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A stochastic Allen-Cahn-Navier-Stokes system with singular potential

We investigate a stochastic version of the Allen-Cahn-Navier-Stokes system in a smooth two- or three-dimensional domain with random initial data. The system consists of a Navier-Stokes equation coupled with a convective Allen-Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. The problem is endowed with a no-slip boundary condition for the (volume averaged) velocity field, as well as a homogeneous Neumann condition for the order parameter. We first prove the existence of analytically weak martingale solutions in two and three spatial dimensions. Then, in two dimensions, we also estabilish pathwise uniqueness and the existence of a unique probabilistically-strong solution. Eventually, by exploiting a suitable generalisation of the classical De Rham theorem to stochastic processes, existence and uniqueness of a pressure is also shown.


Introduction
Modeling the behavior of immiscible (or partially miscible) binary fluids is a very active area of research because of its importance, for instance, in Biology and Materials Science. A well-known and effective approach is the so-called diffuse interface method (see, e.g., [6]). This approach is based on the introduction of an order parameter (or phase field) which accounts for the presence of the fluid components in a sufficiently smooth way, that is, there is no sharp interface separating them but a sufficiently thin region where there is some mixing. More precisely, denoting by ϕ the relative difference between the (rescaled) concentrations of the two components, the regions {ϕ = 1} and {ϕ = −1} represent the pure phases. However, they are separated by diffuse interfaces where ϕ can take any intermediate value, i.e. ϕ ∈ (−1, 1). The interaction between the two components is a competition between the mixing entropy and demixing effects and can be represented by a potential energy density of the form for some 0 < θ < θ 0 . This is known as the Flory-Huggins potential (see [37,55]). Letting O be a (sufficiently) smooth domain of R d , d = 2, 3, the Helmholtz free energy associated with the order parameter is then given by where the first term accounts for the surface energy separating the phases. Here ε > 0 is related to the thickness of the diffuse interface. Then, the functional derivative of E(ϕ) is called the chemical potential and usually denoted by µ, that is, µ = −ε 2 ∆ϕ + F ′ (ϕ).
Here we have assumed that the mobility is constant and equal to the unity. We also recall that, due to the singular behavior of the mixing entropy, the Flory-Huggins potential (1.1) is often approximated with a regular potential like This choice simplifies the mathematical treatment. However, when the total mass of ϕ is conserved (e.g. in (1.5)) one cannot ensure that ϕ takes its values in the physical range [−1, 1]. Here we choose to keep the physically relevant potential also in view of extending our analysis to conserved Allen-Cahn equations where in (1.6) or in (1.9) µ is replaced by µ −μ,μ being the spatial average of µ (see [68], see also [42] and references therein).
When we deal with a two-component fluid mixture, the equation for the phase variable is coupled with an equation for a suitably averaged velocity u of the fluid mixture itself. A well known choice is the Navier-Stokes system subject to a capillary force, known as Korteweg force, which can be represented as µ∇ϕ. More precisely, in the case of an incompressible mixture and taking ε = 1, constant density equal to the unity and constant viscosity ν > 0, we have coupled with ∂ t ϕ + u · ∇ϕ = ∆µ (1.5) or ∂ t ϕ + u · ∇ϕ = −µ (1. 6) in (0, T ) × O, for some given T > 0. Here u represents the volume averaged velocity and π stands for the pressure. System (1.3)-(1.4) coupled with (1.5) is known as Cahn-Hilliard-Navier-Stokes system, if (1.5) is replaced by (1.6) then the system is known as Allen-Cahn-Navier-Stokes system. We recall that the standard boundary conditions are no-slip for u and no-flux for (1.5) or (1.6).
Starting from the pioneering contribution [53], two-phase flow models have then been developed in several works. In particular, we refer to [48] for the Cahn-Hilliard-Navier-Stokes system and to [14] for the Allen-Cahn-Navier-Stokes system (see [4,40,63] for more refined models with unmatched densities and [51,52] for general thermodynamic derivations). The corresponding mathematical analysis of such models has also experienced a remarkable development in the last decades. Concerning the Cahn-Hilliard-Navier-Stokes system with matched densities see [1,43] and references therein (see also [2,3,12,41,44,45] for more general models). Regarding the Allen-Cahn-Navier-Stokes system, we refer to [38,39] for the matched case (see also references therein) and to [34,35,42,54,57,58,60,61] for more refined models. The deterministic description fails in rendering possible unpredictable oscillations at the microscopic level. These include, for example, the environmental noise due to temperature and configurational effects. The most natural way to take into account such factors was first proposed in [21] where a stochastic version of the Cahn-Hilliard equation was introduced (see also [15,16] for nucleation effects). That equation has been analyzed under various assssumptions in a number of contributions (see, for instance, [22,24,25,31,47] and, more recently, [70,72,74]). We also refer to [64,71,73] for related stochastic optimal control problems. The stochastic Allen-Cahn equation has been investigated in the framework of regular potentials (see, for instance, [13,49,50,65] for examples of well-posedness analysis, see also [18,69] for numerical schemes and simulations). The singular potential has been analyzed in [10] (see also [9] for the double obstacle potential), while in [11] the separation property from the pure phases has been established.
Here we analyze a stochastic version of the Allen-Cahn-Navier-Stokes system characterized by two independent sources of randomness, the former acting on the fluid velocity and the latter acting on the order parameter dynamic to incorporate thermal fluctuations. More precisely, on account of (1.3)- (1.4) and (1.6), taking ν = 1 for the sake of simplicity, we consider the following system of stochastic partial differential equations du + [−∆u + (u · ∇)u + ∇π − µ∇ϕ] dt = G 1 (u) dW 1 in ( Here W 1 and W 2 are two independent cylindrical Wiener processes on some (possibly different) separable Hilbert spaces, and G i is a suitable stochastically integrable process with respect to W i , for i = 1, 2. Moreover, n stands for the outward normal unit vector to ∂O.
The presence of random terms in both the equations has been considered in [33] in the case of a smooth potential like (1.2) (see also [46,76,80] for modified models and [26,28,77,79] for random terms only in the Navier-Stokes system). We also remind that the case of Cahn-Hilliard-Navier-Stokes system for a compressible fluid has been studied in [32] (see, e.g., [27,78] for random terms only in the Navier-Stokes system in the case of regular potential).
Here, for system (1.7)-(1.12) with a potential like (1.1), we prove the existence of martingale solutions in dimension two and three, as well as pathwise uniqueness and existence of probabilistically-strong solutions in dimension two. The main difficulties on the mathematical side are two. The former is the presence of noise also in the Allen-Cahn equation with singular potential: this requires some ad hoc ideas based on a suitable compensation between the degeneracy of the noise and the blow up of F ′′ at the endpoints (see ((A3)) below). The latter is the coupling term µ∇ϕ in the Navier-Stokes equation. Indeed, for the Allen-Cahn equation one can recover only a L 2 (0, T ; L 2 (O))-regularity for µ, while for the Cahn-Hilliard equation one gets µ ∈ L 2 (0, T ; H 1 (O)). This results in the necessity to reformulate the first equation for the fluid in an alternative fashion, i.e., without employing µ explicitly.
We recall that, in [33], the authors proved the existence of a (dissipative) martingale solution for a similar problem with a smooth potential. Then, taking advantage of the smooth potential, they used the maximum principle to show that the range of the order parameter remains confined in [−1, 1]. Thus the global Lipschitz continuity of the potential and its derivatives holds. This fact was exploited to prove the weak-strong pathwise and in law uniqueness in dimension three. However, if the potential is given by (1.1), then no global Lipschitz continuity can be achieved unless one can prove that the solution stays uniformly away from the pure phases, but this is not straightforward in the stochastic case (see [11] for the single stochastic Allen-Cahn equation).
Besides the existence and uniqueness of solutions, there are still a number of issues to investigate, which will be object of future work. For example, regularity properties of the solution and existence of analytically-strong solutions are open issues. The low regularity of the chemical potential µ in the Allen-Cahn equation that we have mentioned above seems to make the analysis challenging. Yet, some higher regularity properties have been shown in the deterministic case (see [42]). Their extension to the stochastic case is currently under investigation. Moreover, in the spirit of [11], it would be interesting to establish some random strict separation property from the pure phases. To do this, suitable regularity results might be needed. It also worth pointing out that system (1.7)-(1.12) is the non-conserved version of the model, meaning that the spatial average of ϕ is not preserved during the evolution. The deterministic conserved version is now well-understood (see [42]). Its stochastic counterpart will also be the subject of further analysis. This issue will require a tuning of the diffusion coefficient G 2 (see for example [7]). Finally, we point out that also more general versions of the stochastic Cahn-Hilliard-Navier-Stokes system might be considered on account of the recent advances in the analysis of the stochastic Cahn-Hilliard equation.
The content of this work is structured as follows. In Section 2, we introduce the notation used throughout the work and state the main results. Sections 3 and 4 are devoted to the proof of existence of a martingale solutions and, in dimension two, of a probabilistically-strong solution, respectively.

