Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the $H^{\alpha/2}$-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.


Introduction
In this contribution, we consider a two-phase flow for incompressible fluids of different densities and different viscosities. The two fluids are assumed to be macroscopically immiscible and to be miscible in a thin interface region, i.e., we consider a diffuse interface model (also called phase field model) for the two-phase flow. In contrast to sharp interface models, where the interface between the two fluids is a sufficiently smooth hypersurface, diffuse interface model can describe topological changes due to pinch off and droplet collision.
There are several diffuse interface models for such two-phase flows. Firstly, in the case of matched densities, i.e., the densities of both fluids are assumed to be identical, there is a wellknown model H, cf. Hohenberg and Halperin or Gurtin et al. [HH77,GPV96]. In the case that the fluid densities do not coincide there are different models. On one hand Lowengrub and Truskinovsky [LT98] derived a quasi-incompressible model, where the mean velocity field of the mixture is in general not divergence free. On the other hand, Ding et al. [DSS07] proposed a model with a divergence free mean fluid velocities. But this model is not known to be thermodynamically consistent. In Abels, Garcke and Grün [AGG11] a thermodynamically consistent diffuse interface model for two-phase flow with different densities and a divergence free mean velocity field was derived, which we call AGG model for short. The existence of weak solutions of the AGG model was shown in [ADG13]. For analytic result in the case of matched densities, i.e., the model H, we refer to [Abe09b] and [GMT19] and the reference given there. Existence of weak and strong solutions for a slight modification of the model by Lowengrub and Truskinovsky was proven in [Abe09a,Abe11].
Concerning the Cahn-Hilliard equation, Giacomin and Lebowitz [GL97,GL98] observed that a physically more rigorous derivation leads to a nonlocal equation, which we call a nonlocal Cahn-Hilliard equation. There are two types of nonlocal Cahn-Hilliard equations. One is the equation where the second order differential operator in the equation for the chemical potential is replaced by a convolution operator with a sufficiently smooth even function. We call it a nonlocal Cahn-Hilliard equation with a regular kernel in the following. The other is one where the second order differential operator is replaced by a regional fractional Laplacian. We call it a nonlocal Cahn-Hilliard equation with a singular kernel, since the regional fractional Laplacian is defined by using singular kernel. The nonlocal Cahn-Hilliard equation with a regular kernel was analyzed in [GZ03,G14,GL98,LP11a,LP11b]. On the other hand, the nonlocal Cahn-Hilliard equation with a singular kernel was first analyzed in Abels, Bosia and Grasselli [ABG15], where they proved the existence and uniqueness of a weak solution of the nonlocal Cahn-Hilliard equation, its regularity properties and the existence of a (connected) global attractor. Concerning the nonlocal model H with a regular kernel, where the convective Cahn-Hilliard equation is replaced by the convective nonlocal Cahn-Hilliard equation with a regular kernel, first studies were done by [CFG12,FG12a,FG12b] , see also [FGGS19] and the references there for more recent results. More recently, the nonlocal AGG model with a regular kernel, where the convective Cahn-Hilliard equation is replaced by the convective nonlocal Cahn-Hilliard equation with a regular kernel, was studied by Frigeri [F15] and he showed the existence of a weak solution for that model. The method of the proof in [F15] is based on the Faedo-Galerkin method of a suitably mollified system and the method of passing to the limit with two parameters tending to zero. The method is different from [ADG13] which is based on implicit time discretization and a Leray-Schauder fixed point argument.
In this contribution, we consider a nonlocal AGG model with a singular kernel, where a convective Cahn-Hilliard equation in the AGG model is replaced by a convective nonlocal Cahn-Hilliard equation with a singular kernel. Our aim is to prove the existence of a weak solution of such a system.
In this contribution we consider existence of weak solutions of the following system, which couples a nonhomogeneous Navier-Stokes equation system with a nonlocal Cahn-Hilliard equation: where ρ = ρ(ϕ) :=ρ 1 +ρ 2 2 +ρ 2 −ρ 1 2 ϕ, J = −ρ 2 −ρ 1 2 m(ϕ)∇µ, Q = Ω × (0, ∞). We assume that Ω ⊂ R d , d = 2, 3, is a bounded domain with C 2 -boundary. Here and in the following v, p, and ρ are the (mean) velocity, the pressure and the density of the mixture of the two fluids, respectively. Furthermoreρ j , j = 1, 2, are the specific densities of the unmixed fluids, ϕ is the difference of the volume fractions of the two fluids, and µ is the chemical potential related to ϕ. Moreover, Dv = 1 2 (∇v + ∇v T ), η(ϕ) > 0 is the viscosity of the fluid mixture, and m(ϕ) > 0 is a mobility coefficient. The term J describes the mass flux, i.e., we have It is important to have the term with J in (1.1) in order to obtain a thermodynamically consistent model, cf. [AGG11] for the case with a local free energy.
