Cahn-Hilliard equation on the boundary with bulk condition of Allen-Cahn type

The well-posedness for a system of partial differential equations and dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk $\Omega $ and on the boundary $\Gamma$. The Poisson equation for the chemical potential, the Allen-Cahn equation for the order parameter in the bulk $\Omega$ are considered as auxiliary conditions for solving the Cahn-Hilliard equation on the boundary $\Gamma$. Recently the well-posedness for the equation and dynamic boundary condition, both of Cahn-Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved.


Introduction
In this paper, we treat the Cahn-Hilliard equation [1] on the boundary of some bounded smooth domain. Let 0 < T < +∞ be some fixed time and let Ω ⊂ ℝ d , d = 2 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 ∂ t denotes the partial derivative with respect to time, and ∆ Γ denotes the Laplace-Beltrami operator on Γ (see, e.g., [21,Chapter 3]). Here, the unknowns u Γ and μ Γ : Σ → ℝ 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 ∂ ν on Γ acts on functions μ, u : Q := (0, T) × Ω → ℝ that satisfy the following trace conditions: where μ | Σ and u | Σ are the traces of μ and u on Σ. Moreover, these functions μ and u solve the following equations in the bulk Ω: −∆μ = 0 in Q, (1.4) τ∂ t u − ∆u + W (u) = f in Q, (1.5) where τ > 0 is a positive constant and ∆ denotes the Laplacian.
Note that the nonlinear terms W Γ and W are the derivatives of the functions W Γ and W, usually referred as double-well potentials, with two minima and a local unstable maximum in between. The prototype model is provided by W Γ (r) = W(r) = (1/4)(r 2 − 1) 2 , so that W Γ (r) = W (r) = r 3 − r, r ∈ ℝ, 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 W Γ or W.
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 [5].
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 ε ↘ 0. In the second step, we improve suitable estimates in order to make them independent of λ. Then we can pass to the limit as λ ↘ 0 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.

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 τ > 0, we have −∆μ = 0 a.e. in Q, in Ω, u Γ (0) = u 0Γ a.e. on Γ, (2.5) where f : Q → ℝ, f Γ : Σ → ℝ are given sources, u 0 : Ω → ℝ, u 0Γ : Γ → ℝ are known initial data, β stands for the subdifferential of the convex partβ and π stands for the derivative of the concave perturbationπ of a double well potential W(r) =β(r) +π(r), defined for all r in the domain ofβ. The same setting holds for β Γ and π Γ . Typical examples of the nonlinearities β, π are given by • β(r) = r 3 , π(r) = −r for all r ∈ ℝ, with D(β) = ℝ for the prototype double well potential where c > 0 is a large constant which breaks convexity.
Similar choices can be considered for β Γ , π Γ and the related potential W Γ . 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 β Γ (see the later assumption (A5)) that forces the growth of β to be controlled by the growth of β Γ . A similar approach was taken in previous analyses, see [2, 4-6, 9, 28]. As a remark, τ > 0 plays the role of a viscous parameter. Indeed, if τ = 0, 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 τ = 0 or, in our framework, also study the singular limit as τ ↘ 0. 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.
It is easy to see that problem (P) has a structure of volume conservation on the boundary Γ. Indeed, integrating the last equation in (2.3) on Σ, and using (2.1) and (2.5), we obtain (2.6) hereafter, we put where |Γ| := ∫ Γ 1 dΓ. The space V * denotes the dual of V, and ⟨ ⋅ , ⋅ ⟩ V * ,V denotes the duality pairing between V * and V. Moreover, it is understood that H is embedded in V * in the usual way, i.e., ⟨u, z⟩ V * ,V = (u, z) H for all u ∈ H, z ∈ V. Then we obtain V → → H → → V * , where " → →" stands for the dense and compact embedding, namely, (V , H, V * ) is a standard Hilbert triplet. Under this setting, we define the solution of (P) as follows.

Main theorems
The first result states the existence of the solution. To this end, we assume the following: (A1) f ∈ L 2 (0, T; H) and f Γ ∈ W 1,1 (0, T; H Γ ).
The minimal section β ∘ of β is specified by β ∘ (r) := {r * ∈ β(r) : |r * | = min s∈β(r) |s|} and the same definition applies to β ∘ Γ . These assumptions are the same as in [2,5]. Concerning assumption (A1), let us note that the regularity conditions for f and f Γ 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 τ > 0 plays a role, and if the term τ∂ t u 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 u 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: and assume that the corresponding solutions (u (i) , μ (i) , ξ (i) ) exist. Then there exists a positive constant C > 0, depending on L, L Γ and T, such that for all t ∈ [0, T].

Approximate problem and uniform estimates
In this section, we first consider an approximate problem for (P), and then we obtain uniform estimates. For each ε, λ ∈ (0, 1], we introduce an approximate problem (P; λ, ε) where the proof of the well-posedness of (P; λ, ε) is given in Appendix A.

Proof of the main theorem
In this section, we prove Theorem 2.3.

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 λ ↘ 0. The point of emphasis is the effective usage of the mean value zero function.
We have collected all information which enables us to pass to the limit as λ ↘ 0.
for all z := (z, z Γ ) ∈ V and i = 1, 2. Let us use the notation We take the difference of equations (4.50), test it by (ū,ū Γ ) and integrate over [0, t] with respect to s, obtaining Γ (s)),ū Γ (s)) H Γ ds 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 δ,δ > 0. Now we take advantage of (3.16) Indeed,ū Γ satisfies the zero mean value condition from (4.48). Moreover, by (4.55), we can recover from (4.51) the following inequality: for all t ∈ [0, T], that is, δ := 1/(4C P ) andδ := 1/(2C P ). Then the continuous dependence (2.13) follows from the application of the Gronwall inequality.

A Appendix
We use the same notation as in the previous sections. We also use the following notation of function spaces. Moreover, define an inner product of H by Then we see that the induced norm ‖ ⋅ ‖ H and the standard norm | ⋅ | H are equivalent, because Let us define a linear bounded operator F : Then there exists a positive constant c (ε) P such that see, e.g., [5,Appendix]. Thus, we see that | ⋅ | V ε 0 and the standard | ⋅ | V are equivalent norms of V ε 0 , and F is the duality mapping from V ε 0 to (V ε 0 ) * . We also define the inner product in (V ε 0 ) * by Thanks to [5,Appendix] again, we obtain V ε Proof. Firstly, we see that Next, we infer that Therefore, we deduce that ‖z − P ε z‖ 2 H = ((z − P ε z, z − y + y − P ε z)) H = ((z − P ε z, z − y)) H i.e., (A.1) holds.
Incidentally, we note that another possibility of projection is given bỹ andP ε is actually the projection from H to H ε 0 with respect to the standard norm (cf. (A.1)). However, this choice is not suitable from the viewpoint of the trace condition. Indeed,P ε z ∉ V even if z ∈ V.