Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal.152016, 1077–1105], we considered the conserved phase-field system
in a bounded domain of , , where is a relaxation time, is the viscosity parameter, is the heat capacity, ϕ is the order parameter, u is the absolute temperature and is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at . Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at . To establish the existence of inertial manifolds of dimension independent of the two parameters δ and ε, we require ε to be dominated from above by δ. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn–Hilliard equation, improving and extending some previous results.
We consider, either in a bounded domain Ω of with smooth boundary, or in , , for , the conserved phase-field system for phase transitions (cf. [8,27])
where τ is a relaxation time, δ is the viscosity parameter, ε is the heat capacity, ϕ is the order parameter, u is the absolute temperature and with G a double-well potential. This system of equations describes phase transition processes such as melting or solidification.
If , then System (1.1) reduces to the well-known Cahn–Hilliard equation (cf. [9,39]), namely
which plays an important role in material sciences.
The global attractor is a compact invariant set lying on the phase space which attracts uniformly the trajectories starting from bounded sets when time goes to infinity (see, e.g., [10,41,43]). However, the global attractor may have a complicated fractal structure, even for finite-dimensional dynamical systems, and a reasonably explicit description of the dynamics on the attractor might be out of reach. An inertial manifold is a positively invariant smooth finite-dimensional manifold which contains the global attractor and which attracts the trajectories at a uniform exponential rate (see [12,30,41,43]). It follows that the a priori infinite-dimensional dynamical system reduces, on the inertial manifold, to a finite system of ordinary differential equations. The condition required by almost all of the current theorems to ensure the existence of an inertial manifold is a sufficiently large gap in the spectrum of the linear operator associated with the PDE. This restrictive condition known as “the spectral gap” is not verified for many dynamical systems. In this case, it is still possible to construct an intermediate object between the global attractor and the inertial manifold which is called exponential attractor. This set is a compact and positively invariant set with finite fractal dimension which attracts all the trajectories starting from bounded sets at a uniform exponential rate (see [1,14,18,19,25,31]). The sensitivity of these invariant sets under small perturbations and approximations (including numerical simulations) is an important issue that we will address in this paper. We refer the reader to [34] for some recent developments in the construction of exponential attractors. See also [44] for a recent survey on inertial manifolds and finite-dimensional reduction for dissipative PDEs.
A well-posedness result for a phase-field model including (1.1) with irregular potential and subject to dynamic boundary conditions was proven in [27]. Global and exponential attractors were proven in [32] for Problem (1.1) with . There, the author considered a large class of potential containing polynomials of arbitrary even degree with positive leading coefficients, improving previous results of [6,7]. The same problem with , subject to Neumann boundary conditions, was considered in [3] recently. There the author proved the existence of the global attractor and constructed a robust family of exponential attractors. The author also proved the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimensions. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets at were also proven. More recently in [4], Problem (1.1), with a fixed value of , was analyzed, showing the convergence of the dynamics to the one of the viscous Cahn–Hilliard equation. Let us now mention a few earlier works [2,15–17,22,23,26,33] on non-conserved phase-field systems where convergence to the Cahn–Hilliard equations were proven. In [15–17], the upper and lower semicontinuity of the global attractor were proven. In [33], Dirichlet and Neumann boundary conditions were considered and the authors constructed robust family of exponential attractors on a closed subspace of the phase space . In [2], Dirichlet boundary conditions were considered and an analysis similar to that of [3] was carried out on a closed subspace of . Models with dynamic boundary conditions were investigated in [22,23,26]. Finally, we note that global and exponential attractors as well as some stability results were proven in [35,36] for a 3d conserved phase-field system with viscosity and memory terms and subject to Neumann boundary conditions.
