ON THE CONVERGENCE OF A STOCHASTIC 3D GLOBALLY MODIFIED TWO-PHASE FLOW MODEL

. We study in this article a stochastic 3D globally modiﬁed Allen-Cahn-Navier-Stokes model in a bounded domain. We prove the existence and uniqueness of a strong solutions. The proof relies on a Galerkin approximation, as well as some compactness results. Furthermore, we discuss the relation between the stochastic 3D globally modiﬁed Allen-Cahn-Navier-Stokes equations and the stochastic 3D Allen-Cahn-Navier-Stokes equations, by proving a convergence theorem. More precisely, as a parameter N tends to inﬁnity, a subsequence of solutions of the stochastic 3D globally modiﬁed Allen-Cahn- Navier-Stokes equations converges to a weak martingale solution of the stochastic 3D Allen-Cahn-Navier-Stokes equations.


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, [18]. For instance, this approach is used in [2] to describe cavitation phenomena in a flowing liquid. The model consists of the NS equation coupled with the phasefield system, [6,18,17,19]. In the isothermal compressible case, the existence of a global weak solution is proved in [14]. 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, [18]. The associated initial and boundary value problem was studied in [18] 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 [18] in terms of some model parameters. The dynamic of simple single-phase fluids has been widely investigated although some important issues remain unresolved, [37]. In the case of binary fluids, the analysis is even more complicate and the mathematical studied is still at it Let us recall that stochastic partial differential equations (SPDE) are sometimes used to model physical systems subjected to the influence of internal, external or environmental noises. As noted in [5,4], SPDE can also be used to describe systems that are too complex to be described deterministically, e.g., a flow of a chemical substance in a river subjected to wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, etc. With the development of the theory of stochastic processes, systems such as the Navier-Stokes perturbed by noises have been widely investigated with the goal to better understand the complex phenomena of turbulent flow. The mathematical theory of the stochastic Navier-Stokes equation is very rich, covering a broad area of deep results on existence of solutions, dynamical system feature, ergodicity, and many more. The presence of noise in a model can lead to new and important phenomena. For instance, contrary to the deterministic case, it is known that the 2D Navier-Stokes system driven with a sufficiently degenerate noise has a unique invariant measure and hence exhibits ergodic behavior in the sense that the time average of a solution is equal to the average over all possible initial data, [5].
The aim of this article is to investigate the stochastic version of the GMACNSE studied in [35]. The model include an abstract and general form of random external forces depending eventually on the velocity v of the fluid and the phase function φ. We prove the existence and uniqueness of a strong solution in a three dimensional bounded domain. Here the word "strong" means "strong" both in the sense of the theory of partial differential equations and the theory of stochastic analysis. The proof of the existence relies on the Galerkin approximation, the local monotonicity of the coefficients and some compactness results. Moreover we investigate the asymptotic behavior of the unique solution when the parameter N tends to infinity. This gives the existence of a weak martingale solution for the stochastic 3D AC-NSE.
In [27], the authors considered the Kolmogorov equation associated with the stochastic Navier-Stokes equations in 3D, they proved the existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u m of the Galerkin approximations of the velocity u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. In [16], a 2-dimensional Navier-Stokes equation perturbed by a sufficiently distributed white noise is considered. The authors proved the uniqueness of the invariant measures, strong law of large numbers, and convergence to equilibrium.
The article is organized as follows. In the next section we present the stochastic GMACNSE model and its mathematical setting. The existence and uniqueness a solution is given in Section 3. The proof of the existence relies on a Galerkin approximation as well as some a priori estimates. The asymptotic behavior of the solution is investigated in Section 4. We investigate the convergence of the solution when the parameter N tends to infinity.

