Time periodic solutions of Cahn Hilliard system with dynamic boundary conditions

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1 Time periodic solutions of Cahn Hilliard system with dynamic boundary conditions arxiv:7.3697v [math.ap] Dec 7 Taishi Motoda Graduate School of Education, Kyoto University of Education Fujinomori, Fukakusa, Fushimi-ku, Kyoto 6-85 Japan motoda.math@gmail.com Abstract The existence problem for Cahn Hilliard system with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations by using viscosity approach and the Schauder fixed point theorem in the level of approximate ploblem. One of the key point is the assumption for maximal monotone graphs with respect to their domains. Thanks to this, we obtain the existence result of the weak solution by using the passage to the limit. Key words: solutions. Cahn Hilliard system, dynamic boundary condition, time periodic AMS MOS) subject classification: 35K5, 35A, 35B, 35D3. Introduction In this paper, we consider the following Cahn Hilliard system with dynamic boundary condition and time periodic condition, say P), which consists of the following: u µ = in Q :=, T ),.) t µ = κ u + ξ + πu) f, ξ βu) in Q,.) u = u, µ = µ on Σ :=, T ),.3) u t + νµ µ = on Σ,.4) µ = κ ν u κ u + ξ + π u ) f, ξ β u ) on Σ,.5) u) = ut ) in, u ) = u T ) on.6) where < T < +, is a bounded domain of R d d =, 3) with smooth boundary :=, κ, κ are positive constants, ν is the outward normal derivative on, u, µ stand for the trace of u and µ to, respectively, is the Laplacian, is the Laplace Beltrami operator see, e.g., []), and f : Q R, f : Σ R are given data. Moreover,

2 Time periodic solutions of Cahn Hilliard system in the nonlinear diffusion terms, β, β : R R are maximal monotone graphs and π, π : R R are Lipschitz perturbations. The Cahn Hilliard equation [8] is a description of mathematical model for phase separation, e.g., the phenomenon of separating into two phases from homogeneous composition, the so-called spinodal decomposition. In.).), u is the order parameter and µ is the chemical potential. Moreover, it is well known that the Cahn Hilliard equation is characterized by the nonlinear term β + π. It play an important role as derivative of double-well potential W. As pioneering results, Kenmochi, Niezgódka and Paw low study the Cahn Hilliard equation with constraint by subdifferential operator approach [3]. In addition, Kubo investigates the Cahn Hilliard equation [5]. Recently, in [], it is discussed the strong solution of the Cahn Hilliard equation in bounded domains with permeable and non-permeable walls in maximal L p regularity spaces. In terms of.3).5), we consider the dynamic boundary condition as being u, µ unknown functions such that satisfied trace condition.3). The dynamic boundary condition was treated in recent years, for example, for the Stefan proplem [ 3, 4], wider the degenerate parabolic equation [5,6] and the Cahn Hilliard equation [,,7,9,9]. To the best our knowledge, the type of dynamic boundary condition on the Cahn Hilliard equation like P) is formulated in [8]. As you can see, we consider the same type of equations.).) on the boundary. In other words,.).5) is a transmission problem connecting and. The nonlinear term β + π is also the derivative of double-well potential W. As a well-known example, W = W = /4)r ), namely, W = W = r3 r for r R. This is called the prototype double well potential. Note that we do not have to take differnt nonlinear terms W and W in and on, respectively. However, in problem P), we treat it differently to generalize. In this case, it is necessary to assume the compatibility condition see e.g., [9, ]), stated A4). The other example of W, W is stated later. About the problem P), Colli and Fukao study the Cahn Hilliard system with daynamic boundary condition and initial value condition []. They set a function space that the total mass is zero. This idea is arised from the property of dynamic boundary condition. The property is called the toal mass conservation. Moreover, focusing on.6), the study respect to existence of time periodic solutions of the Cahn Hilliard equation is not much. For example, [6 8,3]. In particular, Wang and Zheng discuss the existence of time periodic solution of the Cahn Hilliard equation with Neumann boundary condition [3]. They employ the method of [4]. Note that they impose two assumptions for a maximal monotone graph, specifically, restricted domains and the following growth condition for maximal monotone graph β : βr) cr for all r R, for some positive constant c. However, the above assumption is too restrictive for some physical applications. In this paper, following the method of [3], we apply the abstract theory of evolution equations by using the viscosity approach and the Schauder fixed point theorem in the level of approximate problem. Moreover, by virtue of the viscosity approach, we can apply the abstract result [4]. Noting that, in [], they do not impose the assumption of growth condition for maximal monotone graphs β and β. Therefore, by setting the

3 Taishi Motoda 3 appropriate convex functional and using the Poincaré Wirtinger inequality, we can also avoid imposing the growth condition even though we consider the time periodic problem. Thanks to this result, we can choose wider kinds of nonlinear diffusion terms β + π and β + π. However, respect to the assumption of restricted domains see A5)), this is a essential point to slove the problem P). Nevertheless, we can infer that the assumption A5) is not necessary when we choose the prototype double well potential. We sketch it Remark 4.. The present paper proceeds as follows. In Section, main theorem and definition of solution are stated. At first, we prepare the notation used in this paper and set appropriate function spaces. Next, we introduce the definition of weak solution of P) and the main theorems are given there. Also, we give the example of double-well potentials. In Section 3, in order to prove convergence theorem, at first, we set convex functionals and consider approximate problems. Next, we obtain the solution of P) ε by using the Schauder fixed point theorem. Finally, we deduce uniform estimates of the solution of P) ε. In Section 4, we prove the existence of weak solutions by passing to the limit ε. Finally, we sketch the case that we choose the derivative of prototype double well potential as nonlinear diffusion terms. A detailed index of sections and subsections follows.. Introduction. Main results.. Notation.. Definition of the solution and main theorem 3. Approximate problem and uniform estimates 3.. Abstract formulation 3.. Approximate problem for P) 3.3. Uniform estimates 4. Proof of convergence theorem Main results. Notation We introduce the spaces H := L ), H := L ), V := H ), V := H ) with standard norms H, H, V, V and inner products, ) H,, ) H,, ) V,, ) V, respectively. Moreover, we set H := H H and V := { z := z, z ) V V : z = z a.e. on }.

