − �(t) + �@t�(t); �|@� =

�‖ − � − g( )− @n�= h2(t) := @t (t)(20)

Indeed, according to (13), (16) and (19), we have

‖h1(t)‖2L2(�) + ‖h2(t)‖2L2(@�)6C(1 + ‖(�(0); (0))‖2D�)eK1t (21)

Applying now the maximum principle (see Appendix A, Lemma A.2) to this problem, wededuce that

‖�(t)‖2L∞(�) + ‖ (t)‖2L∞(@�)6C1(1 + ‖(�(0); (0))‖2D�)eK1t (22)

Finally, having the L∞-estimates (22), we can interpret the non-linearities f(�) and g( ) asexternal forces as well and apply the H 2-regularity theorem (see Appendix A, Lemma A.1)to the linear elliptic system that we obtain. This yields that

‖�(t)‖2H 2(�) + ‖ (t)‖2H 2(@�)6Q(‖(�(0); (0))‖2D�eK1t) (23)

for some monotonic function Q which depends on f, g and M0, but is independent of �, andTheorem 1.1 is proven.



In a next step, we deduce dissipative estimates for the solutions of (5) in the weak energyspace introduced in (8).

Proposition 1.1Let the assumptions of Theorem 1.1 hold. Then, every solution (�(t); (t)) of (5) satis�esthe following estimates:

�‖�(t)‖2L2(�) + ‖�(t)‖2H−1(�) + ‖ (t)‖2L2(@�)

+∫ t+1

t(‖�(s)‖2H 1(�) + ‖ (s)‖2H 1(@�) + ‖F(�(s))‖L1(�) + ‖G( (s))‖2L1(@�)) ds

6C(�‖�(0)‖2L2(�) + ‖�(0)‖2H−1(�) + ‖ (0)‖2L2(@�))e−�t + C1 (24)

where F(v) :=∫ v0 f(u) du, G(v) :=

∫ v0 g(u) du and the positive constants C, C1 and � are

independent of � and t.

ProofLet ��(t) :=�(t) − M0 and � (t) := (t) − M0 be the same as in the proof of Theorem 1.1.Then, recalling that � = 〈�〉 − (−�x)−1N @t� and using the fact that 〈 ��〉=0, we derive from(15) that

1=2@t(‖ ��(t)‖2H−1(�) + �‖ ��(t)‖2L2(�) + ‖ � (t)‖2L2(@�))

+ ‖∇x�(t)‖2L2(�) + ‖∇‖ (t)‖2L2(@�)+ (f(�(t)); ��(t)) + (g( (t)); � (t))@� + �( (t); (t)− M0)@� = 0 (25)

Moreover, due to estimates (17), we obtain

@t(‖ ��(t)‖2H−1(�) + �‖ ��(t)‖2L2(�) + ‖ � (t)‖2L2(@�))

+2�(‖ ��(t)‖2H 1(�) + ‖ � (t)‖2H 1(@�)) + ‖∇x�(t)‖2L2(�) + ‖∇‖ (t)‖2L2(@�)+ (|f(�(t))|; 1 + |�(t)|) + (|g( (t))|; 1 + | (t)|)@�6C (26)

for some positive constants C and � which are independent of � (here, we have implicitlyused the fact that ‖�‖H 1(�)6C(‖∇x�‖L2(�) + ‖ ‖H 1(@�))). Applying Gronwall’s inequality to(26), we have

�‖�(t)‖2L2(�) + ‖�(t)‖2H−1(�) + ‖ (t)‖2L2(@�)

+∫ t+1

t(‖�(s)‖2H 1(�) + ‖ (s)‖2H 1(@�)) ds



+∫ t+1

t((|f(�(s))|; 1 + |�(s)|) + (|g( (s))|; 1 + | (s)|)@�) ds

6C(�‖�(0)‖2L2(�) + ‖�(0)‖2H−1(�) + ‖ (0)‖2L2(@�))e−�t + C1 (27)

In order to deduce (24) from (27), it remains to note that assumptions (3) imply that

|F(v)|6|f(v)|(1 + |v|)− C; |G(v)|6|g(v)|(1 + |v|)− C (28)

for some positive C and for every v∈R (indeed, according to (3), the functions f and g aremonotonic if |v| is large enough). Proposition 1.1 is proven.In a third step, we establish a smoothing-type property for the solutions of (5) as follows.

Theorem 1.2Let the assumptions of Theorem 1.1 hold and let (�; ) be a solution of problem (5). Then,for every T ∈ (0; 1), the following estimate holds:

‖(�(t); (t))‖2D�6QT (�‖�(0)‖2L2(�) + ‖�(0)‖2H−1(�) + ‖ (0)‖2L2(@�)); t ∈ [T; 1] (29)

where the monotonic function QT depends obviously on T¿0, but is independent of t and �.

ProofWe �rst note that, in order to verify (29), it is su�cient to verify only that there exists atleast one time T0 ∈ [T=2; T ] (which can depend on the solution (�(t); (t))) such that

‖@t�(T0)‖2H−1(�) + �‖@t�(T0)‖2L2(�) + ‖@t (T0)‖2L2(@�)6CT−1(1 + �‖�(0)‖2L2(�) + ‖�(0)‖2H−1(�) + ‖ (0)‖2L2(@�)) (30)

for some constant C which is independent of � and (�(t); (t)). Indeed, if (30) is known,then, di�erentiating equation (5) with respect to t and arguing exactly as in the derivation of(12), we verify that the analogue of (30) holds for every t ∈ [T0; 1]. Having this estimate, wecan obtain the analogue of estimate (19) for 〈�(t)〉 (exactly as in (14)–(19)) and then derivethe required H 2-estimates by applying the elliptic regularity theorem to problem (20) (see(21)–(23)). Thus, it only remains to verify (30). To this end, multiplying the �rst equationof (5) scalarly by (−�x)−1N @t�(t), using the fact that 〈@t�(t)〉=0 and the expression for �and integrating by parts, we have

‖@t�(t)‖2H−1(�) =− (�(t); @t�(t))= (�x�(t); @t�(t))− �‖@t�(t)‖2L2(�)− (f(�(t)); @t�(t))

=−1=2@t(‖∇x�(t))‖2L2(�) + 2(F(�(t)); 1))

− �‖@t�(t)‖2L2(�) + (@n�(t); @t�(t))@� (31)



Expressing @n� from the third equation of (5), inserting this expression into the last term offormula (31) and integrating by parts again, we �nd

