A TWO-PHASE FLOW MODEL WITH DELAYS

. In this article, we study a coupled Allen-Cahn-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier- Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of the weak and strong solution when the external force contains some delays. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions.


1.
Introduction. It is well accepted that the incompressible Navier-Stokes (NS) equation governs the motions of single-phase fluids such as air or water. On the other hand, we are faced with the difficult problem of understanding the motion of binary fluid mixtures, that is fluids composed by either two phases of the same chemical species or phases of different composition. Diffuse interface models are well-known tools to describe the dynamics of complex (e.g., binary) fluids, [11]. For instance, this approach is used in [1] to describe cavitation phenomena in a flowing liquid. The model consists of the NS equation coupled with the phase-field system, [2,11,10,12]. In the isothermal compressible case, the existence of a global weak solution is proved in [9]. In the incompressible isothermal case, neglecting chemical reactions and other forces, the model reduces to an evolution system which governs the fluid velocity v and the order parameter φ. This system can be written as a NS equation coupled with a convective Allen-Cahn equation, [11]. The associated initial and boundary value problem was studied in [11] in which the authors proved that the system generated a strongly continuous semigroup on a suitable phase space which possesses a global attractor. They also established the existence of an exponential attractor. This entails that the global attractor has a finite fractal dimension, which is estimated in [11] in terms of some model parameters. The dynamic of simple single-phase fluids has been widely investigated although some important issues remain unresolved, [21]. In the case of binary fluids, the analysis is even more complicate and the mathematical studied is still at it infancy as noted in [11]. As noted in [10], the mathematical analysis of binary fluid flows is far from being well understood. For instance, the spinodal decomposition under shear consists of a two-stage evolution of a homogeneous initial mixture: a phase separation stage in which some macroscopic patterns appear, then a shear stage in which these patters organize themselves into parallel layers (see, e.g. [18] for experimental snapshots).

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This model has to take into account the chemical interactions between the two phases at the interface, achieved using a Cahn-Hilliard approach, as well as the hydrodynamic properties of the mixture (e.g., in the shear case), for which a Navier-Stokes equations with surface tension terms acting at the interface are needed. When the two fluids have the same constant density, the temperature differences are negligible and the diffuse interface between the two phases has a small but non-zero thickness, a well-known model is the so-called "Model H" (cf. [13]). This is a system of equations where an incompressible Navier-Stokes equation for the (mean) velocity v is coupled with a convective Cahn-Hilliard equation for the order parameter φ, which represents the relative concentration of one of the fluids.
Time delays are omnipresent and virtually unavoidable in everyday life. They appear in applications such as physics, biology, epidemics, fluid control, transport and population models. Because of their importance, differential equations with delays have received considerable attention, [15,3,5,6,7,16,8,17,20,22]. In real world applications, to control a system by applying some type of external forces, it is natural that these forces take into account not only the present state of the system, but also its history, [3]. In fact, in many cases it was shown that the presence of a delay term in a differential equations can be a source of instability and even an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay, [16,8]. The presence of a delay term in a differential equations can drastically change the mathematical analysis of the model. For instance, when a delay term, general enough is added in equations such as the 2D Navier-Stokes system or the 2D Allen-Cahn-Navier-Stokes system, special attention is needed to analyse the global asymptotic behavior of the system since one needs to consider a semigroup in a different phase space, [3,19]. In the stochastic setting, the behaviors of a stochastic differential equation with delays and without delays can be very different. More precisely, if a delay term is added to a stochastic differential equation, its solution is no longer Markov and losing the Markovian properties makes the analysis of the system more difficult and complex, [15].
In [5,6,7], the authors studied the NS equations in which the forcing term contains some hereditary features. The model can be used for instance to control a system by applying a force which takes into account not only the present state of the system, but also the history of the solutions. The existence and uniqueness of solutions to the 2D NS equations with delays was investigated in [5] and the asymptotic behavior of the solutions is studied in [6]. The existence of attractors for the 2D NS equations with delays is proved in [7]. In [4], the authors studied the existence of an attractor for the 3D Lagrangian averaged Navier-Stokes−α (3D LAN-α) model with delays. Instead of working directly with the 3D LAN-α model, they proved the existence of attractors for an abstract delay model and then applied the result to the 3D LAN-α model. Motivated by the above works, we study in this article an AC-NS model with delays. We prove the existence and uniqueness of a weak and a strong solutions when the external force contains some delays. Let us note that the coupling between the Navier-Stokes and the Allen-Cahn systems makes the analysis more involved. In [19], we proved the existence of an attractor for the model using the theory of pullback attractors.
The article is divided as follows. In the next section, we introduce the AC-NS model with delays and its mathematical setting. The third section studies the existence of solutions when the delay term satisfies some hypothesis similar to that of [5,6,7]. In the fourth section, we study the asymptotic behavior of the weak solutions when the delay term satisfies some hypothesis used in [20]. The stability of the stationary solutions is analyzed in the fifth section.