Our aim in this paper is to analyze the convergence of the dynamics of Problem (1.1) as both parameters ε and δ go to zero. System (1.1) then can be viewed as a singular perturbation of the Cahn–Hilliard equation, extending results of [20,21,24]. All the estimates carried out in [4] will be repeated here again with the extra condition that they are independent not only of ε but also independent of δ. Of course, this introduced more challenging difficulties, especially in the proofs of the existence and continuity of inertial manifolds. In addition, the continuity properties of the global attractor, exponential attractors and inertial manifolds proven in [3] relied on an estimate similar to (7.16) that required the condition . The proof of this estimate was based on a method devised in [2] that needed to consider solutions starting from a bounded absorbing set of a subspace of . Here we introduce a different method to prove Estimate (7.16) and we only need to consider solutions starting from a bounded absorbing set of a subspace of . Thus, the condition required in [3] is weakened here to , improving the results of [2,3].
This paper is organized as follows. In Section 2, we set the problem. In Sections 3 and 4 we derive a priori estimates and we demonstrate the well-posedness of the problem and the existence of the global attractor and its upper semicontinuity at . Then, in Section 5, assuming that ε is dominated from above by δ, we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimensions. In Section 6, we construct exponential attractors. In the final Section 7, we prove continuity of the exponential attractors at and we also prove continuity properties of the intersection of the inertial manifolds with bounded sets at when .
Functional setting
If W is a Sobolev-type space, then we set
where
Moreover, we set
We denote by the dual space of W.
System (1.1) is subject to the boundary conditions either of Neumann or periodic type
(the symbol denotes the outward normal derivative) if Ω is a bounded domain of , with smooth boundary , or
if .
Let us define the linear unbounded operator, with domain ,
with
which is self-adjoint and nonnegative. If N is restricted to , then it turns to be positive with compact inverse . Moreover, one can define the powers of N for (cf. [43] at page 57). The spaces are Hilbert spaces. In particular, or , , or . The injection is compact whenever . We denote by and the usual norm and scalar product in (and also in ). When r is positive, is a subspace of and
is a norm on which is equivalent to the usual -norm; we endow with the norm
Note that for and any , we have
when is even, and
when is odd.
We consider the problem
where and . We denote the function and we assume that and the following conditions hold (cf., e.g., [24]):
where is arbitrary when and when . For instance, satisfies (2.4)–(2.7). However, we note that in one space dimension, no growth assumption on g is needed.
The space denotes the closure of a metric space in the topology of the complete metric space Y. Furthermore, there exist two positive constants such that
and
For every , we endow the Hilbert space
with the norm
Note that is a norm on which is equivalent to . Sometimes, we will use the equivalent norms
The Hausdorff semi-distance with respect to the metric of E is defined as:
whereas the symmetric Hausdorff distance between A and B is
Then, we set
for some and .
Multiplying the first equation of (2.3) and second one by 1, and integrating over Ω, we find
and
respectively, so that
and
From now on, the same letter c and (and sometimes , ) denote positive constants that may change from line to line, but are always independent of ε, δ and on time (unless explicitly specified).
A priori estimates
All the a priori estimates that we shall derive formally in this paper can be rigorously justified using a Galerkin truncation (cf., e.g., [41, Chapter 11]).
We multiply the first equation of (2.3) by and the second one by u, and integrate over Ω. Summing the resulting equations, we obtain
Multiplying the first equation of (2.3) by and integrating over Ω, we obtain, owing to (2.5),
where and , but we will omit the dependence of constants with respect to . By multiplying the first equation of (2.3) by ϕ and integrating over Ω, we obtain, owing to (2.6),
Summing (3.1), times (3.2) and times (3.3), where are small enough, we deduce
where , and
There exist , independent of ε and δ, such that
and
since
which can be deduced from (2.5) and (2.7). Thus, we deduce from (3.4) that
Now, multiplying the first equation of (2.3) by and and integrating over Ω, we get
and
We multiply the second equation of (2.3) by and integrate over Ω. This yields
When , we have
and
When , we have
and
When , we have
and
Summing (3.6), (3.7) and (3.8), we deduce
where
The global attractor
We prove the following result.