@t(‖∇x�(t)‖2L2(�) + 2(F(�(t)); 1) + ‖∇‖ (t))‖2L2(@�)+2( �G( (t)); 1)@�) + 2‖@t�(t)‖2H−1(�) + 2�‖@t�(t)‖2L2(�) + 2‖@t (t)‖2L2(@�) = 0 (32)

where �G(v) :=G(v) + �=2v2. Multiplying now (32) by t and integrating over [0; t], t ∈ [0; 1],we have

t(‖∇x�(t)‖2L2(�) + 2(F(�(t)); 1) + ‖∇‖ (t)‖2L2(@�) + 2( �G( (t)); 1)@�)

+2∫ t

0s(‖@t�(s)‖2H−1(�) + �‖@t�(s)‖2L2(�) + ‖@t (s)‖2L2(@�)) ds

=∫ t

0(‖∇x�(s)‖2L2(�) + 2(F(�(s)); 1) + ‖∇‖ (s)‖2L2(@�) + 2( �G( (s)); 1)@�) ds (33)

Using �nally inequality (24) in order to estimate the right-hand side of (33), we deduce that∫ t

0s(‖@t�(s)‖2H−1(�) + �‖@t (s)‖2L2(�) + ‖@t (s)‖2L2(@�)) ds

6C(�‖�(0)‖2L2(�) + ‖�(0)‖2H−1(�) + ‖ (0)‖2L2(@�)) (34)

where C is independent of � and (�(t); (t)). Estimates (30) follow immediately from (34)and Theorem 1.2 is proven.

We are now ready to obtain a dissipative estimate for the D�-norm of the solution(�(t); (t)).

Corollary 1.1Let the assumptions of Theorem 1.1 hold. Then, every solution (�(t); (t)) of problem (5)satis�es the following estimate:

‖(�(t); (t))‖D�6Q(‖(�(0); (0))‖D�)e−�t + C∗ (35)

where the positive constants C∗ and � and the monotonic function Q are independent of t,u(t) and �.

Indeed, in order to deduce (35), it is su�cient to use (9) for t61 and (24) and (29) fort¿1.In the sequel, we shall also need the uniform bounds on the solutions in H 3(�)×H 3(@�)

which are formulated in the next theorem.

Theorem 1.3Let the assumptions of Theorem 1.1 hold and let (�(t); (t)) be a solution of problem(5). Then, (�(t); (t))∈H 3(�) × H 3(@�) for every t¿0 and the following estimates



hold:

‖�(t)‖H 3(�) + ‖ (t)‖H 3(@�)6t−1=2Q(‖(�(0); (0))‖D�); t ∈ (0; 1] (36)

where the monotonic function Q is independent of t and �.

ProofWe only give a formal derivation of (36), which can be easily justi�ed in a standard way(using, e.g. t-regularizations of the solutions of (5) via ��(t) :=

∫∞0 K�(t − s)�(s) ds, where

the kernel K� is smooth and satis�es supp K� ⊂ [0; �], ∫∞0 K�(z) dz=1, and then passing to

the limit �→ 0). As in the proof of Theorem 1.1, we di�erentiate Equations (5) with respectto t and set (u(t); v(t); w(t)) := @t(�(t); �(t); (t)). These functions satisfy equations (10). Wemultiply scalarly the �rst, second and third equations of (10) by t(−�x)−1@tu(t), t@tu(t) andt@tw(t), respectively, and take the sum of the equations that we obtain. Then, after standardtransformations, we have

1=2@t(t‖∇xu(t)‖2L2(�) + t‖∇‖w(t)‖2L2(@�) + �t‖w(t)‖2L2(@�))

+t‖@tu(t)‖2H−1(�) + �t‖@tu(t)‖2L2(�) + t‖@tw(t)‖2L2(@�)=1=2(‖∇xu(t)‖2L2(�) + ‖∇‖w(t)‖2L2(@�) + �‖w(t)‖2L2(@�))+(f′(�(t))u(t); t@tu(t)) + (g′( (t))w(t); t@tw(t))@� (37)

We estimate the (most complicated) second term on the right-hand side of (37) as follows:

|(f′(�(t))u(t); t@tu(t))|6Ct‖f′(�(t))u(t)‖H 1(�)‖@tu(t)‖H−1(�)

6t‖@tu(t)‖2H−1(�) +Q(‖�(t)‖H 2(�))‖@t�(t)‖2H 1(�) (38)

and the last term on the right-hand side of (37) can be estimated analogously (the proof ishowever simpler, since the negative norms are not necessary). Integrating now (37) in t andusing estimate (9), we deduce

t(‖@t�(t)‖2H 1(�) + ‖@t (t)‖2H 1(@�))6Q1(‖(�(0); (0))‖D�); t ∈ (0; 1] (39)

for some monotonic function Q1 which is independent of � and t. Having obtained estimate(39), we can rewrite problem (5) as a linear elliptic boundary value problem:{−�x�(t)= h1(t) :=�(t) + �@t�(t)− f(�(t)); �(t)|@� = (t)

−�‖ (t) + � (t) + @n�(t)= h2(t) := − @t (t)− g( (t))(40)

Moreover, according to (9) and (39), we have

‖h1(t)‖H 1(�) + ‖h2(t)‖H 1(@�)6t−1=2Q2(‖(�(0); (0))‖D�); t ∈ (0; 1] (41)

for some monotonic function Q2. Applying now the H 1–H 3 regularity theorem to the linearelliptic problem (40) (see (A8′)), we obtain (36) and �nish the proof of Theorem 1.3.



We conclude this section by verifying the uniqueness of a solution of (5) and the Lipschitzcontinuity with respect to the initial data.

Proposition 1.2Let the assumptions of Theorem 1.1 hold and let the functions (�1(t); 1(t)) and (�2(t); 2(t))be two solutions of (5) such that

〈�1(0)〉= 〈�2(0)〉 (42)

Then, the following estimates hold:

�‖�1(t)− �2(t)‖2L2(�) + ‖�1(t)− �2(t)‖2H−1(�) + ‖ 1(t)− 2(t)‖2L2(@�)6CeKt(�‖�1(0)− �2(0)‖2L2(�) + ‖�1(0)− �2(0)‖2H−1(�) + ‖ 1(0)− 2(0)‖2L2(@�)) (43)

where the positive constants C and K are independent of �, t and the initial data for thesolutions considered.