2.
The AC-NS model and its mathematical setting.
2.1. Governing equations. In this article, we consider a model of homogeneous incompressible two-phase flow with delays. More precisely, we assume that the domain M of the fluid is a bounded domain in 2 . Then, we consider the system in M × (0, +∞).
In (1), the unknown functions are the velocity v = (v 1 , v 2 ) of the fluid, its pressure p and the order (phase) parameter φ. The quantity µ is the variational derivative of the following free energy functional where, e.g., F (r) = r 0 f (ζ)dζ. Here, the constants ν > 0 and K > 0 correspond to the kinematic viscosity of the fluid and the capillarity (stress) coefficient respectively, , α > 0 are two physical parameters describing the interaction between the two phases. In particular, is related with the thickness of the interface separating the two fluid. Hereafter, as in [11] we assume that ≤ α. We endow (1) with the boundary condition v = 0, ∂φ ∂η = 0 on ∂M × (0, +∞), where ∂M is the boundary of M and η is its outward normal. The initial condition is given by The external forcing Q takes into account not only the present state of the system, but also the history of the solutions.

2.2.
Mathematical setting. We first recall from [11] a weak formulation of (1)- (4). Hereafter, we assume that the domain M is bounded with a smooth boundary ∂M (e.g., of class C 2 ). We also assume that f ∈ C 1 ( ) satisfies where c f is some positive constant and k ∈ [1, +∞) is fixed. It follows from (5) that If X is a real Hilbert space with inner product (·, ·) X , we will denote the induced norm by | · | X , while X * will indicate its dual. We set We denote by H 1 and V 1 the closure of V 1 in (L 2 (M)) 2 and (H 1 0 (M)) 2 respectively. The scalar product in H 1 is denoted by (·, ·) L 2 and the associated norm by | · | L 2 . Moreover, the space V 1 is endowed with the scalar product We now define the operator A 0 by where P 1 is the Leray-Helmotz projector in L 2 (M) onto H 1 . Then, A 0 is a selfadjoint positive unbounded operator in H 1 which is associated with the scalar product defined above. Furthermore, A −1 0 is a compact linear operator on H 1 and |A 0 · | L 2 is a norm on D(A 0 ) that is equivalent to the H 2 −norm.
Note that A −1 γ is a compact linear operator on L 2 (M) and |A γ · | L 2 is a norm on D(A γ ) that is equivalent to the H 2 −norm.
We introduce the bilinear operators B 0 , B 1 (and their associated trilinear forms b 0 , b 1 ) as well as the coupling mapping R 0 , which are defined from D( More precisely, we set Note that R 0 (µ, φ) = P 1 µ∇φ. We recall that B 0 , B 1 and R 0 satisfy the following estimates (12) Now we define the Hilbert spaces Y and V by endowed with the scalar products whose associated norms are We also set f γ (r) = f (r) − α −1 γr and observe that f γ still satisfies (7) with γ in place of 2γ since ≤ α. Also its Hereafter, we denote by λ 1 > 0 a positive constant such that Using the notations above, we rewrite (1) where Q : [−r, ∞) × Y → H 1 and ϑ : [−r, 0] → Y are continuous functions satisfying some additional conditions (see (18)). Hereafter, we will also use the notation

Remark 1.
In the weak formulation (14), the term µ∇φ is replaced by A γ ∇φ. This is justified since f γ (φ)∇φ is the gradient F γ (φ) and can be incorporated into the pressure gradient, see [11] for details.
In the case when the delay r is zero, the weak formulation of (14) was proposed and studied in [11,10], where the existence and uniqueness of solution was proved. Hereafter, to simplify the notation, we set K = 1.
We assume that the function τ (t) is differentiable and there exists a constants τ * and r > 0 such that Throughout this article we also suppose that the forcing Q satisfies the local Lipschitz condition.

