APPROXIMATION OF THE TRAJECTORY ATTRACTOR FOR A 3D MODEL OF INCOMPRESSIBLE TWO-PHASE-FLOWS

. In this article we study the relations between the long-time dynamics of the 3D Allen-Cahn-LANS- α model and the exact 3D Allen-Cahn- Navier-Stokes system. Following the idea of [26], we prove that bounded set of solutions of the Allen-Cahn-LANS- α model converge to the trajectory at- tractor U 0 of the 3D Allen-Cahn-Navier-Stokes system as time goes to + ∞ and α approaches 0 + . In particular we show that the trajectory attractors U α of the 3D Allen-Cahn-LANS- α model converges to the trajectory attractor U 0 of the 3D Allen-Cahn-Navier-Stokes as α approaches 0 + . Let us mention that the strong nonlinearity that results from the coupling of the convective Allen- Cahn system and the LANS- α equations makes the analysis of the problem considered in this article more involved. ,


1.
Introduction. In this article we study relations between the long-time dynamics of the 3D Allen-Cahn-LANS-α (AC-LANS-α) model and the exact 3D Allen-Cahn-Navier-Stokes (AC-NS) system. The AC-LANS-α model is derived from the AC-NS system by substituting the Navier-Stokes system by the LANS-α equations. As the LANS-α model is to the Navier-Stokes system, the AC-LANS-α model can be considered as a regularized approximation of the AC-NS system, depending on a small positive parameter α > 0, where in some terms, the unknown velocity function v is replaced by a smoother function u which are related by the elliptic system v = u − α 2 ∆u. For α = 0, the model reduces to the exact AC-NS system.
Let us recall that because the uniqueness theorem for the global weak solutions (or the global existence of strong solutions) of the initial-value problem of the 3D AC-NS system is not proved yet, the known theory of global attractors of infinite dimensional dynamical systems is not applicable to the 3D AC-NS system. This situation is the same for the 3D Navier-Stokes system. Using regular approximation equations to study the classical 3D Navier-Stokes system has become an effective tool both from the numerical and the theoretical point of views. As noted in [26], it was demonstrated analytically and numerically in many works that the LANS-α model gives a good approximation in the study of many problems related to turbulence flows. In particular, it was found that the explicit steady analytical solution of the LANS-α model compare successfully with empirical and numerical experiment appear in Section 5, where we prove that the trajectory attractor U α converges to U 0 as α approaches 0 + in an appropriate topology.
2. Trajectory attractor of the 3D AC-NS system.
In (1), the unknown functions are the velocity u = (u 1 , u 2 , u 3 ) 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 1 > 0 and K > 0 correspond to the kinematic viscosity of the fluid and the capillarity (stress) coefficient respectively, ν 2 , > 0 are two physical parameters describing the interaction between the two phases. In particular, ν 2 is related with the thickness of the interface separating the two fluids. Hereafter, as in [17] we assume that ν 2 ≤ .