ProofWe set (�; �; ) := (�1 − �2; �1 − �2; 1 − 2). These functions satisfy the following problem:

@t�=�x�; @n�|@� =0�= −�x�+ �@t�+ [f(�1)− f(�2)]

@t =�‖ − � − [g( 1)− g( 2)]− @n�; x∈ @�

�|@� =

(44)

We also note that 〈�(t)〉=0 (thanks to assumption (42)). Consequently, multiplying the �rstequation of (44) by (−�x)−1N � and arguing as in the proof of Theorem 1.2, we have

1=2@t(‖�(t)‖2H−1(�) + �‖�(t)‖2L2(�) + ‖ (t)‖2L2(@�)) + ‖∇x�(t)‖2L2(�)+�‖ (t)‖2L2(@�) + ‖∇‖ (t)‖2L2(@�)=− ([f(�1(t))− f(�2(t))]; �(t))− ([g( 1(t))− g( 2(t))]; (t))@� (45)

It remains to note that, due to assumptions (3), f′(v)¿−K and g′(v)¿−K for some positiveconstant K . Consequently, (45) implies

@t(‖�(t)‖2H−1(�) + �‖�(t)‖2L2(�) + ‖ (t)‖2L2(@�))

+2‖∇x�(t)‖2L2(�)62K(‖�(t)‖2L2(�) + ‖ (t)‖2L2(@�)) (46)

Using now the interpolation inequality ‖ · ‖2L2(�)6C‖ · ‖H 1(�) · ‖ · ‖H−1(�) in order to estimatethe �rst term on the right-hand side of (46) and applying Gronwall’s inequality, we �nd (43)and �nish the proof of Proposition 1.2.



2. EXISTENCE OF SOLUTIONS

In this section, we establish the existence of a solution for problem (5). To this end, we �rstneed to study the following linear non-homogeneous problem, which corresponds to (5) withnull boundary conditions:

@t�=�x�; @n�|@� =0; �= −�x�+ �@t� − h(t) (47)

equipped with the null Dirichlet boundary condition �|@� =0 (here, h(t) corresponds to givenexternal forces and satis�es 〈h(t)〉 ≡ 0). It is however more convenient to rewrite this problemas a single equation for �, using the operator (−�x)−1N , as follows:

(�+ (−�x)−1N )@t�=�x� − 〈@n�〉@� + h(t); �|@� =0; �|t=0 =� 0 (48)

As usual, it is convenient to study Equation (48) by using the anisotropic Sobolev spacesW (1;2)

p (�T ), 1¡p¡∞, �T := [0; T ] × � (which are, by de�nition, the spaces of functionswhose t-derivative and x-derivatives up to the order 2 belong to Lp(�T )), associated withsecond-order parabolic operators. The next lemma gives the standard Lp-regularity estimatefor Equation (48).

Lemma 2.1We assume that �¿0, h∈Lp(�T ), with 〈h(t)〉 ≡ 0, and the initial datum � 0 ∈W 2(1−1=p)

p (�) forsome 26p¡∞. Then, problem (48) has a unique solution u(t)∈W (1;2)

p (�T ) and the followingestimate holds:

‖�‖W (1;2)p (�T )

6C(‖� 0‖W 2(1−1=p)p (�) + ‖h‖Lp(�T )) (49)

where the constant C depends on T and �, but is independent of �.

ProofWe �rst apply the operator (�+(−�x)−1N )

−1 to both sides of (48) and transform this equationas follows:

�@t�=�x� −K�+ �(�+ (−�x)−1N )−1h(t); �|@� =0 (50)

where

Kv := 〈@nv〉@� + (�+ (−�x)−1N )−1(−�x)−1N (�xv − 〈@nv〉@�) (51)

We then recall that, according to the classical Lp-regularity theory for the Laplacian (see, e.g.Reference [15] or [16]), we have

‖Kv‖Lp(�)6Cs;p‖v‖W 1+1=p+sp (�); ‖(�+ (−�x)−1N )

−1v‖Lp(�)6Cp‖v‖Lp(�) (52)

for all 1¡p¡∞ and s¿0. Thus, the linear equation (50) is a compact perturbation of aheat equation and, consequently, the existence and uniqueness of a solution can be veri-�ed in a standard way (using e.g. the Leray–Schauder �xed point theorem, see Reference[17]). So, we restrict ourselves to verifying estimate (49) only. To this end, we �rst multiply



Equation (50) scalarly by �x�(t) and use the following estimate:

|(K�;�x�)|6C‖K�‖2L2(�) + 1=4‖�x�‖2L2(�)6C1‖�‖2H 2−�(�) + 1=4‖�x�‖2L2(�)6C1‖�‖2H 1(�) + 1=2‖�x�‖2L2(�) (53)

for some 0¡�¡ 12 (here, we have used (52) with p=2 and s= 1

2 − �). Then, we have

�=2@t‖∇x�(t)‖2L2(�)6C‖∇x�(t)‖2L2(�) + C‖h(t)‖2L2(�)and, applying Gronwall’s inequality to this relation, we infer

‖�(t)‖2H 1(�)6CeKt(‖� 0‖2H 1(�) + ‖h‖2L2(�t)) (54)

We are now ready to �nish the derivation of estimate (49). To this end, we apply the classicalLp-regularity theorem for heat equations to (50), interpreting the term Kv as external forces.Then, using the second estimate of (52), we have

‖�‖W (1;2)p (�T )

6C(‖K�‖Lp(�T ) + ‖h‖Lp(�T ) + ‖� 0‖W 2(1−1=p)p (�)) (55)

We then estimate the �rst term on the right-hand side of (55) by using the �rst estimate of(52) with s= 1

2 − �, a proper interpolation inequality and estimate (54):

‖K�(t)‖Lp(�)6C‖�(t)‖W 2−�p (�)6Cv‖�(t)‖H 1(�) + v‖�(t)‖W 2

p(�) (56)

Fixing v¿0 small enough and inserting (56) into the right-hand side of (55), we infer (49)and �nish the proof of Lemma 2.1.

The next corollary gives the analogue of estimate (49) for non-homogeneous boundaryconditions.

