In this paper, we treat the Cahn–Hilliard equation  on the boundary of some bounded smooth domain. Let be some fixed time and let , or 3, be a bounded domain occupied by a material, where the boundary Γ of Ω is supposed to be smooth enough. We start from the following equations of Cahn–Hilliard type on the boundary Γ:
where denotes the partial derivative with respect to time, and denotes the Laplace–Beltrami operator on Γ (see, e.g., [21, Chapter 3]). Here, the unknowns and stand for the order parameter and the chemical potential, respectively. In the right-hand sides of (1.1) and (1.2), the outward normal derivative on Γ acts on functions that satisfy the following trace conditions:
where and are the traces of μ and u on Σ. Moreover, these functions μ and u solve the following equations in the bulk Ω:
where is a positive constant and Δ denotes the Laplacian.
Note that the nonlinear terms and are the derivatives of the functions and , usually referred as double-well potentials, with two minima and a local unstable maximum in between. The prototype model is provided by , so that , , is the sum of an increasing function with a power growth and another smooth function which breaks the monotonicity properties of the former and is related to the non-convex part of the potential or .
Therefore, we can say that system (1.1)–(1.2) yields the Cahn–Hilliard equation on a smooth manifold Γ, and equations (1.4) and (1.5) reduce to the Poisson equation for μ and the Allen–Cahn equation for u in the bulk Ω, as auxiliary conditions for solving (1.1)–(1.2). In other word, (1.1)–(1.5) is a sort of transmission problem between the dynamics in the bulk Ω and the one on the boundary Γ. With the initial conditions
problem (1.1)–(1.6) becomes an initial value problem of a Poisson–Allen–Cahn system with a dynamic boundary condition of Cahn–Hilliard type, named (P). Indeed, the interaction between μ and u appears only on (1.1)–(1.2), whereas (1.4) and (1.5) are independent equations. As a remark, if , then problem (P) turns out to be a quasi-static system. From (1.1), (1.4) and (1.6), we easily see that the mass conservation on the boundary holds as follows:
Let us mention some related results: the papers [11, 14, 12, 13, 16] consider some quasi-static systems with dynamic boundary conditions (see also [6, 4]), the contributions [3, 5, 8, 9, 15, 17, 20, 28] set a Cahn–Hilliard equation on the boundary as the dynamic boundary condition, and a more complicated system of Cahn–Hilliard type on the boundary with a mass conservation condition is investigated in . Especially, in this paper we will exploit the previous result in , on which equations (1.1)–(1.5) were replaced by
Our essential idea of the existence proof is to be able to pass to the limit as in system (1.8)–(1.12), with (1.6). To be more precise about our arguments, let us introduce a brief outline of the paper along with a short description of the various items.
In Section 2, we present the main results of the well-posedness of system (1.1)–(1.6). A solution to problem (P) is suitably defined. The main theorems are concerned with the existence of the solution (Theorem 2.3) and the continuous dependence on the given data (Theorem 2.4), the second theorem entailing the uniqueness property.
In Section 3, we consider the approximate problem for (P), with two approximation parameters ε and λ, by substituting the maximal monotone graphs with their Yosida regularizations in terms of the parameter λ. Moreover, we obtain uniform estimates with suitable growth order. Here, we can apply the results that have been shown in .
In Section 4, we prove the existence result. The proof is split in several steps. In the first step, we obtain uniform estimates with respect to ε. Then, combining them with the previous estimates of Section 3, we can pass to the limit as . In the second step, we improve suitable estimates in order to make them independent of λ. Then we can pass to the limit as and conclude the existence proof. The last part of this section is devoted to the proof of the continuous dependence.
Finally, in Appendix A, the approximate problem for (P) and some auxiliary results are discussed.
2 Main results
In this section, our main result is stated. At first, we give our target system (P) some equations and conditions as follows: for any fixed constant , we have
where , are given sources, , are known initial data, β stands for the subdifferential of the convex part and π stands for the derivative of the concave perturbation of a double well potential , defined for all r in the domain of . The same setting holds for and .
Typical examples of the nonlinearities β, π are given by
, for all , with for the prototype double well potential
, for all , with for the logarithmic double well potential
where is a large constant which breaks convexity.
, for all , with for the singular potential
where is the indicator function on .