2.
A stochastic GMACNSE model and its mathematical setting.
2.1. Governing equations. In this article, we consider a stochastic version of the GMACNSE a three-dimensional domain. We assume that the domain M of the fluid is a bounded domain in 3 . We first recall the following stochastic AC-NSE . is a sequence of independent one dimensional standard Brownian motions on some complete filtration probability space (Ω, F, P, (F t ) t∈(0,T ) ). If (e k ) k≥1 is an orthonormal basis of l 2 , we may formally define W by taking W = k W k e k . As such W is a cylindrical Brownian motion evolving over l 2 . We recall that l 2 is the Hilbert space consisting of all sequences of square summable real numbers. We define the aux- α k e k . Note that the embedding of l 2 ⊂ U 0 is Hilbert-Schmidt. Moreover, using standard martingale arguments with the fact that each W k is almost surely continuous (see [28]), we have that for almost every ω ∈ Ω, W (ω) ∈ C([0, T ]; U 0 ). See Section 3 for the precise assumptions on the coefficients g = (g 1 , g 2 ) and {σ 1 k ,σ 2 k ; k = 1, ..., ∞}. In (1), the unknown functions are the velocity v = (v 1 , v 2 , v 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., Here, the constants ν 1 > 0, ν 3 > 0 and K > 0 correspond to the kinematic viscosity of the fluid, the mobility constant and the capillarity (stress) coefficient respectively. Here ν 2 , α > 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 [18] we assume that ν 2 ≤ α.
We endow (1) with the boundary condition where ∂M is the boundary of M and η is its outward normal. The initial condition is given by Now, we define the function F N : for some (fixed) N ∈ + .
The following lemma gives some important properties of the map F N (see [7,31] for the proof) 2) Now we consider the following stochastic GMACNSE in M × (0, +∞), where v and (v, φ) U are some norms defined below. As noted in [7] in the case of the GMNSE, the GMACNSE are indeed globally modified. The factors F N ( v ) and F N ( (v, φ) U ) depend respectively on the norms v and (v, φ) U . They prevent large values of v and (v, φ) U dominating the dynamics. Just like the GMNSE, the GMACNSE violates the basic laws of mechanics, but mathematically the model is well defined, [35].

2.2.
Mathematical setting. We first recall from [18] (see also [34,33,36]) a weak formulation of (1)- (5). 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 2 ( ) satisfies where c f is some positive constant.
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 3 : div u = 0 in M}. We denote by H 1 and V 1 the closure of V 1 in (L 2 (M)) 3 and (H 1 0 (M)) 3 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-Helmotz projector in L 2 (M) onto H 1 . Then, A 0 is a self-adjoint 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 also set f γ (x) = f (x) − α −1 ν 2 γx and observe that f γ still satisfies (12) with γ in place of 2γ since ν 2 ≤ α. Also its primitive F γ (x) = endowed with the scalar products whose associated norms are respectively 15) We will also use the following notation: It follows that Hereafter, we set Hereafter, if X is a Banach space, we will denote by X * the dual space of X. To simplify the notations, the duality paring between X and X * will be denoted ·, · and the norm in X * will be denoted · * . 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) = Pµ∇φ.

THEODORE TACHIM MEDJO
We note that To further simplify the presentation, we define the operators A : U → U * , B N : where ·, · denotes the duality pairing between U and U * or between V i and V * i , i = 1, 2. Note that We also set For simplicity we will also set We also set ) then we can rewrite (10) as: Here G = (g 1 , g 2 ) is a mapping from U (resp. H) into U (resp. H) and σ k (.), k ≥ 1 is a sequence of mappings from U (resp. H ) into U (resp. H).
Consider the following hypothesis.
Hereafter, we will denote by c a generic positive constant that depends on the domain M. To simplify the notations, we set (without loss of generality) The next result will be useful in our study of the stochastic 3D GMACNSE.
There exists a constant c 3 > 0 independent of u 1 and u 2 such that Proof. The proof of (34) 1 is given in [7,31]. For the reader convenience, we sketch it here. We first note that
The next lemma shows that B 0 for u 1 , u 2 ∈ U with u 1 U , u 2 U ≤ r.
. With these notations, it is clear that (31) can be written as Hereafter, we assume that the function f γ satisfies the additional conditions: where α 0 > 0 is a positive constant independent of φ 1 , φ 2 . The next lemma shows some strong monotonicity of the operator G.
where the constant c 0 > 0 is independent of u 1 , u 2 .
3. Existence and uniqueness of solutions. In this section, we present one of the main results of the paper.
• I) Uniqueness Let X andX two solutions to (56) starting from the same initial value u 0 . For any ξ > 0, define the stopping time Set Θ(t) = X(t) −X(t). Then by Itô s formula, we have From Lemma 4, there exists a constant c 0 > 0 such that Combining the estimates (66) with (65), we get By Gronwall's inequality, we get for any t ∈ [0, T ] E|Θ(t ∧ τ ξ )| 2 H = 0. And the uniqueness follows by letting ξ → ∞ and Fatou's lemma.
• II) Existence We will use the Galerkin approximation combined with the strong monotonicity of the stochastic 3D GMACNSE. We Assume that u 0 ∈ L 6 (Ω, F 0 ; U). The general case E u 0 2 U < ∞, can also be treated similarly.
Let {e i : i ≥ 1} ⊂ D(A) be a fixed orthonormal basis of H consisting of eigenvectors of A, so that it is also orthogonal in U. Denote π n the orthogonal projection from H onto the finite dimensional space H n := span{e 1 , e 2 , ..., e n }: Thus π n is also the orthogonal projection onto H n in U.
Moreover by Lemma 3, and (32), the maps u ∈ H n → π n G(u) ∈ H n and u ∈ H n → π n σ are respectively locally Lipschitz continuous and Lipschitz continuous. Then by the theory of stochastic differential equations (see [24,32]), there exists a We now prove some a priori estimates of the approximated solution.
Lemma 5. There exists a constant C independent of n such that 2) E sup Proof. The proof is similar to that of Lemma 7 given in Section 4.

