On a transmission problem for equation and dynamic boundary condition of Cahn-Hilliard type with nonsmooth potentials

This paper is concerned with well-posedness of the Cahn-Hilliard equation subject to a class of new dynamic boundary conditions. The system was recently derived in Liu-Wu (Arch. Ration. Mech. Anal. 233 (2019), 167-247) via an energetic variational approach and it naturally fulfills three physical constraints such as mass conservation, energy dissipation and force balance. The target problem examined in this paper can be viewed as a transmission problem that consists of Cahn-Hilliard type equations both in the bulk and on the boundary. In our approach, we are able to deal with a general class of potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double-obstacle potential. Existence, uniqueness and continuous dependence of global weak solutions are established. The proof is based on a novel time-discretization scheme for the approximation of the continuous problem. Besides, a regularity result is shown with the aim of obtaining a strong solution to the system.


Introduction
In this paper, we consider the following initial boundary value problem for a Cahn-Hilliard equation subject to a dynamic boundary condition that is also of Cahn-Hilliard type. Let 0 < T < ∞ be some fixed time and let Ω ⊂ R d , d = 2 or 3, be a bounded domain with smooth boundary Γ := ∂Ω. We aim to find four unknown functions φ, µ : Q := (0, T ) × Ω → R and ψ, w : Σ := (0, T ) × Γ satisfying where ∂ t and ∂ ν denote the partial time derivative and the outward normal derivative on Γ, respectively; ∆ denotes the Laplacian and ∆ Γ denotes the Laplace-Beltrami operator on Γ (see, e.g., [22,Chapter 3]); φ | Γ standards for the trace of φ on the boundary Γ. In view of (1.4), system (1.1)-(1.6) is a sort of transmission problem between the Cahn-Hilliard equation in the bulk Ω and the Cahn-Hilliard equation on the boundary Γ. The nonlinear functions W and W Γ are usually referred as the double-well potentials, with two minima and a local unstable maximum in between. Typical and physically significant examples of such potentials are the so-called classical potential, the logarithmic potential , and the double obstacle potential , which are given, in this order, by where the constants in (1.8) and (1.9) satisfy c 1 > 1 and c 2 > 0, so that W log and W 2obs are nonconvex. The nonlinear terms W ′ (φ) in (1.2) and W ′ Γ (ψ) in (1.6) characterize the dynamics of the Cahn-Hilliard system. In cases like (1.7) and (1.8), W ′ and W ′ Γ denote simply the derivatives of the related potentials; while non-smooth potentials like (1.9) are considered, then W ′ and W ′ Γ denote the subdifferential of the convex part plus the derivative of the smooth concave contribution, i.e., for (1.9) it is Of course, in this case one should replace the equalities in (1.2) and (1.6) by inclusions. In this paper, we are able to handle completely general potentials W and W Γ including all the three cases (1.7)-(1.9) mentioned above.
The system (1.1)-(1.6) was first derived by Liu and Wu [29] in a more general form (see also [33]) on the basis of an energetic variational approach. It describes effective shortrange interactions between the binary mixture and the solid wall (boundary), furthermore, it has the feature that the related model naturally fulfills important physical constraints such as conservation of mass, dissipation of energy and force balance relations. In its current formulation, we see that equations (1.1) and (1.2) yield a Cahn-Hilliard system subject to a no-flux boundary condition (1.3) together with a non-homogeneous Dirichlet boundary condition (1.4), while the dynamic boundary condition (1.5) and equation (1.6) provide an evolution system of Cahn-Hilliard type on the boundary Γ. These two Cahn-Hilliard systems in the bulk and on the boundary are coupled through the trace condition (1.4) and the normal derivative term ∂ ν φ in (1.6).
The total energy functional for system (1.1)-(1.6) given by is decreasing in time (see [29]) and furthermore, system (1.1)-(1.6) can be interpreted as a gradient flow of E(φ, ψ) in a suitable dual space (see [18]). In light of (1.1), (1.3) and (1.5), we easily deduce that the following properties on mass conservation: In this paper, we study the well-posedness of system (1.1)-(1.6) for a weak solution subject to the following initial data φ(0) = φ 0 in Ω, ψ(0) = ψ 0 on Γ. (1.12) Moreover, we also establish a regularity theory in order to obtain a strong solution. In particular, we are able to treat the initial value problem for system (1.1)-(1.6) in a wider class of nonlinearities W and W Γ . Indeed, in the previous contributions, the well-posedness was investigated only in the case of smooth potentials like (1.7) (cf. [29,Remark 3.2] and [18,Remark 2.1]): this is the point of emphasis of our present paper. We would like to mention some related problems in the literature. In 2011, Goldstein, Miranville and Schimperna [21] studied a different type of transmission problem between the Cahn-Hilliard system in the bulk and on the boundary with non-permeable walls (cf. a previous work Gal [15] for the case with permeable walls). Their system can be derived from the same energy functional (1.10) by a variational method, however, the corresponding boundary conditions turn out to be different from (1.3) and (1.5). This also leads to a different property on the mass conservation comparing with (1.11) such that the total (bulk plus boundary) mass is conserved. We refer to [29] for more detailed information on the comparison between these models. In addition, we mention the contributions [6,8,15,21] related to the well-posedness, [9,12,13,16,17,20] for the study of long time behavior and the optimal control problems, [7,14] for numerical analysis and [24] for the maximal regularity theory. Comparing the large number of known results on the previous model [15,21], we are only aware of the recent papers [18,29] that analyze the well-posedness of system (1.1)-(1.6) with (1.12).