For the boundary condition, we assume that u(x, t) and φ(x, t) are periodic in x = (x 1 , x 2 , x 3 ) (with period 2πL) with zero mean, that is The initial condition is given by 2.2. Mathematical setting. As in [17,16,18] we assume that potential function f satisfies lim where c f is some positive constant. It follows from (5) that |f (r)| ≤ c f (1 + |r| 3 ), ∀r ∈ R.
Hereafter, 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 V 1 = {u : u is a vector valued trigonometrical polynomial defined in M, div u = 0, M udx = 0}. (8) 2232
We now define the operator A 0 by where P is the Leray-Helmotz projector in (L 2 (M)) 3 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, i.e., there exists a constant c 1 > 0, depending only on M, such that and so D(A 0 ) is a Hilbert space with the scalar product Hereafter, we set H = V, with the scalar product (u, v) H = ((·, ·)), and U = D(A 0 ), with the scalar product ((u, v)) U = (A 0 u, A 0 v). We also denote by | · | H and · U the associated norms defined on H and U respectively. Then, H and U are two real and separable Hilbert spaces such that U ⊂ H, with the injection being compact and dense. We will identify H with its topological dual H * , but considering U as a subspace of H * , where we identify v ∈ U with the element f v ∈ H * given by We will denote by · U * the norm of U * , by ·, · , the duality product between U * and U. Now, we define the operator A by Then, we have (see [7,6]) We note that owing to the properties of A, we define It is clear that ((·, ·)) A is a scalar product in U whose associated norm is equivalent to the usual norm · U . From now on, without loss of generality, we simply set ((u, v)) U = Au, v , ∀u, v ∈ U and u 2 U = Au, u . It then follows that where λ 1 > 0 is the first eigenvalue of the operator A.
We also note that We denote by H 2 and V 2 the closure of V 2 in L 2 (M) and H 1 (M) respectively. The scalar product in H 2 is denoted by (·, ·) L 2 and the associated norm by | · | L 2 . Moreover, the space V 2 is endowed with the scalar product We define the linear positive unbounded operator A 1 on H 2 by: Note that A −1 1 is a compact linear operator on H 2 and |A 1 · | L 2 is a norm on D(A 1 ) 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( Note that R 0 (µ, φ) = Pµ∇φ.
We can easily check that the operators B 0 , R 0 and B 1 satisfy: Now we define the Hilbert spaces Y and V by endowed with the scalar products whose associated norms are
, it follows from the well known Lions-Magenes lemma (see [24] denotes the space of weakly continuous functions from [0, T ] to Y. Therefore for every t ≥ 0, the value (u, φ)(t) makes sense in the space Y and in particular the initial condition (4) is meaningful.
The weak formulation of (24), (4) was proposed and studied in [17,16], and the existence and uniqueness of solution was proved in the two-dimensional case. In the three-dimensional case, the existence trajectory attractor is proved in [18] when the velocity u and the order parameter satisfy the Dirichlet's and Neumann boundary conditions respectively. In this section, we adapt the results of [18] in the case when u and φ satisfy a periodic boundary condition. [17,16,18], we can check that if (u, φ) satisfies (24) and (25), then (u, φ) ∈ C w (0, T ; Y). Therefore, the initial condition (4) holds weakly.