We add to our system the boundary and initial conditions v| ∂Ω = 0 on ∂Ω × (0, ∞), (1.9) (1.11) Here ∂ n = n · ∇ and n denotes the exterior normal at ∂Ω. We note that (1.9) is the usual no-slip boundary condition for the velocity field and ∂ n µ| ∂Ω = 0 describes that there is no mass flux of the fluid components through the boundary. Furthermore we complete the system above by an additional boundary condition for ϕ, which will be part of the weak formulation, cf. Definition 3.2 below. If ϕ is smooth enough (e.g. ϕ(t) ∈ C 1,β (Ω) for every t ≥ 0) and k fulfills suitable assumptions, then where n x 0 depends on the interaction kernel k, cf. [ABG15, Theorem 6.1], and x 0 ∈ ∂Ω.
The total energy of the system at time t ≥ 0 is given by are the kinetic energy and the free energy of the mixture, respectively, and for all u, v ∈ H α 2 (Ω) is the natural bilinear form associated to L, which will also be used to formulate the natural boundary condition for ϕ weakly. Every sufficiently smooth solution of the system above satisfies the energy identity for all t ≥ 0. This can be shown by testing (1.1) with v, (1.3) with µ and (1.4) with ∂ t ϕ, where the product of Lϕ and ∂ t ϕ coincides with under the same natural boundary condition for ϕ(t) as before, cf. (1.12).
We consider a class of singular free energies, which will be specified below and which includes the homogeneous free energy of the so-called regular solution models used by Cahn and Hilliard [CH58]: where 0 < ϑ < ϑ c . This choice of the free energies ensures that ϕ(x, t) ∈ [−1, 1] almost everywhere. In order to deal with these terms we apply techniques, which were developed in Abels and Wilke [AW07] and extended to the present nonlocal Cahn-Hilliard equation in [ABG15]. Our proof of existence of a weak solution of (1.1)-(1.4) together with a suitable initial and boundary condition follows closely the proof of the main result of [ADG13]. The following are the main differences and difficulties of our paper compared with [ADG13]. Since we do not expect H 1 -regularity in space for the volume fraction ϕ of a weak solution of our system, we should eliminate ∇ϕ from our weak formulation taking into account the incompressibility of v. Implicit time discretization has to be constructed carefully, using a suitable mollification of ϕ and an addition of a small Laplacian term to the chemical potential equation taking into account of the lack of H 1 -regularity in space of ϕ. While the arguments for the weak convergence of temporal interpolants of weak solutions of the time-discrete problem are similar to [ADG13], the function space used for the order parameter has less regularity in space since the nonlocal operator of order less than 2 is involved in the equation for the chemical potential. For the convergence of the singular term Ψ ′ (ϕ), we employ the argument in [ABG15]. The only difference is that we work in space-time domains directly. For the validity of the energy inequality, additional arguments using the equation of chemical potential and the fact that weak convergence together with norm convergence in uniformly convex Banach spaces imply strong convergence are needed.
The structure of the contribution is as follows: In Section 2 we present some preliminaries, we fix notations and collect the needed results on nonlocal operator. In Section 3, we define weak solutions of our system and state our main result concerning the existence of weak solutions. In Section 4, we define an implicit time discretization of our system and show the existence of weak solutions of an associated time-discrete problem using the Leray-Schauder theorem. In Section 5, we obtain compactness in time of temporal interpolants of the weak solutions of time-discrete problem and obtain weak solutions of our system as weak limits of a suitable subsequence.

Preliminaries
denotes the duality product, where X is a Banach space and X ′ is its duak. We write X ֒→֒→ Y if X is compactly embedded into Y . For a Hilbert space H its inner product is denoted by (· , ·) H .