Corollary 2.1We assume that �¿0 and that the function (t) belongs to the anisotropic Sobolev spaceW (1−1=(2p);2−1=p)

p (@�T ) for some 26p¡∞ (here and below, @�T := [0; T ] × @�). Then, theproblem

(�+ (−�x)−1N )@t�=�x� − 〈@n�〉@�; �|@� = ; �|t=0 =0 (57)

possesses a unique solution �∈W (1;p)p (�T ) and the following estimate holds:

‖�‖W (1;2)p (�T )

6C‖ ‖W (1−1=(2p);2−1=p)p (@�T )

(58)

where the constant C depends on � and T , but is independent of . Moreover,∫ t

0(@n�(s); (s))@� ds

=1=2‖�(t)‖2H−1(�) + �=2‖�(t)‖2L2(�) +∫ t

0‖∇x�(s)‖2L2(�) ds¿0 (59)



ProofThere exists a continuous linear extension operator

Tp :W (1−1=(2p);2−1=p)p (@�T )→W (1;2)

p (�T ) such that (Tp )|@�T = (60)

(see e.g. Reference [18]). Moreover, without loss of generality, we can also assume that

〈(Tp )(t)〉 ≡ 0; ∀t¿0; ∀ ∈W (1−1=(2p);2−1=p)p (@�T ) (61)

We now set v :=Tp . Then, the function satis�es Equation (48) with

h(t) := (�+ (−�x)−1N )@tv(t)−�xv(t) + 〈@nv(t)〉@�Applying the Lp-regularity estimate (49) to this equation and using (60) and (61), we obtainestimate (58). In order to deduce (59), it remains to multiply Equation (57) by �(t) and tointegrate in x and t. Corollary 2.1 is proven.

We are now ready to study the linear analogue of the complete system (5), which werewrite, analogous to (49), in the following form:

(�+ (−�x)−1N )@t�=�x� − 〈@n�〉@� + h1(t); �|t=0 =�0

@t =�‖ − � − @n�+ h2(t); |t=0 = 0

�|@� =

(62)

The following lemma is the analogue of Lemma 2.1 for this problem.

Lemma 2.2We assume that �¿0, that the external forces h1 and h2 belong to Lp(�T ) and Lp(@�T ),respectively, for some 26p¡∞ and that 〈h1(t)〉 ≡ 0. We also assume that the initial data �0and 0 belong to W 2(1−1=p)

p (�) and W 2(1−1=p)p (@�), respectively. Then, problem (62) possesses

a unique solution (�(t); (t)) and the following estimates hold:

‖�‖W (1; 2)p (�T )

+ ‖ ‖W (1; 2)p (@�T )

6C(‖�0‖W 2(1−1=p)p (�)

+‖ 0‖W 2(1−1=p)p (@�) + ‖h1‖Lp(�T ) + ‖h2‖Lp(@�T )) (63)

where the constant C depends on � and T , but is independent of (�; ) and (h1; h2).

ProofWe introduce the operator T :W (1−1=(2p);2−1=p)

p (@�T )→W (1;2)p (�T ) as the unique solution of

problem (57) and set v(t) := (T )(t) and �(t) :=�(t)− v(t). Then, system (62) reads, in thenew variables (�; ),

(�+ (−�x)−1N )@t�=�x� − 〈@n�〉@� + h1(t); �|t=0 =�0; �|@� =0@t =�‖ − � − @nv(t) + h2(t)− @n�(t)

|t=0 = 0

(64)



Thus, the �rst equation of (64) is now independent of and can be solved separately thanksto Lemma 2.1, which gives the existence and uniqueness of � and the following estimate:

‖�‖W (1; 2)p (�T )

6C(‖�0‖W 2(1−1=p)p (�) + ‖h1‖Lp(�T )) (65)

So, it only remains to solve the second equation of (64), which has the following form:

@t =�‖ − � − @n(T ) + h(t); |t=0 = 0; h(t) := h2(t)− @n�(t) (66)

We note that, due to (65), h∈Lp(@�T ). Moreover, according to (58) and to a proper tracetheorem, we have

‖@n(T )‖Lp(@�T )6C‖ ‖W (1−1=(2p); 2−1=p)p (@�T )

(67)

Consequently, Equation (66) is again a compact perturbation of the heat equation (but nowon the boundary @�) and, therefore, the existence and uniqueness of a solution can be veri�edin a standard way (using e.g. the Leray–Schauder principle). For the reader’s convenience,we give below the derivation of the Lp-regularity estimate for the solutions of (66). Applyingthe classical Lp-regularity estimate for heat equations to (66) in which the non-local term isinterpreted as external forces and using (67), we have

‖ ‖W (1; 2)p (@�T )

6C(‖ 0‖W 2(1−1=p)p (@�) + ‖ ‖W (1−1=(2p); 2−1=p)

p (@�T )+ ‖h‖Lp(@�T )) (68)

Using now the obvious interpolation inequality

‖ ‖W (1−1=(2p); 2−1=p)p (@�T )

6C‖ ‖L2(@�T ) + ‖ ‖W (1; 2)p (@�T )

(69)

and �xing ¿0 small enough, we �nd

‖ ‖W (1; 2)p (@�T )

6C1(‖ 0‖W 2(1−1=p)p (@�) + ‖ ‖L2(@�T ) + ‖h‖Lp(@�T )) (70)

So, we only need to estimate the L2-norm of the solution (t). To this end, we multiplyEquation (66) scalarly in L2(@�) by (t), integrate in t and use the positivity of form (59).Then, we have

1=2‖ (t)‖2L2(@�) − 1=2‖ 0‖2L2(@�)6∫ t

0(h(s); (s))@� ds (71)

Applying Gronwall’s inequality to this relation, we deduce the required estimate for theL2-norm of and �nish the proof of Lemma 2.2.

We are now ready to verify the existence of a solution for the non-linear problem (5).

Theorem 2.1Let the assumptions of Theorem 1.1 hold. Then, for every �¿0 and every initial datumbelonging to D�, problem (5) has a solution (�(t); (t)) which satis�es all the estimates ofSection 1.

ProofWe verify the existence of a solution by interpreting problem (5) as a non-linear compactperturbation of (62) and by using Lemma 2.2 and the Leray–Schauder principle. To this end,



we consider the following homotopy of Equation (5) to the linear one:

(�+ (−�x)−1N )@t�=�x� − 〈@n�〉@� − s[f(�)− 〈f(�)〉]; �|t=0 =�0

@t =�‖ − � − @n� − sg( ); |t=0 = 0

�|@� =

(72)

Using now Lemma 2.2, we can rewrite (72) as follows:(�

)=M0

(�0

0

)+ sMh

(〈f(�)〉 − f(�)

g( )

)(73)

where M0 : (�0; 0) �→ (�; ) is the solving operator for problem (62) with zero external forcesand Mh : (h1; h2) �→ (�; ) is the solving operator of the same problem, but with null initialdata.We solve Equation (73) in the phase space

:=W (1;2)p (�T )×W (1;2)

p (@�T ); 3¡p¡10=3 (74)

