On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems

The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.


Introduction
The phenomenon of phase separation of incompressible binary fluids in a porous medium can be modeled by means of a diffuse interface approach. Consider a mixture of two fluids occupying a bounded domain Ω ⊂ R d , d = 2, 3, for any time t ∈ (0, T ), T > 0, denote by ϕ the difference of the fluid (relative) concentrations and by u the (averaged) fluid velocity. Assuming that the two fluids have the same constant density, the resulting model is the so-called Cahn-Hilliard-Brinkman (CHB) system (see, e.g., [28,30]) ϕ t + ∇ · (uϕ) = ∆µ µ = −∆ϕ + F ′ (ϕ) −∇ · (ν∇u) + ηu + ∇p = µ∇ϕ + h ∇ · u = 0 (1. 1) in Ω × (0, T ), T > 0. Here ν > 0 is the viscosity coefficient, η > 0 the fluid permeability and p is the fluid pressure. Other constants are supposed to be one for simplicity. The mobility is also assumed to be constant and equal to one, while F stands for a double well potential accounting for phase separation. The average velocity u obeys a modified Darcy's law proposed by H.C. Brinkman in 1947 (see [4]).
System (1.1) endowed with no-slip and no-flux boundary conditions has been analyzed from the numerical viewpoint in [6] (see also [9]). Some theoretical results can be found in [3], where well-posedness in a weak setting as well as longtime behavior of solutions (i.e., existence of the global attractor and convergence to a unique equilibrium) have been investigated. Another interesting issue is the analysis of behavior of solutions when ν goes to zero. Indeed when ν = 0 system (1.1) becomes the so-called Cahn-Hilliard-Hele-Shaw (CHHS) model which is used, for instance, to describe tumor growth dynamics (see, e.g., [26] and references therein, cf. also [8]). This model presents several technical difficulties (cf. [26,32,33], see also [10,9,34] for numerical schemes). For instance, uniqueness of weak solutions is an open issue even in dimension two, as well as the existence of a global strong solution in dimension three for sufficiently general initial data (see [26]). Existence of a global weak solution to the CHHS system is obtained in [3] as limit of solutions to system (1.1) (see also [10,Thm.2.4] for an existence result). In the same paper, the difference of (strong) solutions to (1.1) and the CHHS system is estimated with respect to ν and to the initial data in dimension two. Most of the quoted papers deal with a regular potential F , that is, F is defined on the whole real line (however, see [8] for a singular potential).
In this contribution we want to analyze a nonlocal variant of (1.1) which is obtained by replacing the standard Cahn-Hilliard (CH) equation by its nonlocal version. More precisely, we consider the following nonlocal CHB system in Ω × (0, T ). Here the viscosity may depend on ϕ, while J : R d → R is a suitable interaction kernel and a(x) = Ω J(x − y)dy. This system is endowed with boundary and initial conditions We recall that the nonlocal CH equation can be justified in a more rigorous way from the physical viewpoint (cf. [19], see also [20,21]). Also, the standard CH equation can be interpreted as an approximation of the nonlocal one. The nonlocal CH equation has been analyzed in a number of papers, under various assumptions on the potential F and on the mobility (see, e.g., [1,7,27,17,18,24,25,29], cf. also [22,23] for the numerics). In addition, a series of papers have recently been devoted to the so-called Cahn-Hilliard-Navier-Stokes (CHNS) system in its nonlocal version (cf. [5,11,12,13,14,15,16]). Adapting the techniques devised in [5], we can prove existence of a global weak solution to (1.2)-(1.3). Its uniqueness (for constant viscosity) also holds in dimension three. However, the main goal is the analysis of the vanishing viscosity case where the limi t problem is in Ω × (0, T ), i.e. the nonlocal CHHS system, subject to the boundary and initial conditions in Ω. (1.5) As in [3], we can prove that a solution to (1.4)-(1.5) can be obtained as a limit of solutions to (1.2)-(1.3). In addition, uniqueness holds when ϕ 0 is bounded (and so is ϕ). Here we take advantage of the fact that the nonlocal CH equation is essentially a second-order equation and not a fourth-order equation like in the standard CHHS system. Then, further reasonable assumptions on J allow us to establish some regularity properties of the solutions. These properties help us to estimate the difference, with respect to ν and the initial data, between a solution to (1.2)-(1.3) and a solution to the CHHS system.
The plan of this paper goes as follows. Notation, assumptions and statements of the main results are contained in Section 2. Results concerning existence and regularity for (1. 2 Functional setup and main results

Notation
We set H := L 2 (Ω) and V := H 1 (Ω). We denote by · and (· , ·) the norm and the scalar product in H, respectively, while · stands for the duality between V ′ and V . For every ϕ ∈ V ′ we denote byφ the average of ϕ over Ω, namelȳ ϕ = |Ω| −1 ϕ, 1 . Then we define The linear operator A = −∆ : V 2 ⊂ H → H with dense domain is self-adjoint and non-negative. Moreover, it is strictly positive on V 0 = {ψ ∈ V :ψ = 0} and it for every r ∈ R. Observe that the norm · # defined as is equivalent to the usual norm of V ′ . Besides, let V be the space of divergence-free test functions defined by We shall use the following canonical spaces (see, e.g., [31, Chapter I]) We will still use (· , ·) and · to denote the scalar product in H and the duality between V ′ and V , respectively. Finally, c will indicate a generic nonnegative constant depending on Ω, J, F, and h at most. Instead, N will stand for a generic positive constant which has further dependence on T and/or on some norm of ϕ 0 . The value of c and N may vary even within the same line.
(H6) ν is locally Lipschitz on R and there exist ν 0 , ν 1 > 0 such that 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 Besides, it can be shown that (H3) implies the existence of c 6 , c 7 > 0 such that

Remark 2.6
The convective nonlocal CH equation can formally be rewritten as follows from which the crucial role of (H2) is evident, namely, we are dealing with a convection-diffusion integrodifferential equation.