We assume that (
2.4
)–(
2.7
) hold. If, then (
2.3
) possesses a unique solutionsuch thatfor any. Moreover, if, then
(i) Existence: The existence follows from standard arguments using Galerkin approximations and then passing to the limit (see for instance [43]). If , then the approximate solutions are bounded independently of m (cf. (3.5)), and using weak compactness, we find a subsequence still denoted by and a pair of functions such that , , and
Since g is continuous, we can pass to the limit as in the approximate problem, and is solution to (2.3). From classical compactness theorems, it follows that ϕ is weakly continuous from into , and u is weakly continuous from into . Using (3.1), we can see that the real function is continuous on . We can conclude that ϕ is strongly continuous from into , and u is strongly continuous from into . If , we can proceed like in part (i) to show the existence of a pair of functions solution to (2.3) such that and .
(ii) Uniqueness: Let and be two solutions of (2.3). Setting and , we have , , , , , and the pair satisfies the equations:
By multiplying (4.1) by ϕ and , and integrating over Ω, we obtain
and
We multiply (4.2) by u and find, after integrating over Ω,
When , we have
When , we have
When , we obtain
Adding together equalities (4.3), (4.4) and (4.5), we infer
where
Applying the Gronwall’s lemma to (4.9), we deduce that
hence the result. □
Thanks to Theorem 4.1, we can define the semigroup
where is the solution to (2.3) at time t. The semigroup is strongly continuous. We apply the Gronwall’s lemma to (3.5) and we deduce the existence of an absorbing set for on of the form
where is independent of ε and δ. Note that if , then the constant in (3.5) is bounded from below by a strictly positive constant that does not depend on and , the other constants are also independent of and . Now, let be in a bounded set B of , then there exists depending only on B such that
We also deduce from (3.5) that
Applying the uniform Gronwall lemma to (3.9), we deduce the existence of an absorbing set for on of the form
where is independent of ε and δ. The semigroup restricted to is then uniformly compact. We apply [43, Chapter 1, Theorem 1.1] and we have the following result.
For everyand every, the semigroup, restricted to, has the global attractor, that is,
is compact inand invariant, that is,,;
for any bounded subset, there holds
The semigroup generated by the unperturbed problem (for the variable ϕ alone) possesses the global attractor on (see [43]). We now wish to compare the global attractor with for small values of ε and δ. Observe that, a solution of the unperturbed problem for both variables ϕ and u (at time t) is given by
where
We define
Let us show the existence of a bounded absorbing set for the semigroup in that will be useful in the proof of Theorem 4.3.
Letand. There existsindependent of ε and δ such that the solutionsatisfiesfor any.
Let and set . Since is a bounded absorbing set for in , there exists , independent of ε and δ, such that
Hence
By differentiating the first and second equations of (2.3) with respect to time, we can show that the pair is solution to the problem
where
Second, we also observe that , thus
We multiply (4.13) by and integrating over Ω to obtain
We multiply the second equation of (2.3) by and integrate over Ω. This yields
Adding up (4.17) and (4.18) gives
Owing to (2.6) and noting , we deduce
Integrating (3.1) between 0 and ∞, we find that there exists , independent of ε and δ, such that
We apply the uniform Gronwall’s lemma to (4.20) to obtain
Then, integrating (4.20) between t and , we infer
We multiply (4.13) by and (4.14) by , and integrate over Ω. We obtain
and
Multiplying (4.13) by and integrating over Ω, we obtain
Summing (4.23), (4.24) and ϖ times (4.25), where is small enough, we find after noting that
where
Applying the uniform Gronwall’s lemma to (4.26), we find
Now, from the first and second equations of (2.3), we deduce that
and
Collecting (4.27), (4.28) and (4.29) yield the desired estimate (4.11). Consequently, the semigroup has an absorbing set in of the form
where is independent of ε and δ. □
The global attractor satisfies
for all and . Exploiting the invariance property of the attractor , , we learn that, for any there exists a complete trajectory in such that . In particular, there holds
for all and .
Now, we are in position to prove the following stability result for the global attractor.