4 4 Time periodic solutions of Cahn Hilliard system H and V are then Hilbert spaces with inner products u, z) H := u, z) H + u, z ) H for all u := u, u ), z := z, z ) H, u, z) V := u, z) V + u, z ) V for all u := u, u ), z := z, z ) V. Note that z V implies that the second component z of z is equal to the trace of the first component z of z on, and z H implies that z H and z H are independent. Throughout this paper, we use the bold letter u to represent the pair corresponding to the letter; i.e., u := u, u ). Let m : H R be the mean function defined by mz) := { } zdx + z d + for all z H, where := dx, := d. Then, we define H := {z H; mz) = }, V := V H. Moreover, V, V denote the dual spaces of V, V, respectively; the duality pairing between V and V is denoted, V,V. We define the norm of H by z H := z H for all z H. Now, we define the bilinear form a, ) : V V R by au, z) := κ u zdx + κ u z d for all u, z V. Then, for all z V, z V := az, z) is the norm of V. Also, for all z V, we let F : V V be the duality mapping defined by F z, z V,V := az, z) for all z V. Then, the following the Poincaré Wirtinger inequality holds: there exists a positive constant c P such that z V c P z V for all z V.) see [, Lemma C]). Moreover, we define the inner product of V by z, z ) V := z, F z V,V for all z, z V. Also, we define the projection P : H H by P z := z mz) for all z H, where :=, ). Then, since P is a linear bounded operator, note that the following property holds: let {z n } n N be a sequence in H such that z n z weakly in H for some z, then we infer that P z n P z weakly in H,.) because P is a single-valued operator. Then, we have V H V, where stands for compact embedding see [, Lemmas A and B]).

5 Taishi Motoda 5. Definition of the solution and main theorem In this subsection we define our solution for P) and then we state the main theorem. Firstly, from.),.4) and the property of dynamic boundary condition, it holds the following total mass conservation: ut)dx + u t)d = u)dx + u )d := m for all t [, T ]. To define solutions, we use the following notation: the variable v := u m ; the datum f := f, f ); the function πz) := πz), π z )) for z H. Moreover, we set the space W := H ) H ). Definition.. The triplet v, µ, ξ) is called the weak solution of P) if v H, T ; V ) L, T ; V ) L, T ; W ), µ L, T ; V ), ξ = ξ, ξ ) L, T ; H), and they satisfy µt), z ) v t), z V,V + a µt), z ) = for all z V.3) H = a vt), z ) + ξt) m ξt) ) + π vt) + m ) f, z ) H for a.a. t, T ), and ξ βv + m ) a.e. in Q, ξ β v + m ) a.e. on Σ for all z V.4) with Remark.. v) = vt ) in H..5) We can see that µ := µ, µ ) satisfies µ = κ u + ξ m ξ ) + πu) f a.e. in Q, µ = κ ν u κ u + ξ m ξ ) + π u ) f a.e. on Σ, where u = v + m and u = v + m, because of the regularity v L, T ; W ). Remark.. In.4), this is different from the following definition of [, Difinition.]: ) µt), z = a vt), z ) + ξt) + π vt) + m H ) f, z ) for all z V.6) H for a.a. t, T ). However, by setting µ := µ + m ξ ), µ satisfies µ L, T ; V ) and.6). Hence, in other words, we can employ.6) as definition of P) replaced by.4). We assume that

6 6 Time periodic solutions of Cahn Hilliard system A) f L, T ; V ) and ft) = ft + T ) for a.a. t, T ). A) π, π : R R are locally Lipschitz continuous functions with Lipschitz contants L, L, respectively; A3) β, β : R R are maximal monotone graghs, which is the subdifferential β = R β, β = R β of some proper lower semicontinuous convex functions β, β : R [, + ] satisfying β) = β ) = with domains Dβ) and Dβ ), respectively. A4) Dβ ) Dβ) and there exist positive constant ρ, c > such that β r) ρ β r) + c for all r Dβ ) :.7) A5) Dβ), Dβ ) are bounded domains with non-empty interior and, i.e., Dβ) = [σ, σ ] and Dβ ) = [σ, σ ] for some constants σ, σ, σ and σ with < σ σ < σ σ <. In particular, A3) yields β). The assumption A5) is not imposed []. However, it is essential to obtain uniform estimates in Section 3. This is the greatest difficulties of time periodic problem. Also, the assumption of compatibility of β and β A4) is the same as in [9, ]. Now, we give the example respect to nonlinear diffusion terms under the above assumptions: βr) = β r) = α /) ln + r)/ r)), πr) = π r) = α r for all r Dβ) = Dβ ) =.) and < α < α. for the logarithmic double well potential W r) = W r) = α /){ r) ln r)/) + + r) ln + s)/)} + α /) r ). The condition α < α ensures that W, W have double-well forms see e.g., []). βr) = β r) = I [,] r), πr) = π r) = r for all r Dβ) = Dβ ) = [.] for the singular potential W r) = W r) = I [,] r) r /. βr) = β r) = I [,] r) + r 3, πr) = π r) = r for all r Dβ) = Dβ ) = [.] for the modified prototype double well potential W r) = W r) = I [,] r) + /4)r ) r /. The third example is modified due to the assumption A5). the original prototype double well potential see Remark 4.). Our main theorem is given now. However, we can choose Theorem.. Under the assumptions A) A4), for any given m intdβ ), there exists a weak solution of P) such that { } u)dx + u ) = m. +