Remark 2. As in
For (u, φ) ∈ Y, we define the functional L(u, φ) by: where C e > 0 is a constant large enough to ensure that L(u, φ) is nonnegative. Note that from the property of F, we can find C e > 0 large enough such that Theorem 2.2. Let g ∈ H 1 and (u 0 , φ 0 ) ∈ Y. Then for every T > 0, there exists a weak solution (u, φ) to (24) from the space L ∞ (0, T ; Y) ∩ L 2 (0, T ; V) such that (u, φ)(0) = (u 0 , φ 0 ) and satisfies the energy inequality for all Λ ∈ C ∞ 0 (0, T ; R + ), where C 1 > 0 and ρ 1 > 0 depends on M f , ν 0 and K. Proof. The proof is given in details in [18] when u and φ satisfy respectively the Dirichlet's and Neumann boundary condition. When u and φ satisfy a periodic boundary condition, the proof is similar and we refer the reader to [18] for the details. Proposition 1. For any weak solution (u, φ) of (24), the following inequalities hold for all t ≥ 0 : where Q is a monotone non-decreasing function independent of t and the initial data, C 1 > 0 depends on the parameters of the problem such as M, f, ν 0 , ν 1 and K.

2.3.
Construction of the trajectory attractor of the 3D AC-NS system. We now construct the trajectory space for the 3D AC-NS system (24).
Definition 2.3. The trajectory space K + is the set of all Leray-Hopf weak solution (u, φ) of (24) in the space L 2 loc (R + ; V) ∩ L ∞ loc (R + ; Y) that satisfies the energy inequality (28).
Let us define the spaces F + loc , F + b and the topology Θ + loc . We set (33) In the space F + loc , we define the following local weak convergence topology: By definition, a sequence {(u n , φ n )} ⊂ F + loc converges to (u, φ) ∈ F + loc in the topology Θ + loc as n → ∞ if for each T > 0, the following limit relations hold: Remark 3. The space F + loc equipped with the topology Θ + loc is a Hausdorff Frechet-Uryhson topological vector space with countable base [26].
We consider the linear subspace is compact in the topology Θ + loc . Moreover, the corresponding topological subspace is not metrisable, [26,25].
We consider the translation semigroup We now construct the global attractor for the semigroup {T (h)}. This attractor will be referred to as the trajectory attractor for the AC-NS system (24). We have the following results.
where hereafter Q stands for a monotone, positive non-decreasing function that is independent of time and (u, φ), and R 1 denotes a positive constant depending on M, ν 0 , ν 1 , K and |g| L 2 .
Proof of Proposition 6. The proof of (ii) follows from Proposition 2 and Proposition 7. The proof of the inclusion K + ⊂ F + b follows from (i) by setting h = 0.
We consider a sequence (u n , φ n ) ∈ K + which converges as n → ∞ in Θ + loc to an element (u, φ) ∈ F + loc . Let us prove that (u, φ) ∈ K + . By definition of the topology Θ + loc , for every T > 0, we have (u n , φ n ) (u, φ) weakly star in L ∞ (0, T ; Y) and weakly in L 2 (0, T ; V), . Then passing to a subsequence, still denoted (u n , φ n ), we can assume that . We also note that (u n , φ n ) satisfies in the distribution sense. From the Aubin compactness theorem, we also have Passing to the limit gives It follows that is a weak solution to (24). It remains to prove that (u, φ) satisfies the energy inequality (28). We proceed as in [18]. We recall that (u n , φ n ) satisfies the energy inequality for all Λ ∈ C ∞ 0 (0, T ; R + ). From (48) and the Lebesgue dominant convergence theorem, we have Moreover, we have Hence we deduce that Therefore we can pass to limit in (51) thanks to (52)-(54) to prove that (u, φ) satisfies the energy inequality (28). The proof is finished.
We have defined the trajectory space K + of (24) on R + . We now extend the definition to R. The kernel K 0 of (24) is the set of all weak solution (u, φ)(t), t ∈ R bounded in the space that satisfies the following inequality for all Λ ∈ C ∞ 0 (R; R + ). The norm in F b is defined in a similar way as the norm in F + b replacing R + by R. The same definition also holds for F loc with the topology Θ loc , where the interval [0, T ] is replaced by [−T, T ]. We denote by Π + the restriction operator onto R + . This operator takes a function {φ(t), t ∈ R} to the function Let us now study the translation semigroup {T (h)} acting on the trajectory space K + . We first recall some definitions from [26,25].  The existence of the trajectory attractor for the AC-NS system follows from the following theorem.
Theorem 2.7. Let g ∈ H 1 , then the translation semigroup {T (h)} acting on K + has a trajectory attractor U 0 . The set U 0 is bounded in F + b and compact in Θ + loc .
the set K 0 is bounded in F b and compact in Θ loc .
Proof. This is clear from the above results, see [18].

Governing equations.
We recall that the domain M of the fluid is given by M = (0, L) 3 , where L > 0 is given. The state of the system is described by a pair (u, φ), where u = (u 1 , u 2 , u 3 ) is the velocity field of the fluid and φ is the order parameter. The system of equations for (u, φ) reads: Let us first note that (58) is obtained by coupling the well-known Allen-Cahn equations ∂φ ∂t with the 3D LANS-α system through convection and order parameter. Note that for α = 0, the model (58) reduces to the 3D AC-NS (1). Let us recall from [15] that the positive constant α represents the square of the spacial scale at which the fluid motion is filtered.
For the boundary condition, we assume that u(x, t) and φ(x, t) are periodic in x (with period 2πL) with zero means.
The initial condition is given by 3.2. Mathematical setting. Following the notations of the AC-NS system (1), we define the bilinear operatorB 0 by: Then we have (see [26]) We also note thatB We also have the following properties of the bilinear operatorB 0 , see [26] for details. The operatorB 0 maps V 1 × V 1 to V * 1 and the following inequalities hold.
where Q is a monotone nondecreasing function independent of time and the initial data and R 1 > 0 is a constant depending on g and some physical parameters of the problem.