Similar choices can be considered for and the related potential . What is important in our approach is that the potential on the boundary should dominate the potential in the bulk, that is, we prescribe a compatibility condition between β and (see the later assumption A5) that forces the growth of β to be controlled by the growth of . A similar approach was taken in previous analyses, see [2, 6, 4, 5, 9, 28].
As a remark, plays the role of a viscous parameter. Indeed, if , then equations (2.1) and (2.2) become the stationary problem in Q, namely, the quasi-static system. A natural question arises whether one can investigate also the case or, in our framework, also study the singular limit as . In our opinion, this is not a trivial question and deserves some attention and efforts. For the moment, we can just highlight it as open problem.
2.1 Definition of the solution
We treat problem (P) by a system of weak formulations. To do so, we introduce the spaces , , , , , , with usual norms and inner products; we denote them by , and , , and so on. Concerning these inner products and norms, we use the same notation for scalar and vectorial functions (e.g., typically gradients).
Moreover, we put and . Then and are Hilbert spaces with the inner products
and related norms. As a remark, if , then is the trace of z on Γ, while if , then and are independent. Hereafter, we use the notation of a bold letter like to denote the pair which corresponds to the letter, that is, for .
hereafter, we put
where . The space denotes the dual of , and denotes the duality pairing between and . Moreover, it is understood that is embedded in in the usual way, i.e., for all , . Then we obtain , where “” stands for the dense and compact embedding, namely, is a standard Hilbert triplet.
Under this setting, we define the solution of (P) as follows.
The triplet is called the solution of (P) if , , satisfy
for all and a.a. , and
Taking in the weak formulation (2.8), we see that (2.8) and (2.11) imply the mass conservation (2.6) on the boundary. Moreover, for any , taking in (2.8) and using the trace condition on μ, we deduce that
for a.a. , whence the regularities , allow us to infer the higher regularity . Moreover, the boundedness in gives us the property , as well as
this is the weak formulation of (2.3).
2.2 Main theorems
The first result states the existence of the solution. To this end, we assume the following:
The maximal monotone graphs β and in are the subdifferentials and of some proper lower semicontinuous and convex functions and satisfying , with some effective domains and , respectively. This implies that and .
are Lipschitz continuous functions with Lipschitz constants L and , and they satisfy .
and there exist positive constants such that
where and denote the minimal sections of β and .
and the compatibility conditions , hold.
The minimal section of β is specified by and the same definition applies to . These assumptions are the same as in [2, 5]. Concerning assumption (A1), let us note that the regularity conditions for f and are not symmetric. In fact, we need more regularity for the source term on the boundary, since the equation on the boundary is of Cahn–Hilliard type, while the equation on the bulk turns out to be of the simpler Allen–Cahn type. Of course, here the condition plays a role, and if the term is not present in (2.2), then we would certainly need higher regularity for f.
The second result states the continuous dependence on the data. The uniqueness of the component of the solution is obtained from this theorem. Here, we just use the following regularity properties on the data:
Then we obtain the continuous dependence on the data as follows:
3 Approximate problem and uniform estimates
In this section, we first consider an approximate problem for (P), and then we obtain uniform estimates. For each , we introduce an approximate problem ) where the proof of the well-posedness of ) is given in Appendix A.
3.1 Moreau–Yosida regularization
For each , we define , along with the associated resolvent operators , by
for all , where is the same constant as in assumption (2.12). Note that the two definitions are not symmetric, since in the second one, it is and not directly λ to be used as approximation parameter; let us note that this adaptation comes from the previous work . Now, we easily have . Moreover, the related Moreau–Yosida regularizations of fulfill
It is well known that is Lipschitz continuous with Lipschitz constant and is also Lipschitz continuous with constant . In addition, for each , we have the standard properties
Let us point out that, using [2, Lemma 4.4], we have
for all , with the same constants ϱ and as in (2.12).
Now for each , let and be smooth approximations for and , so that with and with satisfying
where is a positive constant independent of . Indeed, and satisfying (3.3)–(3.6) are given in Appendix A. Hereafter, we use , which satisfies as for . Then we can solve the following auxiliary problem.
As a remark, the trace conditions is included in the regularities (3.7)–(3.8). The strategy of the proof of this proposition is based on the previous work [5, Theorems 2.2, 4.2]. It is given in Appendix A.