7 Taishi Motoda 7 3 Approximate problem and uniform estimates In section, we consider the approximate problem and obtain the uniform estimates to show the existence of weak solutions of P). 3. Abstract formulation In order to prove the main theorem, we apply the abstract theory of evolution equation. To do so, we define a proper lower semicontinuous convex functional ϕ : H [, + ] by κ ϕz) := z dx + κ z d + βz + m )dx + β z + m )d if z V with βz + m ) L ), β z + m ) L ), + otherwise. Next, for each ε, ], we define a proper lower semicontinuous convex functional ϕ ε : H [, + ] by κ z dx + κ z d ϕ ε z) := + β ε z + m )dx + β,ε z + m )d if z V with β ε z + m ) L ), β,ε z + m ) L ), + otherwise, where β ε, β,ε defined as follows are Moreau Yosida regularizations of β, β, respectively: { } β ε r) := inf s R ε r s + βs) = ε r J εr) + β J ε r) ), } { β,ε := inf s R ερ r s + β s) = ερ r J,εr) + β J,ε r) ), where ρ is a constant as in.7) and J ε, J,ε : R R is resolvent operator given by J ε r) := I + εβ) r), J,ε r) := I + ερβ ) r) for all r R. Moreover, β ε, β,ε : R R defined as follows are Yosida approximatioon for maximal monotone operators β, β, respectively: β ε r) := ε r Jε r) ), β,ε r) := r J,ε r) ) ερ for all r R, where J ε, J,ε : R R are resolvent operators. Then, it holds β ε ) = β,ε ) =. It is well known that β ε, β,ε are Lipschitz continuous with Lipschitz constant /ε, /ερ), respectively. Here, we have following properties: β ε r) βr), β,ε r) β r) for all r R.

8 8 Time periodic solutions of Cahn Hilliard system Hence, it holds ϕ ε z) ϕz) for all z H. These properties of Yosida approximation and Moreau Yosida regularizations are as in [5,6,3]. Moreover, thanks to [9, Lemma 4.4], it holds β ε r) ρ β,ε r) + c for all r R 3.) with the same constants ρ and c as in.7). Now, for each ε, ], we also define two proper lower semicontinuous convex functionals ϕ, ψ ε : H [, + ] by κ z dx + κ z d if z V, ϕz) := + otherwise and β ε z + m )dx + β,ε z + m )d if z V, ψ ε z) := + otherwise. Then, from [, Lemma C], the subdifferential A := H ϕ on H is characterized by Az) = z, ν z z ) with z = z, z ) DA) = W V. Moreover, the representation of the subdifferential H ψ ε is given by H ψ ε z) = P β ε z + m ) for all z H. This is proved by the same way as [6, Lemma 3.]. Noting that it holds D H ψ ε ) = H and A is a maximal monotone operator, it follows from the abstract monotonicity methods see e.g., [5, Sect..]) that A + H ψ ε is also a maximal monotone operator. Moreover, by a simple calculation, we deduce that A + H ψ ε ) H ϕ ε. Hence, it follows that, for any z H, H ϕ ε z) = A + H ψ ε ) z) 3.) see e.g., [3]). 3. Approximate problem for P) Now, we consider the following approximate problem, say P) ε : for each ε, ] find v ε := v ε, v,ε ) satisfying εv εt) + F v εt) + H ϕ ε vε t) ) + P π v ε t) + m )) = P ft) in H for a.a. t, T ) 3.3) v ε ) = v ε T ) in H. 3.4) where, for all z H, πz) := πz), π z )) is cut-off function of π, π given by if r σ, πσ )r σ + ) if σ r σ, πr) := πr) if σ r σ, πσ )r σ ) if σ r σ +, if r σ + 3.5)

