Convective nonlocal Cahn-Hilliard equations with reaction terms

We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.


Introduction
Consider an incompressible isothermal and isotropic binary solution of constant molar volume which is initially homogeneous. If the temperature is rapidly quenched under a critical value T c , the two solutes, A and B, begin to separate from each other and the domain is split in A-rich and B-rich subdomains. The early stage of this process in which the solution is called spinodal decomposition. In order to descrive this phenomenon, a well-known mathematical approach was introduced by J.W. Cahn and J.E. Hilliard (see [9]) who introduced an appropriate free energy functional. More precisely, given a bounded domain Ω ⊂ R d with d ≤ 3 occupied by a binary mixture of components A and B with a mass fraction of ϕ A (x) and ϕ B (x), respectively, setting ϕ := ϕ A − ϕ B , the free energy functional is given by The term ε 2 2 |∇ϕ| 2 is a surface tension term and ε is proportional to the thickness of the diffused interface. F is a double well potential energy density which favors the separation of phases, that is, the formation of A-rich and B-rich regions. Although the physically relevant potential F is logarithmic (see [9, (3.1)]), in the literature it is usually approximated, for instance, by the following polynomial function Once an energy is defined, the spinodal decomposition can be viewed as an energy minimizing process. Thus, the evolution of the phenomenon can be described as a gradient flow (cf. [10], see also [22,46]) ϕ t − ∇ · M(ϕ)∇µ L = 0, in Ω × (0, T ), on a given time interval (0, T ), T > 0, where M(ϕ) is called mobility and the chemical potential µ L is the first variation of E, that is, This is the well-known Cahn-Hilliard equation which has been widely studied by many authors. Here we just mention some pioneering works [19,20,43,52] for the case with constant mobility, while for nonconstant (and degenerate at pure phases) mobility the basic reference is [21] (cf. also [49] for nondegenerate mobility). For further references see, for instance, [16,45,46]. However, the purely phenomenological derivation of the Cahn-Hilliard equation is somewhat unsatisfactory from the physical viewpoint. This led Giacomin and Lebowitz to study the problem of phase separation from the microscopic viewpoint using a statistical mechanics approach (see [28] and also [29]). Performing the hydrodynamic limit they deduced a continuum model which is a nonlocal version of the Cahn-Hilliard equation, namely, they found the following in Ω × (0, T ), where µ is defined by in Ω × (0, T ), (1.3) where J : R d → R is a convolution kernel such that J(x) = J(−x) and a(x) := Ω J(x − y)dy. In this case, the free energy functional E is given by It is worth observing that E(ϕ) can be recovered from E(ϕ) using Taylor expansion for the term (ϕ(x) − ϕ(y)) and supposing that J is sufficiently peaked around zero. Thus the original Cahn-Hilliard equation can be seen as an approximation of its local version. Furthermore, as discussed in [30], the corresponding sharp interface model is the same as the local Cahn-Hilliard equation.
The nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential was first studied rigorously in [30]. Then more general results were proven in [26]. The longtime behavior of single solutions was studied in [35,36], while the global longterm dynamics (say, existence of global and exponential attractors) was investigated in [27]. The case of constant mobility and smooth potential was treated in [4,5].
More recently, some works have been devoted to study the nonlocal Cahn-Hilliard equation coupled with the Navier-Stokes system (see [18,24,25]). This is a nonlocal variant of a well-known diffuse interface model for phase separation in two-phase incompressible and isothermal fluids (model H). All such contributions produced, as by-products, further results on the nonlocal Cahn-Hilliard equation.
Taking advantage of some of these results, here we want to introduce and study two nonlocal Cahn-Hilliard equations with reaction terms which have already been proposed and analyzed in the local case: the Cahn-Hilliard-Oono (CHO) equation and the Cahn-Hilliard-Bertozzi-Esedoḡlu-Gillette (CHBEG) equation (see also [17,40] for other types of reaction terms).
Let us introduce the former first.
