A CAHN-HILLIARD-NAVIER-STOKES MODEL WITH DELAYS

. In this article, we study a coupled Cahn-Hilliard-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard 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, [9]. 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, [2,9,8,10]. In the isothermal compressible case, the existence of a global weak solution is proved in [7]. 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, [9]. The associated initial and boundary value problem was studied in [9] 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 [9] in terms of some model parameters. The dynamic of simple singlephase fluids has been widely investigated although some important issues remain unresolved, [15]. In the case of binary fluids, the analysis is even more complicate and the mathematical study is still at its infancy as noted in [9]. As noted in [8], 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. [13] for experimental snapshots). This model has to take 2664 T. TACHIM MEDJO 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. [11]). 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.
In [4,5,6], 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 [4] and the asymptotic behavior of the solutions is studied in [5]. The existence of attractors for the 2D NS equations with delays is proved in [6]. In [3], 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.
In this article, we study an CH-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 Cahn-Hilliard systems makes the analysis more involved.
The article is divided as follows. In the next section, we introduce the CH-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 [4,5,6]. In the fourth section, we study the asymptotic behavior of the weak solutions when the delay term satisfies some hypothesis used in [14]. The stability of the stationary solutions is analyzed in the fifth section.

2.
The CH-NS model and its mathematical setting.

Governing equations.
In this article, we study a 2D non-autonomous Cahn-Hilliard-Navier-Stokes system. 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 (2.1), 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, > 0 and K > 0 correspond to the kinematic viscosity of the fluid, the mobility constant and the capillarity (stress) coefficient respectively. Here , 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 fluids, [8].
A typical example of potential F is that of logarithmic type (see [8]). However, this potential is often replaced by a polynomial approximation of the type F (r) = γ 1 r 4 − γ 2 r 2 , γ 1 , γ 2 being positive constants. As noted in [8], (2.1) 1 can be replaced by The stress tensor ∇φ ⊗ ∇φ is considered the main contribution modeling capillary forces due to surface tension at the interface between the two phases of the fluid. Regarding the boundary conditions for these models, as in [8] we assume that the boundary conditions for φ are the natural no-flux condition where ∂M is the boundary of M and η is the outward normal to ∂M. These conditions ensure the mass conservation. In fact, from ∂ η µ = 0 on ∂M × (0, ∞), we have the conservation of the following quantity where |M| stands for the Lebesgue measure of M. More precisely, we have Concerning the boundary condition for v, we assume the Dirichlet (no-slip) boundary condition v = 0, on ∂M × (0, ∞).
Therefore we assume that there is no relative motion at the fluid-solid interface. 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. We assume that the function τ (t) is differentiable and there exists constants τ * and r > 0 such that We also assume that f ∈ C 2 ( ) satisfies where c f is some positive constant and k ∈ [1, +∞) is fixed. It follows from (2.11) 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 is the Leray-Helmholtz 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.
Hereafter, we set Then we introduce the linear nonnegative unbounded operator on L 2 (M) and we endow D(A N ) with the norm |A N · | L 2 + | · | L 2 , which is equivalent to the H 2 −norm. Also we define the linear positive unbounded operator on the Hilbert space L 2 0 (M) of the L 2 − functions with null mean is equivalent to the H s −norm. Moreover, we set H −s (M) = (H s (M)) * , whenever s < 0.
We will denote by λ 1 > 0 a positive constant such that 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( respectively. More precisely, we set We recall that B 0 , B 1 and R 0 satisfy the following estimates (2.20) We recall that (due to the mass conservation) we have Thus, up to a shift of the order parameter field, we can always assume that the mean of φ is zero at the initial time and, therefore it will remain zero for all positive times. Hereafter, we assume that We set The space Y is a complete metric space with respect to the norm

T. TACHIM MEDJO
We define the Hilbert space V by endowed with the scalar products whose associated norm is Throughout this article, we shall denote by c i , K i , K several positive constants that depend on the data (v 0 , φ 0 ) and Q. We will also denote by c a generic positive constant that depends on the domain M. To simplify the notations, we set (without loss of generality) α = K = 1.
In the case when the delay r is zero, the weak formulation of (2.27) and (2.8) was proposed and studied in [8], and the existence and uniqueness of solution was proved. In this article, we study the pullback asymptotic behavior of solutions (2.27) and (2.8).
We look for (v m , φ m ) ∈ Y m solution to the ordinary differential equations dv m dt , it follows from the theory of ordinary differential equation that this equation has a solution (v m , φ m ), (see also Theorem A1 of [4]). 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 is bounded from below). We can check that (see [8] 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 [14,4,5,6,3], we set and we assume that the inverse G −1 of G is well-defined on [0, +∞). Then we have the following result Proof. By taking the scalar product in H 1 of (3.1) 1 with v m , then taking the scalar product in L 2 (M) of (3.1) 3 with µ m , we derive that (see [9] for the details) Therefore we have by the Bihari inequality (see [14]) we can also check as in [8] that where Q 1 is a monotone non-decreasing function independent on time, the initial condition and m. It follows from (3.11) and (3. Using (3.9), (3.11) and (2.18)-(2.20), we can check that Therefore, we can take a subsequence (still) denoted by (v m , φ m ) such that Thus we have by the compactness theorem (see [15]) that (3.14) Using (3.13)-(3.14) and standard methods as in [15], we can pass to the limit in (3.1) as m → ∞, and we obtain that (v, φ) is a weak solution to (2.27). Let us recall that the passage to the limit in the delay force is obtained as in [14,4].
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 [14]. Let (3.15) We recall that (w, ψ) satisfies (3.16) 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 Multiplying (3.16) 1 by w, (3.16) 3 and (3.16) 2 respectively by A Nμ + ξA N ψ (with ξ > 0 sufficiently small to be selected in the sequel) and A N ψ, respectively, we derive as in [12,9] that where c = c M is a constant that depends only on M and (3.20) It follows that (with ξ small enough such that 1 − cξ > 0) 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.
As a corollary, we have Corollary 1. We suppose that there exist constants a f > 0 and b f ≥ 0 such that 3.1. Strong solution. In this part we discuss the existence and uniqueness of the strong solution to (2.27).
Then there exists a unique strong solution (v, φ) to (2.27) such that Now taking the inner product in H 1 of (3.1) 1 with 2A 0 v m , the inner product in L 2 (M) of (3.1) 2 and (3.1) 3 with 2 A 2 N φ m and 2A 2 Nμ m + 2ζA 3 N φ m respectively (where ζ > 0 is a small parameter to be chosen later). By adding the resulting equalities gives (see [12,9] for the details) where (3.26) Let us set (3.28) It follows from the Gronwall lemma that Therefore, we can take a subsequence (still) denoted (v m , φ m ) such that