9 Taishi Motoda 9 and if r σ, π σ )r σ + ) if σ r σ, π r) := π r) if σ r σ, π σ )r σ ) if σ r σ +, if r σ + 3.6) for all r R, respectively. We establish the above cut-off function by referring to [3]. From now, we show the next proposition. Proposition 3.. unique function Under the assumptions A) A5), for each ε, ], there exists a such that v ε satisfies 3.3) and 3.4). v ε H, T ; H ) L, T ; V ) L, T ; W ) In order to show the Proposition 3., we use the method in [3], that is, we employ the viscosity approach. Conforming to the method, we consider the following problem: for each ε, ] and f L, T ; V ), F + εi)v εt) + ϕ ε vε t) ) = ft) in H for a.a. t, T ), 3.7) v ε ) = v ε T ) in H. 3.8) Now, we can apply the abstract theory of doubly nonlinear evolution equation respect to time periodic problem [4] for 3.7), 3.8) because the operator εi + F and ϕ ε are coercive in H. This is an important assumption to use Theorem. in [4]. It is an advantage of the viscosity approach. Hence, we obtain the next proposition. Proposition 3.. For each ε, ] and f L, T ; V ), there exists a unique function v ε such that 3.7) and 3.8). Hereafter, we apply the Schauder fixed point theorem to prove the existence of the solution of the problem P) ε. To this aim, we set Y := { v ε H, T ; H ) L, T ; V ); v ε ) = v ε T ) }. Firstly, for each v ε Y, we consider the following problem, say P ε ; v ε ): εv εs) + F v εs) + ϕ ε vε s) ) + P π v ε s) + m )) = P fs) in H 3.9) for a.a. s, T ), with v ε ) = v ε T ) in H. Next, we obtain the estimates of the solution of P ε ; v ε ) to apply the Schauder fixed point theorem. Note that we can allow the dependent of ε, ] for estimates of Lemma 3. because we use the Schauder fixed point theorem in the level of approximation.

10 Time periodic solutions of Cahn Hilliard system Lemma 3.. Let v ε be the solution of problem P ε ; v ε ), it holds the following estimates. There exist positive constants C ε, C, C 3ε such that T and v ε s) V ds + T vε t) V + for a.a. t, T ). ε T T v εs) H ds + v εs) C V ds ε, 3.) ) T ) β ε vε s) + m dxds + β,ε v,ε s) + m ds C 3.) β ε vε t) + m ) dx + β,ε v,ε t) + m ) d C3ε 3.) Proof At first, for each v ε Y, note that there exists a positive constant M, depending only on σ, σ, σ and σ, such that π v ε t) + m ) H M for all t [, T ]. 3.3) Now, testing 3.9) at time s, T ) by v εs) and using the Young inequality, we infer that ε v ε s) H + v ε s) + d V ds ϕ ε vε s) ) for a.a. s, T ). Therefore, we have that = P fs) P π v ε s) + m )), v εs) ) H fs) + V v ε s) + M V ε + ε v ε s) H ε v ε s) H + v ε s) + d V ds ϕ ε vε s) ) fs) + M V ε. 3.4) Then, integrating it over, T ) with respect to s and using the periodic property, we see that T ε v ε s) T H ds + v ε s) T V fs) MT ds + ds V ε, which implies that it follows the first estimate 3.). On the other hand, testing 3.9) at time s, T ) by v ε s) and from.), we deduce that d vε s) + ε d ds V vε s) ds H + ϕ ε vε s) ) P fs) P π v ε s) + m )), v ε s) ) H + ϕ ε ) c Pfs) H + vε s) 4c H + c P M + ϕ) P c Pfs) H + vε s) 4 V + c P M + ϕ) c P fs) H + ϕ ε vε s) ) + c P M + ϕ)

11 Taishi Motoda for a.a. s, T ), thanks to the definition of the subdifferential. From the definition of ϕ ε, it follows that d vε s) + ε d ds V vε s) ds H + vε s) 4 V + ) ) β ε vε s) + m dx + β,ε v,ε s) + m d c P fs) H + c P M + + ) β ε m dx + β,ε m ) d for a.a. s, T ). Integrating it over, T ) and using the periodic property, we see that T v ε s) T ) T ) V ds + β ε vε s) + m dxds + β,ε v,ε s) + m dds 4c Pf + 4c L,T ;H ) PT M + T ) ) β m + T β m. Hence, there exist a positive constant C such that the second estimate 3.) holds. Next, for each s, t [, T ] such that s t, we integrate 3.4) over [s, t] with respect to s. Then, by neglecting the first two positive terms, we have ϕ ε vε t) ) ϕ ε vε s) ) + T fs) MT ds + V ε for a.a. s, t [, T ], namely, vε t) ) ) V + β ε vε t) + m dx + β,ε v,ε t) + m d vε s) ) ) V + β ε vε s) + m dx + β,ε v,ε s) + m d + T fs) MT ds + V ε 3.5) for a.a. s, t [, T ]. Now, integrating it over, t) with respect to s, we deduce that t vε t) ) ) V + t β ε vε t) + m dx + t β,ε v,ε t) + m d T vε s) T ) T ) V ds + β ε vε s) + m dxds + β,ε v,ε s) + m dds + T T fs) MT ds + V ε 3.6) for a.a. t [, T ]. In particular, putting t := T and dividing 3.6) by T, it follows that vε T ) ) ) V + β ε vε T ) + m dx + β,ε v,ε T ) + m d T vε s) T V ds + T ) β ε vε s) + m dxds T + T ) T β,ε v,ε s) + m dds + fs) MT ds + T V ε. 3.7)