Existence of solution.
In this section we discuss the global existence of the weak solution and the strong solution for the AC-NS system with delay (14).
We look for (u m , φ m ) ∈ Y m solution to the ordinary differential equations where , it follows from the theory of ordinary differential equation that this equation has a solution (v m , φ m ), (see also Theorem A1 of [5]). Hereafter C denotes a constant independent of m and depending only on data such as M and whose value may be different in each inequality. Finally, c will denote a generic constant.
Hereafter, for any (w, ψ) ∈ Y, we set where α 0 > 0 is a constant large enough and independent on (w, ψ) such that E(w, ψ) is nonnegative (note that F γ is bounded from below). We can check that (see [10] for details) there exists a monotone non-decreasing function C 0 (independent on time and the initial condition) such that Let g be a continuous nonnegative scalar function defined on [−r, +∞) and let R be a continuous positive monotone nondecreasing function defined on [0, +∞). As in [20,5,6,7,4], we set and we assume that the inverse G −1 of G is well-defined on [0, +∞). Then we have the following result.
Theorem 3.1. We suppose that (17)-(18) are satisfied. We also assume that there exists a constant b f ≥ 0 such that Then (14) has a unique weak solution (v, φ) for any initial value Proof. By taking the scalar product in H 1 of (19) 1 with v m , then taking the scalar product in L 2 (M) of (19) 3 with µ m , we derive that (see [11] for the details) It follows from (23) and (24) that Let Then for T ≥ t ≥ 0, we have Therefore we have by the Bihari inequality (see [20]) where Q 1 is a monotone non-decreasing function independent on time, the initial condition and m. It follows from (29) and (28) that A γ φ m is bounded in L 2 (0, T ; H 2 ).
Using (27), (29) and (10)- (12), we can check that Therefore, we can take a subsequence (still) denoted (v m , φ m ) such that Thus we have by the compactness theorem (see [21]) that Using (31)-(32) and standard methods as in [21], we can pass to the limit in (19) as m → ∞, and derive that (v, φ) is a weak solution to (14). Let us recall that the passage to the limit in the delay force is obtained as in [20,5].
For the uniqueness of weak solutions and their continuous dependence (from Y × L 2 (−r, 0, Y) into Y) with respect to the initial data, we proceed as in [20]. Let u 1 = (v 1 , φ 1 ) and u 2 = (v 2 , φ 2 ) be two weak solutions to (14).

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We recall that (w, ψ) satisfies We set w 1 = (v 1 , φ 1 )(t 1 ). Since Q(t, (v, φ)) satisfies a local Lipschitz condition, for any positive constant 0 > 0, there exists L( 0 ) > 0 such that where We can assume without loss of generality that Then as in [11,14], we can check that which gives By the Gronwall lemma, we obtain that This proves the uniqueness of weak solutions and the continuous dependence (in the Y−norm) on the initial data follow.

Global strong solution.
In this part we discuss the existence and uniqueness of the strong solution to (14).
Theorem 3.2. The assumptions are the same as in Theorem 3.
Then, there exists a unique strong solution (v, φ) to (14) such that More precisely, ∀t ∈ [0, T ], we have where M 0 > 0 depends on T as well as the initial condition (v 0 , φ 0 ).
Proof. Let (v m , φ m ) be any fixed approximation to the solution to (14). It follows from the proof of Theorem 3.1 that there exists a constant M 1 = M 1 (T ) > 0 such that Now taking the inner product in H 1 of (14) 1 with 2A 0 v m , the inner product in L 2 (M) of (14) 2 and (14) 3 with 2A 2 γ φ m and adding the resulting equalities gives (see [11] for the details) where Then as in [11], we can check that ( setting 2ν = ν) Let us set (48) It follows from the Gronwall lemma that Using (49) and (10)- (12), we can check that Therefore, we can take a subsequence (still) denoted (v m , φ m ) such that Using (49)-(51), we derive from the compactness theorem (see [21])) that Therefore, by passing to the limit in (19), we derive the desired conclusion. It is clear that (43) follows from (49). The uniqueness of strong solution follows from Theorem 3.3 below.
3.1.1. Continuity in V with respect to the initial data.
where L( 0 ) and Ψ are defined below.