Statement of the main results
Let us introduce the definition of weak solution to (1.2)-(1.3).
and it satisfies Thus the total mass of any weak solution is conserved.
Global existence of a weak solution is given by (Ω) and suppose that (H0)-(H7) are satisfied. Then there exists a weak solution (ϕ, the following energy equality holds for almost every t ∈ (0, T ) Furthermore, we have Weak solutions can be regular provided ϕ 0 is bounded. Indeed we have In particular, we have for some M > 0, independent of ν and T .
If the viscosity ν is constant then we have a continuous dependence estimate , corresponding to the initial data ϕ 1,0 and ϕ 2,0 , respectively.
The limit ν → 0. As a second step in our analysis we study the limit of (1.2)-(1.3) with constant viscosity ν, as ν tends to 0. We recall that the resulting limit system is (1.4)-(1.5) whose weak formulation is given by the following definition.
and it satisfies To analyze (1.4)-(1.5) we replace assumption (H5) with the stronger (H8) η ∈ L ∞ (Ω) and there exists η 0 > 0 such that Furthermore, for the sake of simplicity, we let h = 0. Then we have the following existence theorem Furthermore, the following energy equality holds for almost any t ∈ (0, T ): where E is defined by (2.5).
Next corollary is related to further regularity in the case where η is constant.
In case J is more regular, we gain regularity also for the velocity field u. For the sake of completeness, we first recall the definition of admissible kernel (see [2, Definition 1]).

Existence and regularity for the CHB system
The first part of this section is devoted to prove Theorem 2.2. Then, in the second part, the proofs of Corollary 2.1 and Proposition 2.1 are given.

Proof of Theorem 2.2
The proof will be carried out by means of a Faedo-Galerkin approximation scheme, following closely [5]. We first prove existence of a solution when ϕ 0 ∈ V 2 and h ∈ C([0, T ]; H); then, by a density argument, we will recover the same result for any initial datum ϕ 0 ∈ H with F (ϕ 0 ) ∈ L 1 (Ω) and any h ∈ L 2 ([0, T ]; V ′ ).
We consider the families {ψ j } j∈N ⊂ V 2 and {v j } j∈N ⊂ V respectively eigenvectors of A + I : V 2 → H and of the Stokes operator, which are both self-adjoint, positive and linear. Let us define the n-dimensional subspaces Ψ n := ψ 1 , ..., ψ n and W n := w 1 , ..., w n with the related orthogonal projectors on this subspace P n := P Ψn andP n := P Wn . We then look for three functions of the following form: for every ψ ∈ Ψ n , every w ∈ W n and where ϕ 0n := P n ϕ 0 . By using the definition of ϕ n , µ n and u n , problem (3.1)-(3.5) 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, using ψ = µ n as test function in (3.1) and w = u n in (3.2) we get the following identity: (∇µ n , ∇P n (J * ϕ n )) ≤ 1 4 ∇µ n 2 + ϕ n 2 J 2 W 1,1 , (3.8) By means of (H3), we can deduce the existence of a positive constant α such that (3.10) By using (H6) and Poincaré's inequality, it is easy to show that there exists β > 0 such that and, on account of (H7,) we have Let us now exploit (3.7) in (3.6) and integrate it with respect to time between 0 and t ∈ (0, T * n ). Taking (3.8)-(3.12) into account, we find which holds for all t ∈ [0, T * n ), where and K = 2 J 2 W 1,1 . 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.
We are left to prove the energy identity (2.6). Let us take ψ = µ(t) in equation (2.2). This yields By arguing as in [5, proof of Corollary 2], one can prove the identity which holds for almost every t > 0. Thus (2.6) follows directly from (3.39). ✷
It is now possible to pass to the limit as k → ∞ in the weak formulation of (1.2)-(1.3). We will do that restricting ourselves to the case ψ ∈ W 1,d+ε (Ω) ⊂ V in (2.2) and then recovering the fact that (2.8) holds for every ψ ∈ V by a density argument. Some attention is needed when passing to the limit in the viscous term of the Brinkman equation; as a matter of fact we have which tends to 0 as ν k → 0. The convective term can be treated as follows: where r ≥ 0 is arbitrary. Here the second term vanishes thanks to the boundedness of ϕ and (4.11). The first one goes to 0 thanks to (4.2), (4.8) and the fact that Finally, we can pass to the limit into the the Korteweg force since, for every r ≥ 0, we have and the second term goes to 0 thanks to the boundedness of ϕ and (4.9), while the first one vanishes thanks to (3.22) and (4.8) and the inequality It is easy to see that (2.9) makes sense also for every v ∈ H. Furthermore, thanks to (4.5) we can deduce that (2.8) holds also for every ψ ∈ V by a density argument. Thus, we showed that there is a subsequence of (ϕ k , u k ) converging to a (ϕ, u) which is a weak solution to (1.4)-(1.5). On account of Lemma 2.1 in [26] we can deduce Furthermore, from Theorem 2.4 we have ϕ ∈ L ∞ (0, T ; Ω), which, thanks to (H1), leads to u ∈ L ∞ (0, T ; L p (Ω)) for each p ≥ 1.
Consider now (5.4). Instead of controlling I 1 as in (5.9), we obtain Also, exploiting the estimates for u and arguing as in the proof of Proposition 2.2, thanks to Corollary 2.2 we have Thus we can still prove inequality (5.11) and the proof can be completed arguing as above. ✷ 6 Convergence of solutions as ν → 0 In this section we prove Theorem 2.6.