3.2 A priori estimates
In order to obtain the uniform estimates independent of the approximate parameter ε and λ, we use the following type of Poincaré–Wirtinger inequalities (see, e.g., [29, 22]): there exists a positive constant such that
because we have the compact embeddings for , (see, e.g., [27, Chapter 1, Theorem 16.1]) and , respectively. Next, from the standard theorem for the trace operators of and for (see, e.g., [23, Chapter 2, Theorem 2.24], [29, Chapter 2, Theorem 5.5]), we see that there exists a positive constant such that
for . Moreover, the boundedness of some recovering operator of the trace gives us
There exist two positive constants and , depending on τ but independent of ε and , such that
We test (3.10) at time s by , the time derivative of . Integrating the resultant over leads to
for all . Next, testing (3.12) by and integrating the resultant over , we obtain
for all . Therefore, in order to estimate the right-hand side of (3.26), we prepare the estimate of . Indeed, from the Young inequality, we see that
for all and some . Now, from (A4), we can use the fact that , and then we deduce
Therefore, by taking in (3.27), we have
for some and . Moreover, from the Young inequality, it turns out that
for all . Thus, taking , and using (3.3)–(3.6) and the Gronwall inequality, we see that there exists a positive constant , depending on , , L, , , , and τ, in which as , independent of ε and λ, such that
Moreover, using this, (3.19) implies
By comparison, the following estimates for and are obtained.
Let be the same constant as in Lemma 3.2. Then, for each ε and , the following estimates hold:
From (3.9), we easily see that
Next, we separate as follows:
Then the first term of the right-hand side is estimated as follows:
Moreover, for the second term, we see that
We have obtained the uniform estimates (3.21), (3.22), (3.34), (3.35) provided in Lemmas 3.2 and 3.3 independent of ε and , actually as . They can be used throughout this paper by considering the limiting procedure for each fixed , and next the limiting procedure .
4 Proof of the main theorem
In this section, we prove Theorem 2.3.
4.1 Additional uniform estimate independent of
In this subsection, we obtain additional uniform estimates independent of , which may depend on . Therefore, we use them only in the nest subsection to consider the limiting procedure as .
There exist two positive constants and , depending on but independent of , such that
From the Lipschitz continuity of and , we see from (3.21) that there exists a positive constant , which is proportional to the Lipschitz constants of and of , such that (4.1) holds. Next, let us point out the variational equality, deduced from (3.10) and (3.12):
for all . Then, using the Young inequality along with (3.3), (3.21), (3.22) and (4.1), we deduce the uniform estimate of in . Next, with the help of the Poincaré–Wirtinger inequality (3.14) again, we deduce the uniform estimate of in . Thus, combining the resultant with (3.22), we see that there exists a positive constant , depending on but independent of , such that (4.2) holds. ∎
There exists a positive constant , depending on but independent of , such that
for some positive constant , and owing to both the uniform bounds, we see that
4.2 Passage to the limit as
In this subsection, we pass to the limit in the approximating problem as . Indeed, owing to the estimates stated in Lemmas 3.2, 3.3, 4.1 and 4.2, there exist a subsequence of ε (not relabeled) and some limit functions , and such that
4.3 Proof of Theorem 2.3
In this subsection, we prove the main theorem. To do so, we are going to produce estimates independent of λ, and then pass to the limit as . The point of emphasis is the effective usage of the mean value zero function.
There exists a positive constant , independent of , such that
Recall (2.7) and take
in (4.19). Then we have
Let now be the solution of the following problem:
This problem has one and only one solution, see [5, Appendix, Lemma A] and repeat the proof with the different condition. Moreover, due to (3.14) and (3.15), there exists a positive constant , independent of , such that
and, consequently, we use (4.18) with , to conclude that
Then, from (4.25) and the properties
Now, squaring both sides, we obtain
because for a.a. . Therefore, we integrate (4.28) over with respect to time. Then the right-hand side can be bounded due to (4.20), (4.21) and (4.22). Thus, there exists a positive constant , independent of , such that (4.24) holds. ∎
for a.a. . Then we obtain the following estimate.
There exist two positive constants and , independent of , such that
Taking and in (4.19), we obtain
for a.a. . Thus, (4.21) and (4.29) imply the first estimate of (4.30) for some positive constant independent of . Moreover, by using (3.14) with the above, (4.21) ensures the validity of the second estimate in (4.30). ∎
There exist a positive constant , independent of , such that
which are rigorous due to the regularity of and stated in (4.4). Recalling [2, Lemmas 4.4, 4.5], we observe that the proof is essentially the same as in these lemmas. Therefore, we omit the details for (4.31)–(4.34).