The global attractoris upper semicontinuous at, that is,
We can follow [29] (see also [28]). The proof is based on a contradiction argument. We assume there exists , sequences , and , and a corresponding sequence such that
We set
From previous results in this section, we have that is bounded in , hence it is relatively compact in . Let the complete trajectory such that . Due to the invariance property of the global attractor, it follows that
is a relatively compact set in . Hence, is a precompact set in , and the family of mappings is equicontinuous from into . By Ascoli’s theorem and a classical diagonalization method, it follows the existence of an element such that (up to a subsequence)
for any , and
We have that
Now, exploiting (4.11) we learn that (up to a subsequence)
and
for any . Passing to the limit in (4.32) and (4.33), and by uniqueness of limit, we find that
or
for any . It follows that is a bounded complete trajectory of the semigroup . By characterization of the attractor it is clear that , and by definition . Thus the convergence
implies that
against the initial assumption. □
Inertial manifolds
In this section only, we take or 2, and we assume and is a rational number. In order to prove the existence of an inertial manifold for Problem (2.3)–(2.7), we introduce the prepared problem:
where
and is a function such that is equal to 1 when , and is equal to 0 when , and , . Then we write (5.1) in the following form:
where , , and
The operator is positive and has a discrete spectrum
for and corresponding eigenfunctions
where
are the eigenvalues of N ordered in an increasing sequence and are the corresponding eigenfunctions which is an orthogonal basis of . These eigenvalues and eigenfunctions have the form
and
for
Let . There exists n such that and
Indeed, this is immediate when and the result is due to I. Richards when (see [40]), on account of the assumption made on in the beginning of this section.
Denote
and
We now prove the following lemma.
Provided that n is large enough for (
5.3
) to hold, there existandsuitably small such that the following inequalities are satisfied, for everyand:
(i) The inequality is equivalent to
hence (i), due to (5.3).
(ii) The inequality is equivalent to
Now, the terms and have the same sign, and
which is a positive quantity for every and for some .
Similarly, the terms and have the same sign, and
which is not always positive. Thus the inequality (5.4) holds for any , and for some . In the sequel, the values of and may change, but we will still use the same notations.
(iii) Denote
These are all positive real numbers, whenever (5.3) is satisfied, except when . The quantity is positive when .
The inequality is equivalent to
The terms and have the same sign, and
which is positive when . The inequality
is equivalent to
under the condition that
The inequality (5.8) holds whenever . A computation of (5.7), observing that , gives the equivalent inequality
We have
which is negative, whenever (5.3) holds. The quadratic equation has two positive roots and . Thus
hence (5.9) holds whenever . It follows that (5.6) holds whenever . Now, the inequality may be positive or negative; so that, (5.5) holds whenever and .
The inequality is equivalent to
The inequalities
and
hold for every . It follows that (5.11) holds.
(iv) The quadratic equation has two positive real roots which are , hence (iii), due to (ii).
(v) A computation shows that
due to (i), whenever (5.3) is satisfied, hence (v) holds true for every and . The proof of the lemma is completed. □
We prove the following result.
For every, there exist n (independent of δ and ε) and some positive real numbers,such that the spectral gap conditionholds for everyand.
It is clear that is positive for every k. We have
For every and , the inequality
is equivalent to
which in turn, on account of (v) of Lemma 5.1, is equivalent to
that is,
On account of (iv) of Lemma 5.1, the inequality (5.14) holds for every , hence (5.12) holds true. □
We now prove the existence of an inertial manifold for (2.3).
We assume that,andis rational when. In addition to (
2.4
)–(
2.7
), we assume thatand that ε is dominated from above by δ, that is,Then, there existssuch that, for every, Equation (
2.3
) has an inertial manifoldin, that is,
is a finite-dimensional Lipschitz manifold in;
is positively invariant under, that is,,;
is exponentially attracting, that is, there exists a constantsuch that, for every, there exists a constantsuch that (andare independent of ε and δ)where dist is taken in the norm, with, Γ anddefined by (
5.20
), (
5.21
) and by (
5.24
), respectively.