Let us now describe the contents of the present paper. In Section 2, we state the main well-posedness result for global weak solutions. We consider the problem within a general framework by setting W ′ := β + π and W ′ Γ := β Γ + π Γ , where β and β Γ are maximal monotone graphs with 0 ∈ β(0) and 0 ∈ β Γ (0), while π and π Γ yield the anti-monotone terms that are Lipschitz continuous functions. The main theorems are concerned with the existence of a global weak solution (Theorem 2.1) and the continuous dependence on the given data (Theorem 2.2), which implies the uniqueness.
In Section 3, we study the time-discrete approximate problem for (1.1)-(1.6) with (1.12). We start from the viscous Cahn-Hilliard system by inserting two additional terms, τ ∂ t φ and σ∂ t ψ in the right hand sides of (1.2) and (1.6), respectively, with the parameters τ, σ > 0. Moreover, we take the Yosida approximations β ε and β Γ,ε in place of the maximal monotone graphs β and β Γ and in terms of the parameter ε > 0. Then we apply a time discretization scheme using the approach in [10,11]. We can show the existence of a discrete solution taking advantage of the general maximal monotone theory. After that, we proceed to derive a sequence of uniform estimates. For this purpose, we apply the technique of [5] in order to treat different potentials in the bulk and on the boundary. In the subsequent iterations, we prove the existence results by performing the limiting procedures, with respect to the time step first, then as ε → 0, finally taking the limit as either τ → 0 or σ → 0, or both τ, σ → 0, in order to obtain a partially viscous Cahn-Hilliard system or a pure Cahn-Hilliard system in the limit. The continuous dependence result is then proved by using the energy method.
In Section 4, we discuss the regularity for weak solutions. Returning to the time discrete approximation, we gain some necessary higher order estimates at all the different levels up to the final limits. Thus, we are able to obtain enough regularity as to guarantee a strong solution for the pure Cahn-Hilliard system as well (see Theorem 4.1).
Here, for the reader's convenience, let us include a detailed index of sections and subsections.
Same considerations apply to β Γ , π Γ and W Γ . Since the bulk and boundary potentials are allowed to be different, in order to handle the nontrivial bulk-boundary interaction of the transmission problem, an assumption for the relationship between β and β Γ will be needed. We shall present it later.
Before we state our main theorems, we recall the structure of mass conservation of problem (2.1)-(2.7). Taking z = 1 in (2.8) and integrating from 0 to t with the help of (2.13), we obtain the first equality in (1.11). Analogously, from (2.11) and (2.13) we obtain the second condition in (1.11). Therefore, it is useful to define the following mean value functions: for any z ∈ L 1 (Ω) and z Γ ∈ L 1 (Γ).
Our first result is related to the existence of global weak solutions. The existence of strong solutions will be discussed in Section 4 (see Theorem 4.1).
Our second result is the continuous dependence on the initial data and external sources, which immediately yields the uniqueness of weak solutions: Let sextuplets of functions (φ (i) , µ (i) , ξ (i) , ψ (i) , w (i) , ζ (i) ) be weak solutions of problem (2.1)-(2.7) corresponding to the given data f (i) , g (i) , φ In order to prove Theorem 2.1, we quote the abstract framework as in [25,26] and we also prepare the following function spaces: From the Poincaré-Wirtinger inequalities (see, e.g., [23]), we see that there exists a positive constant C P such that Then, based on the Lax-Milgram theorem, we introduce the operator N Ω : By virtue of these definitions, we can also introduce the norms for all z ∈ V 0 * , equivalent to the usual norm | · | V * , for the elements of V 0 * ; and In this section, we prove the existence of global weak solutions and the continuous dependence with respect to given data. To do so, we introduce an approximate problem for problem (2.1)-(2.7). The idea is based on a time-discretization scheme, the Moreau-Yosida regularization, together with a viscous Cahn-Hilliard approach.
(3.7) Indeed, the terms µ n+1 − µ n in the equation (3.1) and w n+1 − w n in the equation (3.5) play a role of viscosities with the parameter h. In (3.2) and (3.6), f n and g n are known too, defined by In order to approximate the maximal monotone graphs, we recall the Moreau-Yosida regularization (see, e.g., [1,2]). For each ε ∈ (0, 1], we define β ε , β Γ,ε : R → R, along with the associated resolvent operators J ε , J Γ,ε : R → R given by for all r ∈ R, where ̺ > 0 is same as in the condition (2.16). As a remark, the above two definitions are not symmetric, more precisely, the parameter of approximation is not directly ε but ε̺ in the definition of β Γ,ε and J Γ,ε . This is important in order to apply [5,Lemma 4.4], which ensures that for all ε ∈ (0, 1] with the same constants ̺ and c 0 as in (2.16). We also have β for all r ∈ R. Then, we see that β ε and β Γ,ε are Lipschitz continuous with constants 1/ε and 1/(ε̺), respectively. Additionally, we also use the following facts: for all r ∈ R.