(the assumption p¿3 is necessary in order to have the embedding ⊂C(�T )×C(@�T ) andthe condition p¡10=3 is chosen in order to have the embedding H 2(�)⊂W 2(1−1=p)

p (�), whichguarantees that the operator M0 is well-de�ned for every initial datum belonging to D�).We now apply the Leray–Schauder �xed point theorem to Equation (73). To this end,

we �rst note that, due to the compact embedding ⊂C(�T )×C(@�T ) and estimate (63),the operator (�; ) �→Mh(〈f(�)〉 − f(�); g( )) is a compact and continuous operator in .Thus, the Leray–Schauder theory is indeed applicable and we only need to have uniformwith respect to s∈ [0; 1] a priori estimates for the solutions of (73). Indeed, let (�s; s) bea solution of Equation (73). Then, obviously, (�s; s) also solves system (72) and the triple(�s; �s; s), where �s=�s(t) can be found as a solution of

�s(t)− ��x�s(t)=−�x�s(t) + f(�s(t)); @n�s(t)|@� =0 (75)

solves the initial system (5) (in which the non-linearities f and g are replaced by sf and sg,respectively). Moreover, it is not di�cult to verify, arguing as in the proof of Theorem 1.1,that the a priori estimates (9) hold uniformly with respect to s∈ [0; 1]. In particular, theseestimates give

‖�s‖L∞(�T ) + ‖ s‖L∞(�T )6CT (76)

where CT is independent of s. Thus, we also have uniform L∞-estimates on the terms f(�s)and g( s) and, consequently, interpreting these terms as external forces in (72) and applyingthe Lp-regularity estimates (63), we obtain a uniform with respect to s estimate on (�s; s)in . Therefore, according to the Leray–Schauder principle, Equation (72) has a solution forevery s∈ [0; 1]. It remains to note that, for s=1, system (72) is equivalent to (5). Theorem 2.1is proven.

It is not di�cult to extend the existence result of Theorem 2.1 to the case �=0.



Corollary 2.2We assume that the assumptions of Theorem 1.1 hold and that �=0. Then, for every initialdatum belonging to D0, problem (5) has a solution (in the sense de�ned in the Introduction).

ProofLet the initial datum (�0; 0)∈D0. We recall that, in the case �=0, we have �0 = −�x�0 +f(�0) and, according to the de�nition of D0, this expression belongs to H 1(�). Using nowthis fact and Equation (75) for � in the case �¿0, we can easily verify that the initial datumconsidered belongs to D� for every �¿0 and the following uniform estimate holds:

‖(�0; 0)‖D�6Q(‖�0; 0‖D0); �∈ [0; 1] (77)

where the function Q is independent of �. Let (��(t); �(t)), �¿0, be solutions of (5) whoseexistence is veri�ed in Theorem 2.1. Then, according to (77) and Corollary 1.1, these solutionssatisfy

‖(��(t); �(t))‖D�6Q(‖(�0; 0)‖D0)e−�t + C∗ (78)

uniformly with respect to �→ 0. Passing now in a standard way to the limit �→ 0 in Equations(5), we see that the limit triple (�(t); �(t); (t)) satis�es Equations (5) with �=0. Passing�nally to the limit �→ 0 in estimate (78), we verify that the solution (�; ) satis�es (78)with �=0 and Corollary 2.2 is proven.

Thus, the existence of a solution of problem (5) with an initial datum belonging to D� iscompletely veri�ed. We also recall that the uniqueness of this solution has been proven inProposition 1.2. Thus, for every �∈ [0; 1], problem (5) generates a dissipative semigroup St(�)on the phase space D�:

St(�) :D� →D�; St(�)(�0; 0) := (�(t); (t)) (79)

where (�(t); (t)) is the unique solution of (5) with initial datum (�0; 0). The rest of thissection is devoted to the extension of this result to the weak energy spaces introduced in (8).Indeed, according to Proposition 1.2, we have the following uniform Lipschitz continuity inthe L�-norm:

‖St(�)(�1; 1)− St(�)(�2; 2)‖L�6CeKt‖(�1 − �2; 1 − 2)‖L� (80)

for all (�i; i)∈D�, i=1; 2, where the constants C and K are independent of the initialdata. Consequently, the semigroup (79) can be extended in a unique way by continuity toa Lipschitz continuous semigroup acting on the whole space L� by the following standardexpression:

St(�)(�0; 0) := L� − limn→∞ St(�)(�n; n) (81)

where (�0; 0)∈ L�, (�n; n)∈D� and (�n; n)→ (�0; 0) in the norm of L� (here, we haveimplicitly used the obvious fact that D� is dense in L�). Moreover, since the solutions



(�n(t); n(t)) belong obviously to the space C([0; T ]; L�), then the limit function (�(t); (t)):= St(�)(�0; 0) is also continuous with values in L�. Furthermore, passing to the limit n→ ∞in estimate (29) for the solutions (�n(t); n(t)), we obtain the same estimate for the limitfunction (�(t); (t)). Thus, for every t¿0, the extended semigroup St(�) maps L� into D�:

St(�) : L� →D�; t¿0 (82)

Finally, passing to the limit n→ ∞ in Equations (5) and using (29), we can easily establishthat, for every t¿0, the function (�(t); (t)) also satis�es Equations (5). Thus, we haveproven the following result.

Theorem 2.2Let the assumptions of Theorem 1.1 hold. Then, for every initial datum (�0; 0) belonging toL�, problem (5) has a unique solution (�(t); (t)) in the class C([0; T ]; L�) ∩ L∞

loc((0; T ];D�).

Indeed, this solution is given by the formula (�(t); (t)) := St(�)(�0; 0), where the exten-sion of St(�) to L� is de�ned by (81).

3. ROBUST EXPONENTIAL ATTRACTORS

In this section, we construct a robust (as �→ 0) family of exponential attractors for thesemigroups (81) associated with problems (5). In order to do so, we �rst recall that system (5)possesses the conservation law (4) and, consequently, we cannot expect to have a dissipationin the whole phase space L� or D� (the constant C and the function Q in the dissipativeestimate (35) depend on M0) and we need to restrict ourselves to the hyperplanes (4), with aprescribed value of the average 〈�(t)〉. We note however that all the constants in the estimatesobtained in Section 1 are uniform with respect to M0 belonging to some bounded set. So,instead of �xing the prescribed value M0 of the average (4), we will consider below thesemigroups (81) in the phase spaces

L�(M) := L� ∩ {|〈�〉|6M} and D�(M) :=D� ∩ {|〈�〉|6M} (83)

for every M¿0. The following theorem gives a robust family of exponential attractors forthe semigroups (81) restricted to spaces (83).