Lemma 6.
There exists a constant C independent of n such that 2) E sup 3) sup Proof. 1) By Itô's formula, we have Note that for u n = (v n , φ n ), we have We also have − Au n , u n U = −|Au n | 2 H .
From [7,31], we have We also have Using the properties (11) of f γ , we can check that where hereafter q 0 ≥ 1 denotes some integer. The properties of G give The estimates (77))-(82) and (75) yield for some positive integer q 0 > 1. Taking expectation, we get Hence by Gronwall's inequality and Lemma 5, we have for any T > 0, This proves 1).
Taking the supremum over [0, t], we obtain By Burkholder's inequality, we have u n (s) 8 U ((π n σ k (u n (s)), u n (s))) 2 ds Taking the expectation in (88) and using (89), we obtain Applying Gronwall's inequality, we obtain This ends the proof of 3).

4) We have
We also have which gives We also have Similarly, we have For the operator E, we note that for some positive integer q 0 ≥ 1. From (32), we also have It follows that ) and This ends the proof of the lemma.
To complete the proof of the theorem, we need to show that G(s) = G(ũ(s)) andσ k (s) = σ k (ũ(s)) − a.e. on Ω T .
The proof follows the same steps as in [30]. Fix an integer K. Take ϑ ∈ L 2 (Ω T , H K ) where H K is the linear span of e 1 , e 2 , ..., e K . By Itô's formula, 2e −r(s) G(u n (s)), u n (s) ds where Z n = Z 1 n + Z 2 n + Z 3 n with Set r (s) = c + 2c 0 . In view of (58) and (32) we see that Z 1 n ≤ 0. By the weak convergence, it is clear that Also Combining (103)-(108) after cancellations it turns out that As K is arbitrary, by approximation it is seen that (109) holds true for every ϑ ∈ L 2 (Ω T , D(A)). In particular, take ϑ(s) = u(s) in (109) to obtainσ k (s) = σ k (u(s)) for every k ≥ 1. where r λ (s) is defined as r(s) with ϑ replaced by ϑ λ . Dividing (110) by λ we obtain for λ > 0, and for λ < 0. By (58), we have Therefore by the dominated convergence theorem, we get Letting λ → 0 + in (111) and λ → 0 − in (112), we obtain Asθ is arbitrary, we conclude that G(s) = G(u(s)) a.e. on Ω T . Then 4. Convergence to martingale solutions of the stochastic 3D AC-NSE. Let µ 0 be a probability measure on U such that U U 6 dµ 0 (U ) < ∞. Let u 0 be an F 0 -random variable in U with distribution µ 0 . Let u N be the unique strong solution of the stochastic 3D GMACNSE. In this section, we are going to study the asymptotic behavior of u N when N→ ∞.