3.4.
Existence of the trajectory attractors of the 3D AC-LANS-α model.
Corollary 1. The solution (w, φ) of (74), (60) satisfies where Q 1 is a monotone nondecreasing function independent of time and the initial data and R 1 > 0 is a constant depending on g and some physical parameters of the problem.
We consider the Banach space F + b defined by Proposition 5. Let g ∈ H 1 , for any solution (u, φ) of (66), the corresponding couple (w, φ) satisfies Proof. It follows from (75).

3D LANS-α SYSTEM WITH PHASE TRANSITION 2243
Remark 5. Note that the function Q 1 and the constant R 1 are independent of α.
Let us now construct the trajectory attractor for the 3D AC-LANS-α model (74). The trajectory space K + α for the system (74) is defined as follows. Definition 3.3. The trajectory space K + α is the union of all couple (w(t), φ(t)), where (u, φ) is a solution to (66) with an arbitrary initial data (u 0 , φ 0 ) ∈ V 1 × V 2 .
From Theorem 14, it follows that the trajectory space K + α = ∅. Moreover, Proposition 16 implies that K + α ⊂ F + b , for all α > 0. We also consider the topological space Θ + loc introduced in Section 2. Let us recall that F + b ⊂ Θ + loc . We consider the topology Θ + loc on K + α . We prove that the space K + α is closed in Θ + loc . Proposition 6. The space K + α is closed in Θ + loc . Proof. Similar to that of Proposition 8.

The translation semigroup {T (h)} acts on K +
α by the formula: It follows from the definition that T (h)K + α ⊂ K + α for all h ≥ 0. Theorem 3.4. If g ∈ H 1 , then the translation semigroup {T (h)} acting on K + α has a trajectory attractor U α . Moreover, the set U α is bounded in F + b and compact in Θ + loc . Furthermore, we have where K α is the kernel of the system (66).
Proof. We have T (t)K + α ⊂ K + α for all t ≥ 0. The set } is an absorbing set for K + α (see Proposition 16). The ball P 1 is compact in Θ + loc and bounded in F + b . Moreover, P 1 does not depend on α since the function Q 1 and the constant R 1 are independent of α. It follows from general result given in [26,25] that there exists a trajectory attractor U α ⊂ K + α such that U α is bounded in F + b and compact in Θ + loc .