We set
and
We introduce the scalar product in (inspired from [45]) defined by
where and are, respectively, the projections from onto and and the functions and are defined by
with , in and , respectively. Indeed, we have, noting that ,
Thus, for and , noting that ,
we have that
As a consequence, is orthogonal to and to , and the decomposition is orthogonal with respect to the scalar product and we set and . Let and be the unique orthogonal projections onto and , respectively. We now define the norm
From (5.18) and (5.19), we can deduce that there exists c independent of ε and δ such that
for all . Now, we note that , and are bounded continuous functions on , and there exists a constant such that
hence, there exists such that,
(cf. [43, Lemmas 2.1 and 2.2]). Denote
For any , we have
and
for all , where is independent of ε and δ. Thus, the nonlinear function is globally Lipschitz continuous.
Moreover, there exist , independent of ε and δ, such that
for every .
We define the semigroup generated by Equation (5.2) as
where is the solution to (5.2) at time t. The semigroup is continuous with respect to the metric induced by the equivalent norm .
It follows from the existence theorem of inertial manifolds for non self-adjoint operators [43, Chapter 9, Theorem 2.1] (see also [41]) that the semigroup admits an inertial manifold in . More precisely, there exists a Lipschitz mapping such that the graph of defines an inertial manifold
for the semigroup of dimension n, with respect to the metric induced by the norm . Now, we can derive from an inertial manifold for the semigroup of the original problem (2.3). To see this, we first define
where is independent of and , (recall that d is the space dimension). Note that is an absorbing set for as well and there holds , . Then we restrict to the absorbing set for all times . We also have the following relationship between the semigroups and , namely,
where is the matrix
We set
For every and , is an inertial manifold for the semigroup with respect to the metric induced by the norm and . We observe that there holds , , while only satisfies , (cf. [42,45]). □
Contrary to what is written, the constant c in Estimates (48) and (49) in [4] depends on ε. We thus take this opportunity to correct (43) and (44) in [4] by replacing with and with δ. Remark 1 in [4] is null and void.
Let be an element of . We have
where is the positive real solution to the equation
Let
We have
Let such that . If , then
From (5.25) we can deduce that
hence
There holds true
for every , and where
To see this, we just notice that (5.28) is equivalent to
which holds true whenever (due to (5.15))
which in turn, holds true for any .
It follows from (5.27) and (5.28) that
and since , we deduce that
If , then
Inequality (5.22) follows from (5.29) and (5.30).
The other estimate is straightforward. Indeed, we have
hence (5.23). □
Exponential attractors
We first prove a smoothing property for the difference of two solutions to (2.3).
In addition to (
2.4
)–(
2.7
), we assume that. There existindependent of ε and δ such thatfor any,, anyand for any.
We consider two solutions and of (2.3) with initial conditions
, such that . We set , , and . The pair satisfies the problem
We multiplying (6.2) by ϕ and and integrate over Ω. Then we multiply (6.3) by u and integrate over Ω. Adding together the resulting equations, we obtain
On account of (4.12), there exists independent of ε and δ such that
(cf. Estimates from (4.6) through (4.8)). We deduce from (6.5) that
where c and are independent of ε and δ.
Now, we multiply (6.2) by and (6.3) by , and integrate over Ω. Summing the resulting equations we obtain
We have, in light of (4.12) again and taking advantage of the continuous embedding of into ,
where is independent of ε and δ. We get from (6.7) that
We multiply (6.9) by t and obtain
Integrating (6.10) between 0 and t, we can deduce
where c and do not depend on ε and δ. □
We prove the following result.