Theorem 3.1Let the assumptions of Theorem 1.1 hold and let M¿0 be arbitrary. Then, there exists afamily of compact sets M� ⊂⊂D�, �∈ [0; 1], which satisfy the following properties:(1) Finite-dimensionality: the fractal dimension of M� (say, in H 2(�)×H 2(@�)) is �nite

and uniformly bounded with respect to �→ 0:

dimF(M�; H 2(�)×H 2(@�))6C (84)

where the constant C depends on M , but is independent of �.



(2) Invariance: St(�)M� ⊂M�, t¿0.(3) Uniform exponential attraction: there exist a positive constant � and a monotonic func-

tion Q (which are independent of �) such that, for every bounded subset B⊂ L�(M),we have

distH 2(�)×H 2(@�)(St(�)B;M�)6Q(‖B‖L�)e−�t ; t¿0 (85)

where distV denotes the non-symmetric Hausdor� semidistance between sets in thespace V .

(4) The sets M� tend to the limit set M0 as �→ 0 in the following sense:

distsymmH 2(�)×H 2(@�)(M�;M0)6C� (86)

where distsymmV denotes the symmetric Hausdor� distance and the positive constants Cand ∈ (0; 1) depend on M , but are independent of �.

Remark 3.1Due to the parabolic nature of problem (5), we have a standard smoothing property forits solutions (see e.g. (29) and (36)) and, consequently, the concrete choice of the spaceH 2(�)×H 2(@�) is not essential and can be replaced e.g. by D�, or even by H 3(�)×H 3(@�).

Proof of the theoremAs usual (see References [19–22]), we �rst construct the required exponential attractors forthe semigroups (81) with discrete times and then extend the result to continuous times. Tothis end, we �rst introduce the following ball B with su�ciently large radius R in the spaceH 3(�)×H 3(@�):

BR=BR(M) := {(�; )∈H 3(�)×H 3(@�)

‖(�; )‖H 3(�)×H 3(@�)6R; |〈�〉|6M} (87)

Then, obviously, BR ⊂D�(M) for all �∈ [0; 1] and

‖BR‖D�6CR;M (88)

where the constant CR;M depends on R and M , but is independent of �. Moreover, due to thedissipative estimate (35) and the smoothing properties (29) and (36), there exist su�cientlylarge �R= �R(M) and T =T (M) which are independent of � such that the set B :=B �R is anabsorbing set for the semigroup St(�) acting on L�(M) (uniformly for all �¿0) and

St(�)B⊂B; �∈ [0; 1]; t¿T (89)

So, it only remains to construct the required exponential attractors in the phase space B.To this end, we �rst introduce the discrete semigroups Sn(�) := SnT (�), n∈N (which,according to (89), act on the phase space B) and construct the ‘discrete’ exponential attractorsMd

� by using the following exponential attractor’s existence theorem proven inReference [21].



Proposition 3.1Let the discrete semigroups {Sn(�); n∈N} acting on the space B (for every �∈ [0; 1]) satisfythe following properties:

(1) Uniform smoothing property for the di�erence of two solutions:

‖S1(�)(�1; 1)− S1(�)(�2; 2)‖H 1(�)×H 1(@�)

6K‖(�1 − �2; 1 − 2)‖L2(�)×L2(@�); (�i; i)∈B; i=1; 2 (90)

where the constant K is independent of �.(2) Convergence as �→ 0:

‖Sn(�)(�; )− Sn(0)(�; )‖H 1(�)×H 1(@�)6C�eLn; (�; )∈B (91)

where the constants C and L are also independent of �.

Then, there exists a robust family of exponential attractors Md� , �∈ [0; 1], which satisfy the

analogues of properties (1)–(4) of Theorem 3.1, namely:

(1) Finite-dimensionality: the fractal dimension of Md� (in H 1(�)×H 1(@�)) is �nite and

uniformly bounded with respect to �→ 0:

dimF(M�; H 1(�)×H 1(@�))6C (92)

where the constant C is independent of �.(2) Invariance: Sn(�)Md

� ⊂Md� , n∈N.

(3) Uniform exponential attraction: there exist positive constants � and C (which areindependent of �) such that

distH 1(�)×H 1(@�)(Sn(�)B;Md� )6Ce−�n; n∈N (93)

(4) The sets Md� tend to the limit set M

d0 as �→ 0 in the following sense:

distsymmH 1(�)×H 1(@�)(Md� ;M

d0)6C1� (94)

where the positive constants C1 and ∈ (0; 1) are independent of �.Thus, in order to construct the family Md

� of discrete exponential attractors, we only needto verify estimates (90) and (91). We will do so in the next two lemmata.

Lemma 3.1Let the above assumptions hold. Then, for every (�1(0); 1(0)) and (�2(0); 2(0)) belongingto B, the corresponding solutions of problem (5) satisfy the following estimates:

‖�1(T )− �2(T )‖2H 1(�) + ‖ 1(T )− 2(T )‖2H 1(@�)

6T−1CT‖(�1(0)− �2(0); 1(0)− 2(0))‖2L�; T¿0 (95)

where the constant CT depends on T , but is independent of �.



ProofWe �rst note that, due to estimates (35) and (88), we have

‖�i(t)‖H 2(�) + ‖ i(t)‖H 2(@�)6C; i=1; 2; t¿0 (96)

where the constant C is independent of t and �. Moreover, due to the embedding H 2 ⊂C,we have analogous estimates for the L∞-norms, which are necessary in order to handle thenon-linear terms in (5). In order to deduce (95), we �rst need to generalize (using the uniformestimates (96)) Proposition 1.2 to the case where condition (42) is not satis�ed. Indeed, let(�; �; ) := (�1 − �2; �1 − �2; 1 − 2). Then, these functions satisfy Equations (44), but, incontrast to the proof of Proposition 1.2, we now have

〈�(t)〉= 〈�1(0)〉 − 〈�2(0)〉 :=M1;2 �=0 (97)

Nevertheless, introducing the new functions ( ��(t); � (t)) := (�(t) − M1;2; (t) − M1;2), multi-plying the �rst equation of (44) by (−�x)−1N

��(t) and arguing as in the proof of Propositions1.1 and 1.2, we derive the following analogue of (45):

1=2@t(‖ ��(t)‖2H−1(�) + �‖ ��(t)‖2L2(�) + ‖ � (t)‖2L2(@�)) + ‖∇x��(t)‖2L2(�)

+�‖ � (t)‖2L2(@�) + �M1;2〈 � (t)〉@�

+‖∇‖ � (t)‖2L2(@�) =−([f(�1(t))− f(�2(t))]; ��(t))