4.
Convergence of the solutions of the 3D AC-LANS-α model. We formulate and prove the main result of this section concerning the behavior of the solution of the 3D AC-LANS-α model when α approaches 0 + . Theorem 4.1. Let a sequence (w n , φ n ) ⊂ K + αn be given such that
For the proof, we will need the following result.
Lemma 4.2. Let two sequences (u n , φ n )(t) ∈ F + b and {α n } ⊂ (0, 1] be given such that α n → 0 + as n → ∞. We denote by w n = (1 + α 2 A 0 ) 1/2 u n for n ∈ N. We assume that the sequence (w n , φ n ) is bounded in The proof is similar to that of [26] (see also [25,14]). For the reader convenience, we give the details.
We conclude that u n → w strongly in L 2 (0, T ; H 1 ), i.e., w = u and the lemma is proved.
Proof of Theorem 22. We proceed as in [26]. Note that the model considered in this article has a nonlinearity stronger that the one in the 3D LANS-α model studied in [26]. Since The couple (u n , φ n ) is a solution of the original problem (24). The estimates (92),(77) imply that We now prove that (w, φ) is a weak solution of the 3D AC-NS system (24) on the interval [0, T ]. The couple (w n , φ n ) satisfies in the sense of distribution. Here v n = (1 + α 2 n A 0 )u n . From the assumption (83), we have (w n , φ n ) → (w, φ) weakly in L 2 (0, T ; V), (w n , φ n ) → (w, φ) weakly star in L ∞ (0, T ; Y), It follows from (95) that (A 0 w n , A 1 φ n ) → (A 0 w, A 1 φ) weakly in L 2 (0, T ; V), and hence in the topology of D (0, T ; D(A 0 ) * × V * 2 ) as well. From Lemma 23, (95) and the Aubin compactness Theorem, we have (u n , φ n ) → (u, φ) strongly in L 2 (0, T ; Y). Arguing as in the proof of Proposition 8, we can also check that and therefore in D (0, T ; D(A 0 ) * ).
Theorem 5.1. (i) The trajectory attractor U α of the system (66) converges in the topology Θ + loc as α → 0 + to the trajectory attractor U 0 of the AC-NS system (100): , t ≥ 0}, 0 < α ≤ 1, be the bounded set of solutions of (104) that satisfy Then the set of shifted solutions {T (h)B α } converges to the trajectory attractors U 0 of the AC-NS system (100) in the topology Θ + loc as h → +∞ and α → 0 + : Proof. We proceed as in [26]. It is enough if we prove (ii), which implies (i) by taking B α = U α = T (h)U α , h ≥ 0. We assume that (108) fails to hold. Then there is a neighborhood Θ(U 0 ) of U 0 in Θ + loc and two sequences α n → 0 + , h n → +∞ as n → +∞ such that T (h n )B αn is not a subset of Θ(U 0 ). Hence, there exist a couple (w αn , φ αn ) ∈ B αn such that X αn ≡ (w αn , φ αn ) ∈ B αn and the functions is a solution to (104) on the interval (−h n , +∞) with α = α n since (w αn (t + h n ), φ αn (t + h n )) is a solution for t + h n ≥ 0 and the system (104) is autonomous. Moreover, it follows from (107) that This inequality implies that the sequence {(U αn , Φ αn ), n ∈ N} is weakly compact in the space for every T > 0, if we consider α n with the indices n such that h n ≥ T. Therefore, for every fixed T > 0, we can choose a subsequence {α n } ⊂ {α n } such that {W αn = (U αn , Φ αn ), n ∈ N} converges in Θ −T,T . Thus using the well known Cantor diagonal procedure, we can construct a couple of function W (t) = {(U (t), Φ(t)), t ∈ R} and a subsequence {α n } ⊂ {α n } such that weakly in Θ −T,T , as n → +∞ for every T > 0. From (110), we obtain that In particular, we have We can now apply Theorem 22, where we assume that all the functions U α n , Φ α n are defined in [−T, +∞) instead of [0, +∞). From (110) and (114), we conclude that W (t) = (U (t), Φ(t)) is a weak solution of the 3D AC-NS system for all t ∈ R and W (t) = (U (t), Φ(t)) satisfies the energy inequality. Therefore W (t) = (U (t), Φ(t)) ∈ K 0 , where K 0 is the kernel of the system (100). Since Π + K 0 = U 0 and W (t) ∈ K 0 , we have Π + W ∈ U 0 . On the other hand from (110), we have In particular, for n large enough, we have This contradicts (109) and therefore (108) is proved.
6. Convergence to equilibria. In this section, we consider the 3D α-regularized system (3.4), (3.10) in the absence of external forces, i.e., g = 0. We aim to show that the analysis in [17,Section 5] can be extended to the regularized system for each α > 0. In particular, we can show that each trajectory of such a system converges to a single equilibrium provided that f is real analytic. We first recall some standard implications from Section 3 (see Definition 13 and Theorem 14) and [23].
Proposition 7. Let f ∈ C 2 (R) satisfy assumption (2.6) and α > 0. Problem It is easy to see that the semigroup S α (t) has a (strict) global Lyapunov functional defined by the free energy, namely, In particular, we have, for all t > 0, As a result of (118) and [23, Propositions 3.6, 3.7] we can immediately infer that (S α (t), Z) is a gradient system with precompact trajectories (see also [17]).
Remark 7. Let us assume that instead of (2.6), f obeys the following assumptions: lim inf |s|→+∞ f (s) > 0, f (1) ≥ 0 and f (−1) ≤ 0, where the order parameter φ is normalized in such a way that the two pure phases of the fluid are −1 and +1, respectively. Then, arguing as in the proof of [17, Theorem