There is a value h * ∈ (0, 1], depending on τ and σ, such that for every Proof. Define ∆ N : W → H be the Laplace operator, subject to the homogeneous Neumann boundary condition. From (3.1) and (3.3), we infer that where I − ∆ Γ is a linear operator from H 2 (Γ) ⊂ H Γ to H Γ . As a consequence, equation (3.2) can be rewritten as and the condition (3.6) becomes gives a maximal monotone operator A from its domain D( e. on Γ}. This also implies that the subdifferential of J in H at (z, z Γ ) coincides with A(z, z Γ ) = (−h∆z, h∂ ν z − h∆ Γ z Γ ). Next, we define another operator B : H → H by with its domain D(B) = H. Then, we see that B is Lipschitz continuous and monotone provided that h is sufficiently small compared to τ and σ, namely h ∈ (0, h * ] where h * L < τ /2 and h * L Γ < σ/2: of course, B is also coercive. Hence, from general theory of the maximal monotone operator [1, pp. 35-36, Corollaries 2.1 and 2.2], we conclude that Ran(A + B) = H. This implies that for sufficiently small h ∈ (0, h * ], for each φ n , µ n ∈ H and ψ n , w n ∈ H Γ given by the previous step, there exists a unique pair (φ n+1 , ψ n+1 ) ∈ V solving (3.12) and (3.13), where the uniqueness is a consequence of the strict coerciveness of B. Next, we can recover µ n+1 ∈ W and w n+1 ∈ H 2 (Γ) from (3.10) and (3.11), respectively. By comparison in the equations (3.2) and (3.6), we also deduce that φ n+1 ∈ H 2 (Ω) and ψ n+1 ∈ H 2 (Γ), using the elliptic regularity theory (see, e.g., [31,Lemma A.1]). Thus, we can complete the proof of Proposition 3.1 by iterating from n = 0 to n = N − 1. ✷ According to the standard manner, we now define the following piecewise linear functions and step functions: and analogously forμ h ,μ n ,ψ h ,ψ h ,ŵ h ,w h , g h . Then, we have the following useful properties: for some suitable function space X. Indeed, (3.15) is clear from the definition, the equality (3.16) is obtained from the direct calculation as follows: . Concerning the inequality (3.14), invoking the convexity and Jensen's inequality we obtain that constructed above solve the following polygonal approximate problem of the viscous Cahn-Hilliard system: for every h ∈ (0, h * ]. By virtue of the definitions of f n and g n we see that {f h } h>0 and {g h } h>0 are bounded in L 2 (0, T ; V ) and L 2 (0, T ; V Γ ), respectively. Indeed, from the Hölder inequality we infer that and a similar result holds for {g h } h>0 . In the next subsection we will proceed to derive necessary uniform estimates for problem (3.17)-(3.23).

3.2.
A priori estimates and limiting procedure. Hereafter, we derive uniform estimates that are independent of h = T /N for problem (3.17)-(3.23). We also take care of the dependence with respect to τ, σ, ε ∈ (0, 1].
In these cases, we can keep the smoothness of the time derivative, more precisely, Proof of Theorem 2.2. We now prove a continuous dependence estimates with two weak solutions corresponding to the initial data satisfying (2.17), (2.18), and the sources We take the difference of (2.8) and choose z := 1 to obtain that a.e. in (0, T ), whence for all t ∈ [0, T ]. Then, we can take z := N Ω (φ (1) −φ (2) ) as a test function in the difference of (2.8) and obtain for all t ∈ [0, T ]. By operating on the difference of (2.11) in the same way, that is, z Γ := 1 first and z Γ := N Γ (ψ (1) − ψ (2) ) second, we obtain the similar formula for all t ∈ [0, T ]. Next, we multiply the difference of the equalities in (2.9) by φ (1) − φ (2) and integrate the resultant with respect to space and time. Using (2.10) and (2.12), we infer that for all t ∈ [0, T ]. Then, we take the sum of (3.83), (3.84) and combine with the above equality. Thanks to the monotonicity of β, β Γ , the Lipschitz continuity of π, π Γ , and the Poincaré-Wirtinger inequality, we deduce that for all t ∈ [0, T ]. Here, we observe that and similarly, for all t ∈ [0, T ]. Therefore, applying the Gronwall lemma and invoking the equivalences of norms, we conclude the proof of Theorem 2.2. As an immediate sequence, the continuous dependence implies the uniqueness of the weak solution obtained in Theorem 2.1. ✷ The continuous dependence estimate can be extended to the viscous or partially viscous cases, with the following modification: there exists a constant C such that the inequality holds for τ ≥ 0 and σ ≥ 0.