In [31] the authors introduce a reaction between the molecules or the polymers in the mixture in order to stabilize and control the length of the patterns during spinodal decomposition. Let us suppose that the mixture consists of molecules or polymers of type A and B and that a chemical reaction A B is induced during the spinodal decomposition. Here Γ 1 and Γ 2 are, respectively, the temperature dependent forward and backward reaction rates. By defining ϕ as the difference between the concentrations, the following Cahn-Hilliard type equation is proposed (see [31]) This equation is known as the CHO equation and was introduced in [2] (cf. [48], however) for phase separation in diblock copolymers (see also [47]). It can be viewed as a gradient flow generated by the energy functional: where m ∈ R, σ and ε are two positive constants and G is the Green function of −∆ with either periodic or no-flux boundary conditions, i.e., This energy differs from the one in (1.1) just in the last term, which is nonlocal, and prefers rapid oscillations between the two polymers (see [50] for the one-dimensional case). It has been observed that far from the domain border, pattern are not influenced by the boundary conditions. The most commonly used ones are either periodic or no-flux. The latter can be mathematically expressed as follows: ∂ϕ ∂n = 0 and ∂µ L ∂n = 0, on ∂Ω × (0, T ).
The CHBEG equation can be viewed as a generalized version of the CHO equation. It has been introduced by Bertozzi et al. to model image inpainting, i.e., the restoration process of damaged images and super-resolution (see [6,7]) with regard to binary images or images with high contrast. As in the case of phase separation, the double well potential tends to separate the mixture in pure phases, so it does with binary images by separating the two colors of the image without many shades.
The CHBEG equation reads as follows Here Ω is a two-dimensional domain representing the whole image, with no-flux boundary conditions. The function F is still a double well potential, with two minima ϕ 1 and ϕ 2 , that is, the two colors of the image. The function h represents the image that needs to be restored (h(x) = ϕ i (x) where the color is ϕ i , i = 1, 2. The function λ is a penalization parameter which forces the restored image ϕ to be equal to h outside the damaged region.
The numerical simulations carried out in [6] show that the implemented method is faster then its main competitors and it gives satisfactory results. Existence and uniqueness of a weak solution to the Cahn-Hilliard equation for the CHBEG equation have been proven in [7], while the longtime behavior of its solutions has recently been analyzed in [15].
On account of what we have observed concerning the derivation of the nonlocal Cahn-Hilliard equation, it seems natural to study the nonlocal variants of CHO and CHBEG equations.
More precisely, we consider first a nonlocal CHO equation with a transport term which accounts for a possible flow of the mixture at a certain given (averaged) velocity field u : Ω × (0, T ) → R d , that is in Ω × (0, T ). (1.6) where µ is given by (1.3) and g is a given external source. Here we have set all the constants equal to one but m which has now been denoted byφ * on account of (1.5). Then, in the same spirit, we introduce the nonlocal version of the CHBEG equation, that is, As for the CHO equation, we include a transport term which can be used to improve the inpaint of the image and an external source. Therefore we consider the following initial and boundary value problems (1.9) The first goal will be to define a notion of weak solution to problems (1.8) and (1.9) and to establish their well-posedness under suitable conditions. This will be done in Sections 2 and 3. Then, in Sections 4 and 5, we will analyze nonlocal CHO and CHBEG equations equipped with no-flux boundary condition as dynamical systems on suitable phase spaces. In particular, we will prove the existence of a (bounded) absorbing set as well as of the global attractor. The former result will require some care (especially for the CHBEG equation) to handle the bound on the spatial average of ϕ which is not conserved in general.
We also expect to prove further results. for instance, the existence of more regular solutions, the regularization of weak solutions and the existence of exponential attractors as shown in [27] (see also [23]). These issues will possibly be the subject of a forthcoming paper. A more challenging question appears to be the following: does a solution converge to a single equilibrium? This happens for the nonlocal Cahn-Hilliard equation (see, e.g., [27] and references therein), but it is open even for equation (1.4).
2 Nonlocal CHO equation: well-posedness of (1.8) Here we show that there exists a unique weak solution to problem (1.8) under suitable assumptions on u, g, J and F . We also prove that any weak solution satisfies an energy equality and it depends continuously with respect to the initial datum in a suitable norm. We adapt the strategy used in [18].