Let the assumptions of Proposition
6.1
hold. For everyand every, the semigrouppossesses an exponential attractorin, that is,
is compact and positively invariant, that is,,;
the fractal dimension ofis uniformly bounded with respect to ε and δ, that is,(M is independent of ε and δ);
there exists a constantsuch that, for every bounded subset, there exists a constantsuch that (andare independent of ε and δ)
We set
with , . Since is a bounded absorbing set for on , there is a time independent of ε and δ such that
For every , we have
We now deduce from the second equation of (2.3) that
Integrating (3.9) between 0 and t, we get, owing to (4.12),
It follows from the two previous inequalities that
On the one hand, we have
On the other hand, it follows from (6.1) that
Hence, we conclude with
Estimates (6.1) and (6.12) are the requirements for the existence of an exponential attractor, according to [19,34]. This shows the existence of exponential attractors on . Then, like in [24], we can extend the basin of attraction to the whole phase-space by using the transitivity property of the exponential attraction. □
Continuity properties
Here we wish to compare the dynamics of Problem (2.3) with the one of its singular limit which corresponds to the well-known Cahn–Hilliard equation.
We start this section by showing the existence of smooth absorbing sets.
In addition to (
2.4
)–(
2.7
), we assume that. Then, for everyand, the semigrouprestricted tohas an absorbing set of the formwhereis independent of ε and δ.
Let be in a bounded set B of . We learnt from Section 4 that there exists (depending only on B) such that , . Integrating (3.9) between t and , we deduce that
We multiply the first equation of (2.3) by and , the second one by , and integrate over Ω. Summing the resulting equations and noting that
we obtain
Applying the uniform Gronwall lemma to (7.2), on account of (7.1), there exists (depending only on B) such that
Then integrating (7.2) between t and we deduce that
Finally, we multiply the first equation of (2.3) by and the second one by , and integrate over Ω. Summing the resulting equations and noting that
we deduce
Applying the uniform Gronwall lemma to (7.4), due to (7.3), there exists (depending only on B) such that
hence the result. □
Now, we prove the following.
Let the assumption of Proposition
7.1
hold. For everyand, there exists, independent of ε and δ, such thatfor all.
Let . We set , . There exists , independent of ε and δ, such that
Multiplying the second equation of (2.3) by and integrating over Ω, we obtain
From the first equation of (2.3), we find
Substituting (7.8) into (7.7), we find
We have, owing to (4.12) again,
There exist (independent of δ) such that
and
for any . Therefore, we get from (7.9)
We first multiply (7.12) by , then integrate between s and , for any . This yields
Integrating now (7.13) between t and with respect to s, we deduce
since , hence the result. □
Estimates of the difference of two solutions
We denote .
We show the following estimate.
Let the assumption of Proposition
7.1
hold. Then, there exist, c and, all independent of ε and δ, such that
for any, anyand any.
for any, any.
Let us take in . We set
The functions and satisfy the uniform estimate (7.6) and
We now set and . The pair of functions satisfies the following problem:
We multiply (7.18) by and (7.19) by , and integrate over Ω. We obtain
and
Adding together (7.21) and (7.22), we find
We will show that
Applying the Gronwall’s lemma to (7.23), we obtain, owing to (7.24),
Then integrating (7.23) between 0 and t, we obtain, by virtue of (7.25) and (7.24) again,
Now, multiplying (7.19) by R, and integrating over Ω, we obtain
since , . From (7.18), we deduce that
Substituting (7.28) into (7.27), we deduce
We have , , and we deduce from (7.10) that
We get from (7.11) that
and
We have, on account of (7.25), (7.6) and (7.17)
We then deduce from (7.29) that
We multiply (7.30) by t and obtain
Integrating (7.31) between 0 and t, we find
We get, by virtue of (7.24) and (7.26),
hence
On the other hand, we have (cf. (7.25) and (7.26) again)
so that
Now, we multiply (7.18) by and integrate over Ω. Summing the resulting equation with (7.27), we obtain like in (7.23),
We apply the Gronwall’s lemma to (7.34) between and and find
It follows from (7.35), thanks to (7.32) and (7.33),
Applying the Gronwall’s lemma to (7.34) between s and t, we obtain
for any given and any . Let , independent of ε and δ, be such that . This latter estimate, with , combined with (7.36) gives
hence (i). Finally, estimates (7.5) and (7.37) yield (ii). □
First, we observe that the function is solution to the problem
where ,
and, satisfies
We multiply (7.38) by ω and integrate over Ω, and find
hence
and then
The result follows from (7.41) and (7.44), recalling that . □
A robust family of exponential attractors
Sufficient conditions ensuring the existence of a robust family of exponential attractors that are continuous with respect to the metric induced by the norm were given by the main theorem of [25]. Later, this theorem was extended to a convergence result with respect to the metric induced by in [5]. Let us recall this theorem.