12 Time periodic solutions of Cahn Hilliard system Hence, combining the second estimste 3.) and 3.7), we see that v ε T ) ) ) V + β ε vε T ) + m dx + β,ε v,ε T ) + m d C T + T fs) MT ds + V ε. Moreover, from the periodic property, we infer that vε ) ) ) V + β ε vε ) + m dx + β,ε v,ε ) + m d C T + T fs) MT ds + V ε. 3.8) Now, let s be in 3.5). Then, owing to 3.8), we deduce that vε t) ) ) V + β ε vε t) + m dx + β,ε v,ε t) + m d C T + f L,T ;V ) + MT ε for a.a. t [, T ]. Therefore, there exisits a positive constant C 3ε such that the final estimate 3.) holds. In terms of 3.), the key point to prove the estimate is exploiting 3.3). 3.3) is arised from the form of cut-off functions 3.5), 3.6). The form of cut-off functions depend on the assumption A5) essentially. However, considered the same estimate in [, Lemma 4.], They do not impose the assumption. They use the Gronwall inequality to obtain the estimate because the initial value is given data. On the other hand, we can not obtain it even though we use the Gronwall inequality, becasuse the initial value is not given. For this reason, it is nacessary to impose A5). This is a difficult point to solve the time periodic problem. Now, we show the existence of solutions of approximate problem P) ε. Proof of Proposition 3. To this aim, we apply the Schauder fixed point theorem. To do so, we set { Y := v ε Y ; sup v ε V + ε } v ε M H t [,T ],T ;H ) ε, where M ε is a positive constant and be determined by Lemma 3.. Then, the set Y is non-empty compact convex on C, T ; H ). Now, from Proposition 3., for each v ε Y, there exists a unique solution v ε of P ε ; v ε ). Moreover, from Lemma 3., it holds v ε Y. Here, we define the mapping S : Y Y such that, for each v ε Y, corresponding v ε to the solution v ε of P ε ; v ε ). Then, the mapping S is continuous on Y with respect to topology of C, T ; H ). Indeed, let { v ε,n } n N Y be v ε,n v ε in C, T ; H ) and

13 Taishi Motoda 3 {v ε,n } n N be the sequence of the solution of P ε ; v ε,n ). From Lemma 3., there exist a subsequence {n k } k N, with n k as k, and v ε H, T ; H ) L, T ; V ) such that v ε,nk v ε weakly star in H, T ; H ) L, T ; V ). 3.9) Hence, from 3.9) and the Ascoli Alzela theorem see e.g., [3]), there exists a subsequence not relabeled) such that as k. Also, we have v ε,nk v ε in C[, T ]; H ) 3.) v ε,n k v ε weakly in L, T ; H ) 3.) as k. Because we have v ε,nk ) = v ε,nk T ), it holds v ε ) = v ε T ) in H. Hereafter, we show that this v ε is the solution of P ε ; v ε ). Since v ε,nk is the solution of P ε ; v ε,nk ), we see that T P fs) P π vε,nk s) + m )) εv ε,nk s) F v ε,n k s), ηs) v ε,nk s) ) H ds T ) T ϕ ε ηs) ds ϕ ε vε,nk s) ) ds 3.) for all η L, T ; H ), thanks to the definition of subdifferential ϕ ε. follows from v ε,nk v ε in C, T ; H ) that Moreover, it P π v ε,nk + m ) ) P π v ε + m ) ) in C[, T ]; H ). 3.3) Thus, on account of 3.9) 3.3), by taking the upper limit as k in 3.) and using we infer that T lim inf k T ϕ ε vε,nk s) ) ds T ϕ ε vε s) ) ds, P fs) P π vε s) + m )) εv ε s) F v εs), ηs) v ε s) ) H ds T ) T ϕ ε ηs) ds ϕ ε vε s) ) ds for all η L, T ; H ). Hence, we see that the function v ε is the solution of P ε ; v ε ). As a result, it follows from the uniqueness of the solution of P ε ; v ε ) that S v ε,nk ) = v ε,nk v ε = S v ε ) in C[, T ]; H ) as k. Therefore, the mapping S is continuous with respect to C[, T ]; H ). Thus, from the Schauder fixed point theorem, there exists a fixed point on Y, namely, the problem P) ε admits a solution v ε. Finally, from the fact that ϕ ε v ε ) L, T ; H ),

14 4 Time periodic solutions of Cahn Hilliard system which implies v ε L, T ; W ). Now, we consider the chemical potential µ := µ, µ ) by approximating. For each ε, ], we set the approximate sequence µ ε s) := εv εs) + ϕ ε vε s) ) + π v ε s) + m ) fs) 3.4) for a.a. s, T ). From 3.), we can rewrite 3.4) as µ ε s) = εv εs) + Av ε s) + P β ε vε s) + m ) + π v ε s) + m ) fs) 3.5) for a.a. s, T ). Then, we rewrite 3.3) as F v εs) + µ ε s) ω ε s) = in V for a.a. s, T ), where ω ε s) := m π v ε s) + m ) fs) ) for a.a. s, T ). Therefore, we have P µ ε = µ ε ω ε L, T ; V ) and ω ε L, T ). Then, it holds µ ε L, T ; V ) and for a.a. s, T ). v εs) + F P µ ε s) = in V 3.6) 3.3 Uniform estimates In this subsection, we obtain uniform estimates independent of ε, ]. We refer to [3] to obtain uniform estimates. Lemma 3.. T vε s) V ds + There exists a positive constant M, independent of ε, ], such that T ) T β ε vε s) + m dxds + β,ε v,ε s) + m ) dds M. 3.7) Proof From 3.5), 3.6) and the assumption A3), note that π, π is grobally Lipschitz continuous on R. We denote the Lipschitz constant of π, π by L, L, respectively. Moreover, we can take primitive functions π and π of π and π satisfying π v ε s) ) dx, π v,ε s))d