Let M ⊆ R d be measurable. As usual L q (M ), 1 ≤ q ≤ ∞, denotes the Lebesgue space, . q its norm and (. , .) M = (. , .) L 2 (M ) its inner product if q = 2. Furthermore L q (M ; X) denotes the set of all f : M → X that are strongly measurable and q-integrable functions/essentially bounded functions. Here X is a Banach space. If M = (a, b), we denote these spaces for simplicity by L q (a, b; X) and L q (a, b). Recall that f : [0, ∞) → X belongs L q loc ([0, ∞); X) if and only if f ∈ L q (0, T ; X) for every T > 0. Furthermore, L q uloc ([0, ∞); X) is the uniformly local variant of L q (0, ∞; X) consisting of all strongly measurable f : [0, ∞) → X such that Then the orthogonal projection onto L 2 (0) (Ω) is given by For the following we denote Because of Poincaré's inequality, H 1 (0) (Ω) is a Hilbert space. More generally, we define for s ≥ 0 for every open and bounded subset Ω ′ with Ω ′ ⊂ Ω.
The Banach space of all bounded and continuous f : I → X is denoted by BC(I; X). It is equipped with the supremum norm. Moreover, BU C(I; X) is defined as the subspace of all bounded and uniformly continuous functions. Furthermore, BC w (I; X) is the set of all bounded and weakly continuous f : ; X) and we set H 1 (0, T ; X) = W 1 2 (0, T ; X) and H 1 uloc ([0, ∞); X) := W 1 2,uloc ([0, ∞); X). Finally, we note: Lemma 2.1. Let X, Y be two Banach spaces such that Y ֒→ X and X ′ ֒→ Y ′ densely. Then For a proof see e.g. Abels [Abe09a].

Properties of the Nonlocal Elliptic Operator L
In the following let E be defined as in (1.14). Assumptions (1.6)-(1.8) yield that there are positive constants c and C such that This implies that the following norm equivalences hold: cf. [ABG15, Lemma 2.4 and Corollary 2.5].
In the following we will use a variational extension of the nonlocal linear operator L (see (1.5)) by defining L :
We will also need the following regularity result, which essentially states that the operator L is of lower order with respect to the usual Laplace operator. This result is from [ABG15, Lemma 2.6].
belongs to H 2 loc (Ω) and satisfies the estimate where C is independent of θ > 0 and g.
For the following let φ : [a, b] → R be continuous and define φ(x) = +∞ for x ∈ [a, b]. As in [ABG15, Section 3] we fix θ 0 and consider the functional The following characterization of ∂F θ (c) is an important tool for the existence proof.
Moreover, the following estimates hold for some constant C > 0 independent of c ∈ D(∂F θ ) and θ 0.

Weak Solutions and Main Result
In this section we define weak solutions for the system (1.1)-(1.4), (1.9)-(1.11) together with a natural boundary condition for ϕ given by the bilinear form E, summarize the assumptions and state the main result.
, be a bounded domain with C 2 -boundary. The following conditions hold true: (3.1) A standard example for a homogeneous free energy density Ψ satisfying the previous conditions is given by (1.15). Since for solutions we will have ϕ(x, t) ∈ [−1, 1] almost everywhere, we only need the functions m, η on this interval. But for simplicity we assume m, η to be defined on R.
Remark 3.4. Using e.g. ϕ∇µ ∈ L 2 (0, ∞; L 2 (Ω)) one can consider this term in (3.2) as a given right-hand side and obtain the existence of a pressure such that (1.1) holds in the sense of distributions in the same way as for the single Navier-Stokes equations, cf. e.g. [Soh01].

Approximation by an Implicit Time Discretization
Let Ψ be as in Assumption 3.1. We define Ψ 0 : [−1, 1] → R by Ψ 0 (s) = Ψ(s)+κ s 2 2 for all s ∈ [a, b]. Then Ψ 0 : [−1, 1] → R is convex and lim s→±1 Ψ ′ 0 (s) = ±∞. A basic idea for the following is to use this decomposition to split the free energy E free into a singular convex part E and a quadratic perturbation. In the equations this yields a decomposition into a singular monotone operator and a linear remainder. To this end we define an energy E : given by This yields the decomposition Moreover, E is convex and E = F 0 if one chooses φ = Ψ 0 and F 0 is as in Subsection 2.1. This is a key relation for the following analysis in order to make use of Theorem 2.3, which in particular implies that ∂E = ∂F 0 is a maximal monotone operator.
Remark 4.1. (i) As in [ADG13] we obtain the important relation for all ψ ∈ C ∞ 0,σ (Ω) to (4.2), which will be used to derive suitable a-priori estimates. (ii) Integrating (4.3) in space one obtains Ω ϕ dx = Ω ϕ k dx because of div v = 0 and the boundary conditions.
The following lemma is important to control the derivative of the singular free energy density Ψ ′ (ϕ).