We have collected all information which enables us to pass to the limit as .
Proof of Theorem 2.3.
which imply that
due to the maximal monotonicity of β and . Finally, we can pass to the limit as in the variational equality (4.18) as well as in (4.35)–(4.36) obtaining (2.8)–(2.10), and in the initial conditions (4.17) obtaining (2.11). Thus, we arrive at the conclusion. ∎
4.4 Proof of Theorem 2.4
Proof of Theorem 2.4.
Let us note that the solutions , , of (P) satisfy
for all and . Let us use the notation
We take the difference of equations (4.50), test it by and integrate over with respect to s, obtaining
for all . We start by discussing the last term in the left-hand side of (4.51). Let be a solution of the problem
for all , with
a.e. in . Now, testing (4.52) by , we have
for all . On the other hand, considering the difference of (4.49) and testing then by , we infer that
for all . Therefore, we can differentiate (4.52) with respect to time, then test by and integrate the resultant over with respect to s, obtaining
for all . Thus, we have the contribution
on the left-hand side and
by the monotonicity,
and this will be treated by the Gronwall inequality,
due to the compactness inequality (3.18), also for this term we will use the Gronwall inequality (the last two terms can be simply treated by the Young inequality), and finally
for all , that is, and . Then the continuous dependence (2.13) follows from the application of the Gronwall inequality. ∎
We use the same notation as in the previous sections. We also use the following notation of function spaces. For each fixed , define a subspace of by , where ,
Moreover, define an inner product of by
Then we see that the induced norm and the standard norm are equivalent, because
Next, we define , with for all , where we use , a bilinear form defined by
Let us define a linear bounded operator by
Then there exists a positive constant such that
see, e.g., [5, Appendix]. Thus, we see that and the standard are equivalent norms of , and is the duality mapping from to . We also define the inner product in by
Thanks to [5, Appendix] again, we obtain .
The operator defined by
is the projection from to with respect to -norm, namely,
Firstly, we see that
Next, we infer that
Therefore, we deduce that
i.e., (A.1) holds. ∎
We also easily obtain the following conditions:
Incidentally, we note that another possibility of projection is given by
and is actually the projection from to with respect to the standard norm (cf. (A.1)). However, this choice is not suitable from the viewpoint of the trace condition. Indeed, even if .
Note that . Multiplying the above equation by the solution , integrating it over and using the Young inequality, we have
for all . Thus, there exist a subsequence of ε (not relabeled) and some limit function such that
that is, there exist a subsequence of ε (not relabeled) and some limit function such that
From (A.4) we see that
therefore it is enough to prove that there exists a positive constant such that
Testing (A.4) by and integrating the resultant with respect to time, we obtain
for all . Therefore, using the Gronwall inequality, we see that there exists a positive constant , depending on and but independent of , such that
for all . ∎
For each , the solution of the following elliptic system
The strategy of the proof is similar to the one of [7, Lemma A.1]. By virtue of [5, Lemma C], there exists such that satisfies system (A.5)–(A.6). Moreover, assumption (A2) gives us the required regularity . Next we show (3.5). Indeed, testing equation (A.5) by and the second equation in (A.6) by , adding them, and using the Young inequality, we obtain
as . Moreover, we have
which entails that
for all . Indeed, in the case , because of the fact that , we see from (3.2) that
In the case , we have
Then it turns out that
Therefore, using the Young inequality, we deduce
Thus, we obtain that there exists a positive constant such that
as , because, weakly in and as . Then, from (A.6), we can also infer that
Now, if we test (A.6) by , then we obtain
hence, from (3.1), there exists a positive constant , independent of , such that
Thus, we get the conclusion. ∎
Proof of Proposition 3.1.
For each , let us consider the Cauchy problem for the following equivalent evolution equation:
and we define by
Then we see that φ is proper, lower semicontinuous and convex on , and the subdifferential on is characterized by with . The Cauchy problem (A.7)–(A.9) can be solved by applying [5, Sections 4.2–4.4], based on the abstract theory of doubly nonlinear evolution equation , because all assumptions for and in order to obtain the strong solution are satisfied. ∎
The authors are grateful to the anonymous referees for reviewing the original manuscript and for many valuable comments that helped to clarify and refine this paper.
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About the article
Published Online: 2018-07-20
Published in Print: 2019-03-01
The present paper benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations”, the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia for PC; the support of the JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant Number 17K05321 for TF.
Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 16–38, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0055.
© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0