(3.31)
Thus we have by the compactness theorem (see [15])) that (v m , φ m ) → strongly in L 2 (0, T ; V). 3.1.1. Continuity in V with respect to the initial data.
Theorem 3.3. Let (v i , φ i ), i = 1, 2 be two strong solutions corresponding to the initial conditions (v 0 i , φ 0 i ). Let the initial data on the time interval (−r, 0) be denoted Φ 1 , Φ 2 respectively. Then we have where L( 0 ) and Ψ are defined below.
Proof. Let We set w 1 = (v 1 , φ 1 )(t 2 ). Since Q(t, (v, φ)) satisfies a local Lipschitz condition, for any positive constant 0 > 0, there exists L( 0 ) > 0 such that where Ω 1 (t 2 , 0 ) is defined in (3.18). We can assume without loss of generality that . Then (w, ψ) satisfies (3.16). We multiply (3.16) 1 by A 0 w and (5.41) 2 by A 2 N ψ and (3.16) 3 by A Nμ + ξA N ψ (with ξ small enough to be selected later). Adding the resulting equations, we derive that and for a suitable monotone non-decreasing function independent of time Q 2 , (see [9] for details), 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:

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T. TACHIM MEDJO Corollary 2. We 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 (5.4), 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 CH-NS with delay (2.27). In this section we assume that there exist two constants a f > 0 and b f ≥ 0 such that the forcing term Q satisfies Then we have the following asymptotic behavior of weak solutions.
ρ > 0 is a positive number such that and hereafter M 0 denotes a suitable monotone non-decreasing function independent of time.
Proof. We take the scalar product in H 1 of (2.27) 1 with v and the scalar product in L 2 (M) of (2.27) 3 with 2ξφ, ( ξ > 0) and adding the resulting relationships to derive as in [9] that dE dt + κE(t) = ∧ 1 (t), (4.58) where 59) and  for any y ∈ , where c f , c * , c f and c f are positive, sufficiently large constants that depend only on f. From (4.61), we derive 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 (4.59)-(4.64) that which gives where θ > 0 is a positive number such that Note that if (4.54) is satisfied, we can find θ > 0 small enough such that (4.69) holds.

5.
Stability of stationary solutions. Hereafter, we study the stability of stationary solutions to (2.27). We first prove the existence of stationary solution to (2.27) 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 (2.27) converge exponentially to this unique stationary solution. In this section, we assume that the delay term is given by where 3) for some fixed constant L 1 > 0.
A stationary solution to (2.27) is a (v * , φ * ) such that 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 N 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 (5.5).
Lemma 5.1. We assume that Q 0 satisfies (5.2). Then any solution (v m , φ m ) to (5.5) satisfies the estimate where κ 1 is given by (5.10) 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 (5.5), we can easily check that which gives

8) and
ν v m It follows that which gives for a suitable monotone non-decreasing function independent of m. Note that from (2.12), |A Theorem 5.2. Suppose that Q 0 satisfies (5.2)-(5.3). We also assume that Then there exists at least one solution to (5.5).
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 (5.16) Multiplying (5.15) 1 and (5.15) 2 by w and A N ψ respectively and using (5.16), we derive that (5.18) 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 [9], we can prove that (v * , φ * ) is a weak solution to (5.4).

5.2.
Some a priori estimates on (v * , φ * ). Hereafter, we assume that f satisfies the additional condition where κ 0 > 0 is a fixed constant. We 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. We assume that (5.20) is satisfied. a Then any solution (v * , φ * ) to (5.4) satisfies the following estimate: which gives (assuming (5.20)) where and (w, ψ) V = 0 assuming (5.22), where κ 1 = min(ν, 2 ), and the theorem is proved.

(5.43)
If we choose θ such that Conclusion. In this article, we study a CH-NS model with delays in a two-dimensional domain. 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. There are still some interesting topics related to the CH-NS with delays. For instance, one can study the existence of a pullback attractor. We can also consider the case when the delay appears not only in the forcing terms, but also in the convective and the diffusion terms.