15 Taishi Motoda 5 for a.a. s, T ), respectively. Now, we test 3.3) at time s, T ) by v ε s) and use the Young inequality. Then, we deduce that for a.a. s, T ). Namely, we have d v ε s) + ε d v ds V ε s) ds H + ϕ ε vε s) ) P fs) P π v ε s) + m )), v ε s) ) H + ϕ ε ) c Pfs) + vε s) H 4c H + c P M + ϕ) P vε s) 4 V + c Pfs) + c H PM + ϕ) ϕ ε vε s) ) + c Pfs) + c H PM + ϕ) d vε s) + ε d ds V vε s) ds H + vε s) 4 V + ) ) β ε vε s) + m dx + β,ε v,ε s) + m d c Pfs) + c H PM + ϕ) for a.a. s, T ). Integrating it over, T ) and using the periodic property, we see that T vε s) V + T ) T β ε vε s) + m dxds + c P f L,T ;H) + c PT M + T ϕ). β,ε v,ε s) + m ) dds This yields that the estimete 3.7) holds. Lemma 3.3. There exists a positive constant M, independent of ε, ], such that ε T v εs) H ds + T v ε s) M V ds. Proof We test 3.3) at time s, T ) by v εs). Then, by using the Young inequality, we see that ε v ε s) H + v ε s) + d V ds ϕ ε vε s) ) + d π ) d v ε s) + m dx + ds ds = P fs), v εs) ) H fs) + V v ε s) V π v,ε s) + m ) d

16 6 Time periodic solutions of Cahn Hilliard system for a.a. s, T ). This implies that ε v ε s) H + v ε s) + d V ds ϕ ε vε s) ) + d π ) d ) v ε s) + m dx + π v,ε s) + m d ds ds fs) 3.8) V for a.a. s, T ). Therefore, by integrating it over, T ) with respect to s and using the periodic property, we can conclude. Lemma 3.4. There exists a positive constant M 3, independent of ε, ], such that vε t) V + π ) ) v ε t) + m dx + π v,ε t) + m d M3 3.9) for a.a. t [, T ]. Proof For each s, t [, T ] such that s t, we integrate 3.8) over [s, t]. Then, by neglecting the first two positive terms, we see that ϕ ε vε t) ) + π ) ) v ε t) + m dx + π v,ε t) + m d ϕ ε vε s) ) + π ) ) T v ε s) + m dx + π v,ε s) + m d + fs) V ds for a.a. s, t [, T ]. Now, Integrating it over, t) with respect to s, it follows that t v ε t) ) ) V + t β ε vε t) + m dx + t β,ε v,ε t) + m d T vε s) T ) T ) V ds + β ε vε s) + m dxds + β,ε v,ε s) + m dds T + π ) T ) v ε s) + m dxds + π v,ε s) + m dds + T T for a.a. t [, T ]. Here, Note that we have fs) V ds 3.3) πr) r L πτ) dτ r τ dτ + = L r + π) r r π) dτ

17 Taishi Motoda 7 for all r R. and similarly, π r) L r + π ) r for all r R. Then, by using the Young inequality, we infer that π ) ) L v ε s) + m dx vε s) + m + π) vε s) + m dx L v ε s) + m dx + π) L L vε s) dx + m + π) 3.3) L for a.a. s [, T ]. Similarly, we have ) π v,ε s) + m d L v,ε s) d + m + L π ) 3.3) for a.a. s [, T ]. Thus, on account of 3.3) 3.3), we deduce that t vε t) ) ) V + t β ε vε t) + m dx + t β,ε v,ε t) + m d T vε s) T ) T ) V ds + β ε vε s) + m dxds + β,ε v,ε s) + m dds + L vε s) dx + L v,ε s) d + T T fs) V ds + M 4 ) T + Lc P vε s) T ) V ds + β ε vε s) + m dxds T ) T T + β,ε v,ε s) + m dds + fs) V ds + M 4 for a.a. t [, T ], where L := max{ L, L } and M 4 := m + π) + m + π ). L L In particular, putting t := T and dividing it by T, it follows that vε T ) ) ) V + β ε vε T ) + m dx + β,ε v,ε T ) + m d ) T T + Lc P vε s) V ds + T ) β ε vε s) + m dxds T + T ) T β,ε v,ε s) + m dds + fs) T V ds + M 4 T. 3.33)

18 8 Time periodic solutions of Cahn Hilliard system Combining 3.7) and 3.33), there exists a positive constant M 3 such that v ε T ) ) ) V + β ε vε T ) + m dx + β,ε v,ε T ) + m d M3. From the periodic property, we have ϕ ε vε ) ) = v ε ) ) V + β ε vε ) + m dx + β,ε v,ε ) + m ) d M ) Now, integrating 3.8) by, t) with respect to s, it follows from 3.3) 3.3) that ϕ ε vε t) ) + π ) ) v ε t) + m dx + π v,ε t) + m d ϕ ε vε ) ) + π ) ) T v ε ) + m dx + π v,ε ) + m d + fs) V ds ) + Lc P ϕε vε ) ) + T fs) V ds + M ) for a.a. t [, T ]. Therefore, by virtue of 3.34) 3.35), there exists a positive constnat M 3 such that the estimate 3.9) holds. Lemma 3.5. There exists a positive constant M 4, independent of ε, ], such that T ) δ βε vε s) + m ds + δ L ) for some positive constants δ. T ) β,ε v,ε s) + m ds M L ) ) Proof To show this lemma, we can employ the method of [, Lemma 4., 4.3], because of imposing same assumptions as [] for β, β and being m intdβ ). Therefore, we can also exploit the following inequalities stated in [7, Sect. 5]: for each ε, ], there exist two positive constants δ and c such that β ε r)r m ) δ βε r) c, β,ε r)r m ) δ β,ε r) c for all r R. Hence, it follows that βε uε s) ), v ε s) ) δ H β ε uε s) ) dx c + δ β,ε u,ε s) ) d c 3.37) for a.a. s, T ). On the other hand, we test 3.3) at time s, T ) by v ε s). Then, from 3.), we see that εv ε s), v ε s) ) H + v εs), v ε s) ) V + Av ε s), v ε s) ) H + P β ε uε s) ), v ε s) ) H fs) π u ε s) ), v ε s) ) H. 3.38)