−([g( 1(t))− g( 2(t))]; � (t))@� (98)

In contrast to the proof of Proposition 1.2, we cannot estimate the non-linear terms in (98) byusing the facts that f′(v)¿−K and g′(v)¿−K . Instead, we now have the uniform estimates(96) (and analogous estimates for the L∞-norms). So, we can estimate these terms as follows:

−([f(�1(t))− f(�2(t))]; ��(t))− ([g( 1(t))− g( 2(t))]; � (t))@�

6C(M 21;2 + ‖ ��(t)‖2L2(�) + ‖ � (t)‖2L2(@�)) (99)

Inserting this estimate into the right-hand side of (98) and using Gronwall’s inequality, weinfer

‖( ��(t); � (t))‖2L�+∫ T

0(‖∇x�(t)‖2L2(�) + ‖∇‖ (t)‖2L2(@�)) dt

6CT (‖ ��(0); � (0))‖2L�+ |M1;2|2)6C ′

T‖(�(0); (0))‖2L�(100)

where the constants CT and C ′T are independent of �. We are now ready to verify estimate

(95). To this end, we multiply the �rst equation of (44) by t(−�x)−1N @t�(t) and, arguing as



in the derivation of (37), we have

1=2@t(t‖∇x�(t)‖2L2(�) + t‖∇‖ (t)‖2L2(@�) + �t‖ (t)‖2L2(@�))

+t‖@t�(t)‖2H−1(�) + �t‖@t�(t)‖2L2(�) + t‖@t (t)‖2L2(@�)

= 1=2(‖∇x�(t)‖2L2(�) + ‖∇‖ (t)‖2L2(@�) + �‖ (t)‖2L2(@�))

+([f(�1(t))− f(�2(t))]; t@t�(t)) + ([g( 1(t))− g( 2(t))]; t@t (t))@� (101)

Estimating the last two terms on the right-hand side of (101) as follows:

([f(�1(t))− f(�2(t))]; t@t�(t)) + ([g( 1(t))− g( 2(t))]; t@t (t))@�

6Ct(‖�(t)‖2H 1(�) + ‖ (t)‖2L2(@�)) + 1=2(t‖@t�(t)‖2H−1(�) + t‖@t (t)‖2L2(@�)) (102)

(where we have again used estimates (96)), integrating in t and using (100), we infer

T (‖�(T )‖2H 1(�) + ‖ (T )‖2H 1(@�))6CT‖(�(0); (0))‖2L�(103)

which �nishes the proof of the lemma.

Since, obviously, ‖(�; )‖L�6C‖(�; )‖L2(�)×L2(@�), where C is independent of �, thenassumption (90) of Proposition 3.1 is veri�ed.

Lemma 3.2Let (��(t); �(t)) and (�0(t); 0(t)) be two solutions of Equations (5) with positive � andwith �=0, respectively. We also assume that these solutions have the same initial datum,belonging to the set B. Then, the following estimates hold:

‖��(t)− �0(t)‖2H 1(�) + ‖ �(t)− 0(t)‖2H 1(@�)6C�2eKt (104)

where the constants C and K are independent of � and t.

ProofWe set ( ��; ��; � ) := (��−�0; ��−�0; �− 0). Then, these functions satisfy the following systemof equations:

@t��=�x ��; @n ��|@� =0

��=−�x��+ �@t

��+ [f(��)− f(�0)] + �@t�0

@t� =�‖ � − � � − [g( �)− g( 0)]− @n

��; x∈ @�

�|@� = �

(105)



Multiplying now the �rst equation of (105) by (−�x)−1N @t�� and arguing exactly as in the

derivation of (101), we obtain

1=2@t(‖∇x��(t)‖2L2(�) + ‖∇‖ � (t)‖2L2(@�) + �‖ � (t)‖2L2(@�))

+‖@t��(t)‖2H−1(�) + �‖@t

��(t)‖2L2(�) + ‖@t� (t)‖2L2(@�)

= ([f(��(t))− f(�0(t))]; @t��(t)) + ([g( �(t))− g( 0(t))]; @t

� (t))@�

+�(@t�0(t); @t��(t)) (106)

We now recall that, according to (9) and (35), we have

∫ T

0‖@t�0(t)‖2H 1(�) dt6C(T + 1) (107)

where C is independent of �. Estimating the non-linear terms on the right-hand side of (106)by using (102), applying Gronwall’s inequality and estimating the last term on the right-handside of (106) by (107), we obtain estimate (104) and �nish the proof of Lemma 3.2.

Thus, all the assumptions of Proposition 3.1 are veri�ed and, consequently, the discretesemigroups {Sn(�); n∈N} possess a robust family of exponential attractors M� on B whichsatisfy properties (1)–(4) of Proposition 3.1. Moreover, since B gives uniform absorbing setsfor these semigroups in L�(M), then these attractors attract exponentially all the boundedsubsets of L�(M). So, the required family of exponential attractors is constructed for discretetimes. In order to extend this result to continuous times, we use the standard formula

M� :=⋃

t∈[T;2T ]St(�)Md

� (108)

(see e.g. References [19,20]). Then, since the semigroups St(�) are uniformly Lipschitz con-tinuous on [T; 2T ]×B in the norm of L� (the Lipschitz continuity with respect to the initialdata was in fact veri�ed in Proposition 1.2 and Lemma 3.1 and the Lipschitz continuity withrespect to t follows from the fact that (@t�(t); @t (t))∈ L� if (�(t); (t))∈D�), arguing in astandard way (see e.g. Reference [21]), we deduce that family (108) of exponential attractorssatis�es assumptions (1)–(4) of Theorem 3.1, but with the H 2(�)×H 2(@�)-norm replacedby the L�-norm.In order to extend the results obtained to the required H 2(�)×H 2(@�)-norm, it remains

to recall that, according to Theorems 1.2 and 1.3, the semigroups St(�) possess a uniformsmoothing property from L� into H 3(�)×H 3(@�) and to use the obvious interpolation in-equality:

‖(v; w)‖H 2(�)×H 2(@�)6C‖(v; w)‖1=4L�· ‖(v; w)‖3=4H 3(�)×H 3(@�) (109)

where the constant C is also independent of �. Thus, Theorem 3.1 is proven.