TWO-PHASE FLOW WITH DELAYS 3285
It follows from (65)-(66) that and Then for a fixed time t > 0, the Lipschitz continuous dependence (in the V− norm) on the initial data follow.
As a corollary, we have: We assume that (17)- (18) are satisfied. We also suppose that there exist constants a f > 0 and b f ≥ 0 such that Then for every (v 0 , φ 0 ) ∈ V and ϑ ∈ L 2 (−r, 0; Y), there exists a unique strong solution of the system (93), which depends continuously (from V × L 2 (−r, 0; Y) into V) on the initial data.
4. Exponential behavior of weak solutions. In this part we discuss the exponential behavior of weak solutions to the AC-NS with delay (14).
Theorem 4.1. The assumptions are the same as in Corollary 2. Let κ, α 1 , λ 1 , c 1 be given respectively by (80), (81), (13) and (79) below. We also assume that Then we have the following asymptotic behavior of weak solutions. where ρ > 0 is a positive number such that and hereafter M 0 denotes a suitable monotone non-decreasing function independent of time.
Proof. As in [11], we can check that where and for any y ∈ , where c f , c * , c f and c f are positive, sufficiently large constants that depend only on f. From [11], we also note that where C m depends on the shape of M, but not its size and c 1 is given by Let us choose κ ∈ (0, 1) as From now on, c i will denote a positive constant independent on the initial data and on time. Let us set It follows from (75)-(80) that which gives Let where θ > 0 is a positive number such that Note that if (70) is satisfied, we can find θ > 0 small enough such that (85) holds.

Stability of stationary solutions.
Hereafter, we study the stability of stationary solutions to (14). We first prove the existence of stationary solution to (14) when the delay has a special form, provided that viscosity ν and the physical parameter are large enough. Then, we prove that all weak solutions to (14) converge exponentially to this unique stationary solution. We assume that the delay term is given by where for some fixed constant L 1 > 0.
A stationary solution to (14) is a (v * , φ * ) such that 5.1. Existence and uniqueness of stationary solution. Let {(w i , ψ i ), i = 1, 2, 3, · · · } ⊂ V be an orthonormal basis of Y, where {w i , i = 1, 2 · · · }, {ψ i , i = 1, 2 · · · } are eigenvectors of A 0 and A γ respectively. We set Y m = span{(w 1 , ψ 1 ), · · · (w m , ψ m )}. For fixed (U, Φ) ∈ V m , We consider the following approximating problems: We will see that for each m, we may apply a fixed point theorem to the map T m (restricted to a suitable subset ∧ m ⊂ V m ) to ensure that we can obtain the existence of a solution (v m , φ m ) to (94).
Lemma 5.1. We assume that Q 0 satisfies (90)-(92). Then any solution (v m , φ m ) to (94) satisfies the estimate where κ 1 is given by (99) below and M 1 (·) is a suitable monotone non-decreasing function independent of m.
Proof. If (v m , φ m ) ∈ V is a solution to (94), we can easily check that which gives Let It follows that for a suitable monotone non-decreasing function independent of m. Note that from Theorem 5.2. Suppose that Q 0 satisfies (90)-(92). We also assume that Then there exists at least one solution to (94).
Then ∧ m is a compact and convex subset of V m . It is also clear that T n maps ∧ m into itself. To prove the existence of solution, we apply the Brouwer fixed point theorem to the restriction of T m to ∧ m . Therefore it remains to check that T m is continuous.
For the continuity of T m , we proceed as follows. Let (v 1 , φ 1 ) = T m (U 1 , Φ 1 ) and . Then from (94) can check that we (w, ψ) satisfies Note that

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Multiplying (104) 1 and (104) 2 by w and A γ ψ respectively and using (105), we derive that which gives and the continuity of the mapping T m follows. Note that M 2 (·, ·) denotes a suitable monotone non-decreasing function independent of m. It follows that there exists a fixed point (v m , φ m ) of T m in ∧ m . Therefore we can extract a subsequence (still) denoted (v m , φ m ) that converges to (v * , φ * ) strongly in V. Using the same argument as in [11], we can prove that (v * , φ * ) is a weak solution to (93).

5.2.
Some a priori estimates on (v * , φ * ). Hereafter, we assume that f γ satisfies the additional condition where κ 0 > 0 is a fixed constant. We will derive some explicit a priori estimates in the V−norm and under some additional assumptions, we prove the uniqueness of solutions. In particular, we assume that > 0 is larger enough such that Theorem 5.3. Suppose that Q 0 satisfies (90)-(92). We also assume that (108)-(109) are satisfied. Then any solution (v * , φ * ) to (93) satisfies the following estimate: Moreover if then the solution to (93) is unique.

Asymptotic behavior.
Hereafter, we assume that satisfies for some fixed constant L 1 > 0.