19 Taishi Motoda 9 Hence, from 3.37) 3.38) and the maximal monotonicity of A, by squaring we have δ βε uε s) ) dx + δ β,ε u,ε s) ) ) d 3c + ) +9 fs) + H π uε s) ) + ε H v ε s) ) vε H s) H + 3 v ε s) V vε s) V for a.a. s, T ). Therefore, from the Lipschitz continuity of π, π and Lemma 3.4, by integrating it over, T ) with respect to s, there exists a positive constant M 4 such that the estimate 3.36) holds. Lemma 3.6. There exists a positive constants M 5, independent of ε, ], such that T µ ε s) ds M V ) Proof Firstly, by using the Lipschitz continuity of π, π and the Hölder inequality, it follows from.) and Lemma 3.4 that there exists a positive constnat M5 such that m π v ε s) + m )) { } ) ) π vε s) + m dx + π v,ε s) + m d + { L vε s) + + L m H + π) + L v,ε s) + L H m + π ) } { vε + M 5 s) } V + + M 5 M 3 + ) =: M5 3.4) for a.a. s, T ). Therefore, owing to 3.4) we deduce that m µε s) ) = m π vε s) + m ) fs) ) M fs) L + ) + f s) ) ) L ) =: ˆM5 for a.a. s, T ). Next, from.), 3.6) and the fact P µ ε s) = µ ε s) mµ ε s)) for a.a. s, T ), we deduce that T µ ε s) T ds P µ V ε s) T ds + V m µε s) ) ds V T c P P µε s) V ds + + ) T T c P v ε s) + T + ) V ds ˆM 5. m µε s) ) ds Thus, from Lemma 3.3, there exist a positive constant M 5 such that the estimate 3.39) holds.

20 Time periodic solutions of Cahn Hilliard system Lemma 3.7. There exists a positive constant M 6, independent of ε, ], such that T ) βε vε s) + m ds + H 4ρ T ) βε v,ε s) + m ds M H ) Proof From the definition of µ ε, we can infer that µ ε = ε t v ε κ v ε + β ε v ε + m ) mβv ε + m )) + πv ε + m ) f a.e. in Q, 3.4) µ,ε = ε t v,ε + κ ν v ε κ v,ε + β,ε v,ε + m ) mβv ε + m )) + π v,ε + m ) f a.e. on Σ. 3.43) Now, it follows from 3.36) that there exists a positive constant M 6 such that m β ε vε s) + m )) ) βε vε + m L + ) + ) ) β,ε v,ε s) + m L ) ) M ) for a.a. s, T ). Moreover, we test 3.4) at time s, T ) by β ε v ε s) + m ) and exploit 3.43). Then, on account of the fact β ε v ε +m )) = β ε v,ε +m ), by integrating over we deduce that ) κ β ε vε s) + m v ε s) ) dx + κ β ε v,ε s) + m v,ε s) d + ) βε vε s) + m + ) ) β H,ε v,ε s) + m βε v,ε s) + m d fs) + µ ε s) εv εs) π v ε s) + m ), βε vε s) + m ))H + m β ε vε s) + m )), β ε v ε s) + m ) ) H + ) )) f s) + µ,ε s) εv,εs) π v,ε s) + m, βε v,ε s) + m H + m β ε vε s) + m )), β ε v,ε s) + m ) ) 3.45) H for a.a. s, T ). Now, from 3.), since the both sign of β ε r) and β,ε r) is same for all r R, we infer that ) ) ) ) β,ε v,ε s) + m βε v,ε s) + m d = β,ε v,ε s) + m βε v,ε s) + m d ρ ) βε v,ε s) + m c d. 3.46) ρ Also, it holds ) β ε vε s) + m vε s) dx, β,ε v,ε s) + m ) v,ε s) d. 3.47)