APPENDIX A: THE MAXIMUM PRINCIPLE AND L∞-BOUNDS ON THESOLUTIONS FOR AN AUXILIARY ELLIPTIC PROBLEM

In this section, we formulate and prove several estimates which play a fundamental rolefor obtaining L∞-bounds on the solutions of the initial Cahn–Hilliard problem. Since theexistence of a solution has been veri�ed in a more complicated situation in Section 2, werestrict ourselves only to the derivation of the corresponding a priori estimates.We start with the study of the following linear problem:{−�xu= h1(x); x∈�; u|@� =�

−�‖�+ ��+ @nu= h2(x); x∈ @�(A1)

where � is some positive constant. We start with the classical elliptic H 2-regularity estimatesfor this problem.

Lemma A.1Let the functions h1 and h2 belong to the spaces L2(�) and L2(@�), respectively. Then, thefollowing estimates hold:

‖u‖H 2(�) + ‖�‖H 2(@�)6C(‖h1‖L2(�) + ‖h2‖L2(@�)) (A2)

for some positive constant C.

ProofWe �rst multiply the �rst equation of (A1) scalarly in L2(�) by u, integrate by parts and �ndthe expression for the term (@nu; u)@� = (@nu; �)@� from the second equation of (A1). Then,after obvious transformations, we have

‖∇xu‖2L2(�) + ‖∇‖�‖2L2(@�) + �‖�‖2L2(@�) = (h1; u) + (h2; �)@� (A3)

and, consequently, using the fact that ‖u‖H 1(�)6C(‖∇xu‖L2(�) + ‖�‖H 1(@�)), we have

‖u‖H 1(�) + ‖�‖H 1(@�)6C(‖h1‖L2(�) + ‖h2‖L2(@�)) (A4)

for some positive constant C.Applying now the H 2-regularity theorem to the �rst equation of (A1) with Dirichlet bound-

ary conditions, we have

‖u‖H 2(�)6C(‖h1‖L2(�) + ‖�‖H 3=2(@�)) (A5)

Analogously, applying this theorem to the second equation of (A1), we deduce

‖�‖H 2(@�)6C(‖h2‖L2(@�) + ‖@nu‖L2(@�)) (A6)

Moreover, due to a proper interpolation inequality, we obtain, for every 0¡s¡1=2,

‖@nu‖L2(@�)6Cs‖u‖H 3=2+s(�)6C‖u‖H 1(�) + ‖u‖H 2(�) (A7)

where the positive constant can be arbitrarily small. Combining estimates (A5)–(A7), wehave

‖u‖H 2(�)6C1(‖h1‖L2(�) + ‖h2‖L2(@�)) + C ′‖u‖H 2(�) + C ′‖u‖H 1(�) (A8)



which, together with (A4), gives the required estimate for the H 2-norm of u. The H 2-normof � can then be found from estimates (A6) and (A7). Lemma A.1 is proven.

Corollary A.1Let the assumptions of Lemma A.1 hold and let, in addition, the external forces h1 and h2belong to Hs(�) and Hs(@�), respectively, where s¿0 is such that s+ 1=2 does not belongto N. Then, the solution (u; �) of problem (A1) belongs to the space Hs+2(�)×Hs+2(@�)and the following estimates hold:

‖u‖Hs+2(�) + ‖�‖Hs+2(@�)6C(‖h1‖Hs(�) + ‖h2‖Hs(@�)) (A8′)

Indeed, estimate (A8′) can be derived exactly as for (A5)–(A8), except that, instead ofusing the L2–H 2 regularity estimate for linear elliptic equations, one needs to use its analoguefor Hs-spaces.We are now ready to formulate the main result of this section on L∞-bounds on the

solutions of the following non-linear second-order elliptic problem:{−�xu+ f(u)= h1(x); u|@� =�

−�‖�+ ��+ @nu+ g(�)= h2(x)(A9)

Lemma A.2Let the functions h1 and h2 be as in Lemma A.1 and the functions f and g satisfy assumptions(3). Then, every solution of problem (A9) satis�es the following estimates:

‖u‖L∞(�) + ‖�‖L∞(@�)6Cf;g + C(‖h1‖L2(�) + ‖h2‖L2(@�)) (A10)

where the constants Cf;g and C are independent of the solution (u; �).

ProofAs usual, the L∞-estimates (A10) are based on the maximum principle. However, we cannotapply it directly to problem (A9), since the external forces do not belong to L∞. In order toovercome this di�culty, we introduce the function ( �u; ��) solution of the simpli�ed problem(A9) with f= g=0. Then, due to estimate (A2) and the embedding H 2 ⊂C,

‖ �u‖L∞(�) + ‖ ��‖L∞(@�)6C(‖h1‖L2(�) + ‖h2‖L2(@�)) := �K (A11)

We also introduce the function (u; �) := (u − �u; � − ��) which obviously solves the followingproblem:

−�xu+ f(u+ �u)=0; u|@� = �

−�‖�+ ��+ @nu+ g(�+ ��)=0(A12)

In contrast to (A9), all the terms in Equations (A12) belong to L∞ and we can apply themaximum principle. To this end, we �x the constant Cf;g¿0 such that{

sgn v · f(v)¿0; f′(v)¿0

sgn v · g(v)¿0; g′(v)¿0(A13)



for all v =∈ [−Cf;g; Cf;g] (Cf;g exists thanks to conditions (3)), and consider the following testfunction which is independent of x:

U = :=Cf;g + �K (A14)

where the constant �K is the same as in (A11). Then, (A13) and (A11) imply that (U;) isa supersolution of (A12) and, consequently, the function ( �U; �) := (u−U; ��−) satis�es thefollowing di�erential inequalities:

−�x �U + [f(u+ �u)− f(U + �u)]60; �U |@� = �

−�‖ � + � � + @n �U + [g(�+ ��)− g( + ��)]60(A15)

We also recall that, due to (A13),

[f(w + �u)− f(U + �u)]¿0; [g(w + �u)− g( + ��)]¿0 (A16)

for all w¿U =. Multiplying now the �rst inequality of (A15) scalarly in L2(�) by �U+ :=max{ �U; 0}, integrating by parts and using (A16), we have

‖∇x �U+‖2L2(�) + ‖∇‖ �+‖2L2(@�) + �‖ �+‖2L2(@�)60 (A17)

which immediately implies that �U+ = �+ ≡ 0 and, consequently,(u(x); �(x))6(U + �u(x);+ ��(x)) (A18)

which, together with (A11) and (A14), gives the required upper bound on the solution(u; �). The lower bound can be obtained analogously by using the test function (−U;−).Lemma A.2 is proven.

ACKNOWLEDGEMENTS

The authors wish to thank Professor S. Zheng for interesting discussions. The �rst author also wishes tothank him for having invited him to Fudan University. This research was partially supported by INTASProject No. 00-899 and CRDF Grant No. 10545.

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Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

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