We will prove existence of weak solutions with the aid of the Leray-Schauder principle. In order to obtain a suitable reformulation of our time-discrete system we define suitable L k , F k : Moreover we define for w = (v, ϕ, µ) ∈ X. By construction w = (v, ϕ, µ) ∈ X is a solution of (4.2)-(4.4) if and only if

In [ADG13, Section 4.2] it is shown that
′ is invertible and that for every f ∈ L 2 (Ω) −div(m(P h ϕ k )∇µ) + Ω µ dx = f in Ω , ∂ n µ| ∂Ω = 0 (4.12) has a unique solution µ ∈ H 2 n (Ω). This follows from the Lax-Milgram Theorem and elliptic regularity theory. Moreover, in [ADG13, Section 4.2] the estimate is shown. Because of Theorem 2.3, ∂F h is maximal monotone and therefore is invertible. Moreover, (I + ∂F h ) −1 : L 2 (Ω) → H 1 (Ω) is continuous, which can be shown as in the proof of Proposition 7.5.5 in [Abe07]. Since now a nonlocal operator is involved we provide the details for the convenience of the reader. Let f l → l→∞ f in L 2 (Ω) such that f l = u l + ∂F (u l ) and f = u + ∂F (u) be given. Then u l → u in H 1 (Ω) since We introduce the following auxiliary Banach spaces in order to obtain a completely continuous mapping in the following. Because of the considerations above L −1 k : Y → X is continuous. Because of the compact embedding Y ֒→֒→ Y , L −1 k : Y → X is compact. Next we show that F k : X → Y is continuous and bounded. To this end one uses the estimates: Note that P h ϕ k and therefore ρ(P h ϕ k ) belong to H 2 (Ω)). More precisely: (Ω), one has to estimate a term of the form ρ(P h ϕ k )∂ l v i v j in L 3 2 (Ω), which are a product of functions in L ∞ (Ω), L 2 (Ω) and L 6 (Ω). Therefore the term is bounded in L 3 2 (Ω). Moreover, there are terms of the form ∂ l ρ(P h ϕ k )v i v j in L 3 2 (Ω), where each factor belongs to L 6 (Ω).
(ii) To estimate (div J)v in L 3 2 (Ω) one has terms of the form m ′ (P h ϕ k )∂ i P h p k ∂ j µv l and of the form m(P h ϕ k )∂ i ∂ j µv l . For the first type of terms the first factor is in L ∞ (Ω) and the other three are in L 6 (Ω), which yields the bound in L 3 2 (Ω). The second type are products of functions in L ∞ (Ω), L 2 (Ω) and L 6 (Ω).
The estimates of the other terms are more easy and left to the reader. These estimates show the boundedness of F k . Using analogous estimates for differences of the terms, one can show the continuity of F k : X → Y .
We will now apply the Leray-Schauder principle on Y . To this end we use that L k (w) − F k (w) = 0 for w ∈ X is equivalent to (4.14) Therefore we define K k := F k • L −1 k : Y → Y . We remark that K k is a compact operator since L −1 k : Y → X is compact and F k : X → Y is continuous. Hence (4.14) is equivalent to the fixed-point equation Now we have to show that there is some R > 0 such that: To this end we assume that f ∈ Y and 0 ≤ λ ≤ 1 are such that f = λK k f . Let w = L −1 k (f ). Then
Altogether we conclude for some C k independent of (v, ϕ, µ). Using ϕ L ∞ ≤ 1, Korn's inequality, (2.2), and the fact that η, m and a are bounded from below by a positive constant, we obtain (4.20) In order to estimate µ L 2 , we distinguish the cases λ ∈ [ 1 2 , 1] and λ ∈ [0, 1 2 ). In the case λ ∈ [ 1 2 , 1], we simply use 1 2 | Ω µ dx| ≤ λ| Ω µ dx| and conclude as in the proof of Lemma 4.2 together with (4.20) from (4.18) that In the case λ ∈ [0, 1 2 ) we conclude directly from (4.20) that Ω µ dx ≤ C k . Thus (4.20) can be improved to (4.21) With the help of (4.13) we can estimate µ H 2 (Ω) and derive Using (4.18) we also have ∂F h (ϕ) L 2 (Ω) ≤ C k . Altogether we conclude Finally we can estimate f = L k (w) in Y by using that f − λF k L −1 k (f ) = 0 implies f = λF k (w) together with the boundedness of F k : X → Y . Thus we obtain Thus the condition of the Leray-Schauder principle is satisfied, which proves the existence of a solution.
5 Proof of Theorem 3.3