21 Taishi Motoda Moreover, by using the Young inequality, the Lipschitz continuity of π, π and 3.44), there exists a positive constant ˆM 6 such that fs) + µε s) εv εs) π v ε s) + m ), βε vε s) + m )) + m β ε vε s) + m )), β ε vε s) + m ))H ) βε vε s) + m + 4 H fs) + 4 µε s) + 4ε H H v ε s) + 4 ) π vε s) + m H H + m βε vε s) + m )) H ) βε vε s) + m + ˆM H 6 fs) + µε s) + ε H H v ε s) + vε s) + ) H H + M ) H and f s) + µ,ε s) εv,εs) π v,ε s) + m ), βε v,ε s) + m )) H + m β ε vε s) + m )), β ε v,ε s) + m ) ) H ) β ε v,ε s) + m + ρ f 4ρ H s) + ρ µ H,ε s) + ρε v H,εs) H +ρ ) π v,ε s) + m + ρ H m βε vε s) + m )) H ) β ε v,ε s) + m + ρ 4ρ M H 6 +ρ ˆM 6 f s) H + µ,ε s) H + ε v,ε s) H + v,ε s) H + ) 3.49) for a.a. s, T ). Thus, from Lemma 3.3, 3.4 and.), by combining from 3.45) 3.49) and integrating them over, T ), we can conclude existence of the constant M 6 satisfying 3.4). Lemma 3.8. There exists a positive constant M 7, independent of ε, ], such that T κ v ε s) T ds + v H ε s) T H 3 ds + ν v ) ε s) ds M H ) This lemma is proved exactly the same as in [, Lemmas 4.4] because the necessary uniform estimates to prove it is obtained by Lemmas 3.3, 3.4, 3.6 and 3.7. Sketching simply, comparing in 3.4) we deduce that v ε L,T ;H) is uniformly bounded. Moreover, by using the theory of the elliptic regularity and the trace theory see e.g., [7, Theorem 3., p..79 and Theorem.5, p..6], respectively), we can conclude that 3.5) holds. Lemma 3.9. There exists a positive constant M 8, independent of ε, ], such that T ) β,ε v,ε s) + m ds M H )

22 Time periodic solutions of Cahn Hilliard system Proof We test 3.43) at time s, T ) by β,ε v,ε s) + m ) and integrating it over. Then, by using the Young inequality and the Lipschitz continuity of π, there exists a positive constant M 8 such that ) κ β,ε v,ε s) + m v,ε s) d + ) β,ε v,ε s) + m H = ) )) f s) + µ s) εv,εs) ν v ε s) π v,ε s) + m, β,ε v,ε s) + m H ) β,ε v,ε s) + m + m )) β H ε vε s) + m H + M f 8 s) + µ s) + ε H H v,ε s) + ν v H ε s) + v,ε s) + ) H H ) β,ε v,ε s) + m + M H 6 + M f 8 s) + µ s) + ε H H v,ε s) + ν v H ε s) + v,ε s) + ) H H for a.a. s, T ). Noting that it holds ) κ β,ε v,ε s) + m v,ε s) d. 3.5) Thus, on account of Lemma 3.3, 3.4, 3.6 and 3.8, by integrating 3.5) over, T ), we can find a positive constant M 7 such that the estimate 3.5) holds. Lemma 3.. There exists a positive constant M 9, independent of ε, ], such that T vε s) ds M W 9. This lemma is also proved the same as in [, Lemmas 4.5]. The key point to prove it is that we can obtain the uniform estimates of v,ε L,T ;H ) by comparing in 3.43). We omit the proof. 4 Proof of convergence theorem In this section, we obtain the existence of weak solution of P) by performing passage to the limit for the approximate problem P) ε. The convergence theorem is also nearly the same [, Sect. 4]. The different point from [] is that the component of the weak solution of P) satisfies.4) and the periodic property.5). Thanks to the previous estimates Lemmas from 3.3 to 3., there exist a subsequence {ε k } k N with ε k as k and some limits functions v H, T ; V ) L, T ; V ) L, T ; W ), µ H, T ; V ), ξ L, T ; H) and ξ L, T ; H ) such that v εk v weakly star in H, T ; V ) L, T ; V ) L, T ; W ), 4.53)

23 Taishi Motoda 3 ε k v εk strongly in H, T ; H ), µ εk µ weakly in L, T ; V ), β εk u εk ) ξ weakly in L, T ; H), 4.54) β,εk u,εk ) ξ weakly in L, T ; H ) 4.55) as k. Owing to 4.53) and a well-known compactness results see e.g., [3]), we obtain v εk v strongly in C[, T ]; H ) L, T ; V ) 4.56) as k. This yeilds that u εk u := v + m strongly in C[, T ]; H ) L, T ; V ) 4.57) as k. Therefore, from 4.57) and the Lipschitz continuity of π, π, we deduce that πu εk ) πu) strongly in C[, T ]; H). Hence, by passing to the limit in 3.5) and 3.6), we obtain.3) and the following weak formulation: µt), z ) H = a vt), z ) + ξt) m ξt) ) + π vt) ) f, z ) H for all z V 4.58) for a.a. t, T ) where ξ := ξ, ξ ), because of the property.) of linear bounded operator P. Now, we can infer v + m Dβ) and v + m Dβ ). Hence, from the form 3.5) and 3.6), we deduce that πv + m ) = πv + m ) a.e. in Q and π v + m ) = π v + m ) a.e. on Σ. This implies that we obtain.4) replaced by 4.58). Moreover, it follows from 4.56) that v) = vt ) in H. Also, due to 4.54), 4.55), 4.57) and the monotonicity of β, from the fact [5, Prop.., p. 38] we obtain ξ βv + m ) a.e. in Q, ξ β v + m ) a.e. on Σ. Thus, we complete the proof of Theorem.. Remark 4.. In the previous sections, we impose the restricted assumption A5) respect to the domains Dβ) and Dβ ). This is an essential assumption if we treat the multivalued β, β. However, refferring to [8], we can avoid the assumption A5) when focusing only on prototype double potential, namely when we choose βu) = u 3, β u ) = u 3, πu) = u and π u ) = u in.) and.5). The reason why we can avoid it is that we need not to use 3.3) to obtain estimates. The method to do so is used in [8]. Concretely, it is using the Hölder inequality mainly. We can employ the same method even if we consider the problem with dynamic boundary condition. Therefore, we can prove Theorem. without the assumption A5) in the case of the prototype double potential.

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