Let,,,,,be Banach spaces such that,. Set,,and endow them with the following normsrespectively, where, with the convention that,, and. Letdenote a closed ball inof radiusand centered at zero. Consider a one-parameter family of strongly continuous semigroupsacting on the phase-space, for eachand. Then assume that there exist,,, and(all independent of ε and δ) such that, setting, the following conditions hold:
There exists a mapwhich is Hölder continuous of exponent α, uniformly with respect to δ. Hereis endowed with the metric topology of.
There exists, independent of ε and δ such thatandis uniformly bounded (with respect to ε and δ) in the-norm. Moreover, setting, the mapsatisfies, for every,where
For any, there holdHere the “lifting” mapis defined byandis the projection onto the first component when, and the identity map otherwise.
The mapis Lipschitz continuous onendowed with the metric topology of, with a Lipschitz constant independent ofand.
The mapis Hölder continuous of exponent ϑ (we do not require uniformity with respect to ε and δ), whereis endowed with the metric topology of.
Then there exists a family of exponential attractorsonwith the following properties:
attractswith an exponential rate which is uniform with respect to ε and δ, that is,for someand some.
The fractal dimension ofis uniformly bounded with respect to ε and δ, that is,
The familyis Hölder continuous with respect to ε and δ, that is, there exist a positive constantandsuch thatfor all. In addition, there exist a positive constantandsuch thatfor all,, and
Here ω, τ, σ andare independent of ε and δ, and they can be computed explicitly.
1. Only Conditions 2, 4, 5 and not necessarily uniformly bounded (with respect to ε) in the -norm, are needed in the construction of exponential attractors that satisfy (i) and (ii) of Theorem 7.1 (cf. [19,34] as mentioned in the proof of Theorem 6.1).
2. To prove (iii) of Theorem 7.1, again Conditions 2 and not necessarily uniformly bounded (with respect to ε) in the -norm is required. In addition, it is sufficient to have, in place of Condition 3
(cf. [25]).
From now on, we set
where is independent of ε and δ and such that for all . We will always assume that . Then we have that
Note that is a bounded absorbing set for as well.
We will denote
We prove the following result.
Let the assumption of Proposition
7.1
hold. Then, there existand, all independent of ε and δ, and a family of exponential attractorsenjoying all the properties of Theorem
6.1
and such thatandwhereis an exponential attractor for.
On account of Theorem 7.1, we let , , , and we check all the Assumptions 1–5. To verify Assumption 1, we show that there exists a constant c such that
for any φ and ψ in . Assumption 2 is given by Estimates (6.1) and (7.5). There holds, in light of Estimate (7.16) and the fact that ,
for any and any . Hence, Assumption 3 is verified. Assumptions 4 and 5 were already proven in Theorem 6.1. This shows the existence of exponential attractors on that satisfy (7.46) and (7.47). Then, like in Theorem 6.1, we can extend the basin of attraction to the whole phase-space . Taking and relying on Estimate (7.15) instead of (7.16), we obtain the existence of a family of exponential attractors on that satisfy (7.45). □
Continuity of inertial manifolds
We end this paper by showing continuity properties for the intersection of the inertial manifolds with the bounded absorbing set (which are in fact exponential attractors). Firstly, we note that there exists a Lipschitz mapping such that the graph of defines an inertial manifold
for the unperturbed “prepared problem”:
(see, e.g., [38]). Here P is the unique orthogonal projection in onto the space spanned by (cf. Section 5), and .
Let
and
for any . Like in the case of exponential attractors, we denote
We prove the following result.
Let the assumptions of Theorem
5.1
hold. Letand. Then, there existsindependent of ε such that,for every.