Notation
The space L 2 (Ω) will be denoted H and the space H 1 (Ω) will be denoted V . Furthermore we will set: By · and (· , ·) will be denoted respectively the norm and the scalar product in L 2 (Ω). The linear operator A = −∆ : V 2 → H is self-adjoint and non-negative. Moreover, it is strictly positive if restricted to H 0 = {ψ ∈ H :ψ = 0}. Then we will also set A −1/2 (·) and we observe that the norm · # defined as is equivalent to the usual norm of V ′ . Furthermore, c will indicate a generic nonnegative constant depending on J, F, Ω, σ,φ * , g, λ, h and u at most. If it is necessary to distinguish between such constants, then we will use C i . Occasionally we will point out if such constants also depend on other quantities but time. If there is a further dependence on T and/or the initial datum ϕ 0 , then we shall use N. The value of c and N may vary even within the same line.

Assumptions and statements
Let us introduce first the assumptions following [4] and [18] (cf. also [39]): open, bounded and connected with a smooth boundary.
Remark 2.1. Assumption (H2) implies that the potential F is a quadratic perturbation of a strictly convex function. Indeed F can be represented as Here a * = a L ∞ (Ω) and observe that a ∈ L ∞ (Ω) derives from (H1).

Remark 2.2.
Since F is bounded from below, it is easy to see that (H4) implies that F has polynomial growth of order p ′ , where p ′ ∈ [2, ∞) is the conjugate index to p. Namely there exist c 4 > 0 and c 5 ≥ 0 such that Remark 2.3. The potential (1.2) satisfies all the hypotheses on F .

Remark 2.4. It is easy to show that (H4) implies
Remark 2.5. Take for simplicity u = g = 0 in equation (1.6) and observe that it can formally be rewritten as follows from which the crucial role of (H2) is evident. This remark also holds for (1.7).
We can now give our definition of weak solution to problem (1.8): we have
Remark 2.7. Observe that if we choose ψ = 1 in (2.3) and we set f :=ḡ + σφ * , then we obtain d dtφ Thus the total mass is not conserved in general.
Existence and uniqueness for problem (1.8) is given by: (Ω) and suppose(H0)-(H8) are satisfied. Then, for every T > 0 there exists a unique weak solution ϕ to problem the following energy equality holds for almost every t ∈ (0, T ): Furthermore, we have: Let the hypotheses (H0)-(H8) hold. Then the weak solution to (1.8) continuously depend on ϕ 0 with respect to the · # -norm.

Proof of Theorem 2.1 and of Proposition 2.1
This section is divided into three parts. The first part is devoted to the existence of a weak solution. The energy identity (2.6) is shown in the second one and the continuous dependence estimate is deduced in the third.

Existence of a solution
Existence of a weak solution is carried out following closely [18]. We will first prove the existence of the solution when ϕ 0 ∈ V 2 . Then we will recover the existence of a solution when ϕ 0 ∈ H such that F (ϕ 0 ) ∈ L 1 (Ω) by a density argument. We also assume u ∈ C([0, T ]; (L ∞ (Ω) ∩ H 1 0 (Ω)) d ) and g ∈ C([0, T ]; V ′ ) for the moment. As usual in the Faedo-Galerkin method, we consider the family {ψ j } j ⊂ V 2 of the eigenvectors of the self-adjoint, positive, linear operator A + I : V 2 → H. Let us define the n-dimensional subspace Ψ n := ψ 1 , ..., ψ n and consider the orthogonal projector on this subspace P n := P Ψn . We then look for two functions of the following form: for every ψ ∈ Ψ n , and where ϕ 0n := P n ϕ 0 . By using the definition of ϕ n , problem (2.7)-(2.10) becomes equivalent to a Cauchy problem for a system of ordinary differential equations in the n unknowns b (n) i . Thanks to (H2), the Cauchy-Lipschitz theorem yields that there exists a unique solution b (n) ∈ C 1 ([0, T * n ]; R n ) for some maximal time T * n ∈ (0, +∞]. Let us show that T * n = +∞, for all n ≥ 1. Indeed, taking ψ = µ n as test function in (2.7) we get the following identity: which can be rewritten as follows d dt It is easy to see that By means of (H3), we can deduce the existence of a positive constant α such that α < 2c 1 − J L 1 (Ω) . Therefore we have Thanks to Remark 2.4 and to the identity (P n (−J * ϕ n + aϕ n ), 1) = (−J * ϕ n + aϕ n , 1) = 0 we get (2.14) Thus we obtain By using (H5), it is easy to show that On account of (H8) and using the Poincaré-Wirtinger inequality, we have Besides, from (2.14) we get that On the other hand, thanks to (H6), we have We now exploit the convexity of the function F (s) + a(x)s 2 (see Remark 2.1) If we integrate (2.11) with respect to time between 0 and t ∈ (0, T * n ), taking (2.12)-(2.13), (2.15), (2.16), (2.19), (2.20) into account, we find which holds for all t ∈ [0, T * n ), where . Here, we have used the fact that that ϕ 0 and ϕ 0,n are supposed to belong to V 2 . We point out that M and K do not depend on n.
Thus, inequality (2.21) entails that T * n = +∞, for all n ≥ 1. As a consequence, (2.7)-(2.10) has a unique global-in-time solution. Furthermore, we obtain the following estimates, holding for any given 0 < T < +∞: where N is independent of n.
for every ψ ∈ V and almost everywhere in (0, T ). Choosing ψ = A −1 (ϕ −φ), we obtain (2.47) The convective term can be estimated as follows while the convolution term can be controlled in this way Furthermore, on account of (H2), we have On the other hand, recalling Remark 2.4, we have Thus we infer the estimate Using (2.48)-(2.50), it follows from (2.47) that Therefore we obtain and Gronwall's lemma entails the continuous dependence estimate which provides uniqueness.
3 Nonlocal CHBEG equation: well-posedness of (1.9) Here we generalize Theorem 2.1 to problem (1.9). We use a fixed point theorem. Uniqueness is proven under stronger assumptions on F .
We are able to prove uniqueness of weak solutions under stronger assumptions on F , namely, (I8) There exist c 6 > 0 and c 7 ≥ 0 such that Furthermore, we require the existence of a constant c 8 ≥ 0 such that Remark 3.2. It is easy to show that (I8) implies (H3). Moreover, it is straightforward to check that (I8) is satisfied by the standard double well potential (1.2) used in [6] (see also [7] and [15]).
We have:

Proof of Theorem 3.1
We employ the Schauder fixed point theorem in one of his many variants. Let us consider the following problem Adding and subtractingμ to µ in the last term of the energy equality (3.6) yields Observe now that and (cf. (2.14)) Thus (3.7) and (3.8) yield Similarly, we have By exploiting (2.16), on account of (3.9) and (3.10), we obtain from (3.6) the following inequality Furthermore, by arguing as in the proof of Theorem 2.1 to obtain (2.13), we know that (H3) entails that there is α > 0 such that Thus, integrating (3.11) with respect to time between 0 and t ∈ (0, T ), we get (cf.(I7) and (I8)) Then, an application of the Gronwall lemma provides Using (H3) once more to control F (ϕ), we end up with (cf. (3.12))

Λ is continuous and compact
Let {φ m } be a bounded sequence in X T * and consider ϕ m = Λ(φ m ). Then, it is easy to prove that every φ m satisfies the energy equality (3.6) with G(φ m ) and µ m = aϕ m + F ′ (ϕ m ) − J * ϕ m . Also, by arguing as in the previous subsection, we find that φ m also satisfies (3.11) for every m ∈ N. In particular, we have that 13) where N is independent of m.
As a direct consequence of (3.13), we have that In particular, the strong convergence of {ϕ m } proves that Λ(B R (T * )) is compact in X T * . In order to prove the continuity of Λ we just assume that {φ m } converges to some φ in X T * . Then, on account of the above bounds and of the uniqueness for problem (3.5), we have that Λ(φ m ) converges to Λ(φ). We can thus conclude that Λ has a fixed point ϕ which is a local weak solution.

The local solution ϕ is global
We know that ϕ satisfies the energy identity (3.3) on some maximal interval (0, T 0 ).
Then, on account of Theorem 2.1 and Proposition 2.1, for any ϕ 0 ∈ L 2 (Ω) such that F (ϕ 0 ) ∈ L 1 (Ω), there exists a unique global weak solution ϕ. As a consequence we can define a semigroup on a suitable phase space.
We will show that the dynamical system (Y, S(t)) is dissipative, that is, it has a bounded absorbing set. Then, following the strategy outlined in [24], we prove that the same system has the (connected) global attractor. Proof. We adapt [18, proof of Corollary 2]. By exploiting (2.15), (2.16), (2.19) and (2.20) in the energy equality (2.6) (with g = 0), we obtain

Bounded absorbing sets
Observe that On the other hand, taking ψ = 1 in (2.3), we get Therefore, recalling Remark 2.2, we deduce that for every bounded set B of Y there exists t 0 = t 0 (B) ≥ 0 and a positive constant c B such that Then, on account of Remark 2.1, using (4.4)-(4.4) we get where H(t) = C B for all t ∈ [0, t 0 ] and H(t) = C 4 for all t > t 0 . Therefore we have By putting together (4.3) and (4.6) we obtain Then, on account of (H9), we find Finally, we combine (4.2) with (4.7) and obtain Then the thesis follows from Gronwall's lemma applied for t ≥ t 0 .

Global attractor
and, thanks to Remark 2.1, we obtain This implies that Y is connected. Hence the global attractor is also connected (see [3,Corollary 4.3]).
These further restrictions are due to the peculiar difficulty of this equation, that is, the uniform control of |φ(t)| (see [15] for the local case). Indeed, the time dependent average cannot be easily controlled by |φ(0)| as in the case of CHO equation. Clearly (I9) entails (H2)-(H4) and (I8). Such assumptions ensures that for any ϕ 0 ∈ Y, where there is a unique global weak solution ϕ owing to Theorems 3.1 and 3.2. Thus we can define a semigroup S(t) : Y → Y by setting ϕ(t) := S(t)ϕ 0 . Here Y is equipped with the metric (4.1). As in the previous section, we will show that the dynamical system (Y, S(t)) has a bounded absorbing set. Then, the same argument used to prove Theorem 4.1 will lead to the existence of the (connected) global attractor. However, a crucial preliminary step is the uniform (dissipative) bound of |φ(t)| for all t > 0.

Bounded absorbing sets and the global attractor
Thanks to Proposition 5.1 and arguing as in the proof of Proposition 4.1, we can prove the following