Some a priori estimates.
Lemma 7. We have the following a priori estimates on u N E sup E sup where the constants c 1 , c 2 , c 3 and c 4 are independent of N .
Proof. By Itô's formula, we get Note that for u N = (v N , φ N ) ∈ U, we also have (see (24), (32), (26) and (57)) and The properties of G also give It follows that Using the estimates (121)-(127), we arrive at Taking the supremum over [0, T ], we get Raising both sides to the power p 2 for p≥ 2, then taking expectations, we obtain with the Minskowski inequality and Fubini's theorem E sup

THEODORE TACHIM MEDJO
For the stochastic term, we use the Burkhölder-Davis-Gundy inequality Applying the above estimate to (130), we obtain Since Letting p = 4 and p = 2, we obtain the estimates (116) and (118). The estimate (128) implies Taking the supremum over [0, T ], raising both sides to the power 2 then taking expectation, we obtain with Minkowski's inequality and Fubini's theorem For the stochastic term, we have This together with (136) implies the estimate (119). The proof of Lemma 7 is complete.

4.2.
Estimates in fractional Sobolev spaces. We will apply the compactness result based on fractional Sobolev spaces in Lemma 12 (of the Appendix) with For this purpose, we will need the following estimates on fractional derivatives of u N .
where the constants k 1 , k 2 and k 3 are independent of N .

Compactness arguments for
With the estimates independent of N , we can establish the compactness of the family (u N , W ). For this purpose, we consider the following phase spaces: We then define the probability laws of u N and W respectively in the corresponding phase spaces: and This defines a family of probability measures µ N = µ N u × µ W on the phase space X . We now prove that this family is tight in N . More precisely: Lemma 9. Consider the measures µ N on X defined according to (156) and (157). Then the family {µ N } N is tight and therefore weakly compact over the phase space X .
Proof. We can use the same technic as in the proof of Lemma 4.1 in [12]. The main idea is to apply Lemma 12 (of the Appendix) and Chebychev's inequality to (138)-(140).
Strong convergence as N → ∞. Since the family of measures {µ N } associated with the family (u N , W ) is weakly compact on X , we deduce that µ N converges weakly to a probability µ on X up to a subsequence. We can apply the Skorokhod embedding theorem (see Theorem 2.4 in [28]) to deduce the strong convergence of a further subsequence, that is: There exists a probability space (Ω,F,P), and a subsequence N k of random vectors (ũ N k ,W N k with values in X such that (i) (ũ N k ,W N k have the same probability distributions as (u N k , W N k ).
(ii) (ũ N k ,W N k ) converges almost surely as N k → ∞, in the topology of X , to an element (ũ,W ) ∈ X , i.e.
(iv) All the statistical estimates on u N k are valid forũ N k , in particular, the estimates (116)-(119) hold. (v) Each pair (ũ N k ,W N k ) satisfies (56) as an equation in H, that is The following lemma proves thatũ N k ,ũ is weakly continuous with value in H
Combining the strong convergence (158), the estimate (118) and the Vitali convergence theorem, we getũ N k →ũ strongly in L 2 (Ω; L 2 (0, T ; H)), and, thus possibly extracting a new subsequence denoted in the same way to save notation, one has alsõ u N k →ũ for almost all ω, t with respect to the measure dP ⊗ dt.
Fix ϑ ∈ D(A). Using the weak convergence (162), we can pass to the limit in the linear term.
For the proof of (165), we introduce the abbreviations as in [7], Reasoning as in the proof of the convergence of the 3D globally modified Navier-Stokes equations studied in [13], the second term of this equality tends to 0, that is For the first term, we get We therefore infer from (118) that |σ(ũ N k | L2(l 2 ,H) is uniformly integrable for N k in L q (Ω × (0, T )) for any q ∈ [1, 4). With the Vitali convergence theorem, we deduce that for all such q ∈ [1, 4), σ(ũ N k ) → σ(ũ) in L q (Ω, L q (0, T ; L 2 (l 2 , H))).
In particular, we get the convergence in probability of σ(ũ N k ) in L 2 (0, T ; L 2 (l 2 , H)).
Thus along with the convergence (159), we apply Lemma 14 (of the Appendix) and deduce that t 0 σ(ũ N k )dW N k → t 0 σ(ũ)dW in probability in L 2 (0, T ; H).