We were inspired, in part, by the proof of Theorem 8.6 in [11]. Firstly, we can deduce from (5.16) through (5.20) that there exist and (independent of ε) such that
where
For any , we denote by the following Banach space (cf., e.g., [11,37])
with norm
We consider an element of , and we set
where and is given by (4.10) but with g replaced by . Thus, satisfies the following problem:
where
We will prove that, there exists , independent of ε, such that
and, for every , there holds
where
Let be a solution to (5.2). The function satisfies the equation
We write , where and ; and we have that
(cf., e.g., [11,12,43]). Since is a finite-dimensional subspace of , we can choose such that . On account of (7.68), and the fact that , , we can deduce that , , and then , . Similarly, we can show that , . Hence, , due to (7.52). Thus, we deduce from (7.53) that
We have, for ,
and
We infer from (7.51), (7.52) and (7.54) that
that is,
We then deduce from (7.56), with an appropriate choice of n and , that
that is,
Observing that
we deduce from (7.58) that
that is,
We can deduce from (7.50)
Estimates (7.50), (7.59) and (7.60) eventually imply the lower semicontinuity estimate (7.49). □
A corollary of Theorem 7.3 is the following: there exist independent of ε such that
for every , where .
We have , , with and where satisfies and is the solution to
with , where and
It is known that and
(cf. [12]). We have
and
From (7.62), we can also deduce that
We take the -scalar product of (7.62) with and we obtain
hence
We then deduce that
and therefore
We now apply the Gronwall lemma to (7.67) between t and 0, , and find
Collecting (7.64), (7.65) and (7.68), we find
hence the first part of (7.51) holds.
Let us now prove the second part of (7.51). We remind that the unperturbed problem satisfies , so that . We observe that is solution to the linearized problem:
where
For each ,
for some . On account of (5.3), there exists a time varying -finite-dimensional invariant manifold for problem (7.70) of the form (cf. Theorem 3.1, Chapter 3 of [13])
where is solution to:
We have, for all ,
and
On the one hand, we deduce from (7.72), (7.74) and (7.76) that (cf. (7.65))
On the other hand, we take the -scalar product of (7.74) with to obtain
and deduce
We have
on account of (5.3). Therefore,
and then
Note that, due to uniqueness of the solutions and , we have
and therefore
Like (7.69) we can deduce
In particular, , hence the second part of (7.51) holds. □
We have
where
When , we can now see that the sign of is the same as that of
which is positive.
Let us now show the other inequality. We have
where
When , we can see that the sign of is the same as that of
which is negative. The proof of (7.52) is completed. □
Let
and
We observe that and are in fact exponential attractors for the semigroups and , respectively.
We end this subsection by showing the following theorem.
Let the assumptions of Theorem
5.1
and Proposition
7.1
hold. Then, for anythere exist,and a timesuch thatandLetand. Then, for anythere exist,and a timesuch that
Let . Since is an inertial manifold for and is Hölder continuous (see (7.48)), it follows that
This shows that, if , then there exist belonging to and depending only on η such that
We now choose and (which only depends on η) such that , by virtue of (7.15), since . For any and , we have
We deduce from (7.87) and (7.88) that
This result is uniform in and we then obtain the upper semicontinuity (7.83). Note that
where is independent of ε, δ and , .
Proceeding like above, but relying on (7.16) instead of (7.15) since
we prove (7.84). Observe that in (7.84) the distance is taking in the norm and not in .
Let us now prove (7.85). Let . There exists a sequence in that converges to , in the norm , as ε goes to zero, due to (7.61). Since , it follows that is uniformly bounded with respect to ε in the norm (cf. (7.5)), there exists a subsequence which we still denote by and which converges to some , in the norm , due to (7.84). Clearly, and , and this limit is independent of the subsequence chosen. Consequently, the whole sequence converges to , hence the result. □
Footnotes
Acknowledgements
The authors are grateful to the King Fahd University of Petroleum and Minerals for its support. The authors are also grateful to the referees for their helpful and careful reading of previous versions of this manuscript.
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