Preliminaries and main results
2.1. Functional setting and notation. For any (real) Banach space E, its (topological) dual is denoted by E * and the duality pairing between E * and E by ·, · E * ,E . If E is a Hilbert space, then the scalar product of E is denoted by (·, ·) E . For every couple of separable Hilbert spaces E, F the space of Hilbert-Schmidt operators from E to F is denoted by the symbol L 2 (E, F ) and endowed with its canonical norm , P) be a filtered probability space satisfying the usual conditions (namely it is saturated and right-continuous), with T > 0 being a prescribed final time. We will use the symbol L = to denote identity in law for random variables. Throughout the paper, W 1 and W 2 are independent cylindrical Wiener process on some separable Hilbert spaces U 1 and U 2 , respectively. For convenience, we fix once and for all two complete orthonormal systems {u 1 j } j∈N on U 1 and {u 2 j } j∈N on U 2 . We denote by P the progressive sigma algebra on Ω × [0, T ]. For every s, r ∈ [1, +∞] and for every Banach space E the symbols L s (Ω; E) and L r (0, T ; E) indicate the usual spaces of strongly measurable Bochner-integrable functions on Ω and (0, T ), respectively. For all s, r ∈ [1, +∞) we write L s P (Ω; L r (0, T ; E)) to stress that measurability is intended with respect to P. For all s ∈ (1, +∞) and for every separable and reflexive Banach space E we also define which yields by [30,Thm. 8.20.3] the identification L s w (Ω; L ∞ (0, T ; E * )) = L s/(s−1) (Ω; L 1 (0, T ; E)) * .
In case of distribution-valued processes, for every s ∈ [1, +∞), r ∈ (0, +∞), and q ∈ (1, +∞] we set . We now define the functional spaces endowed with their standard norms · H , · V 1 , and · V 2 , respectively. As usual, we identify the Hilbert space H with its dual through the Riesz isomorphism, so that we have the variational structure , with dense and compact embeddings (both in the cases d = 2 and d = 3). We will also denote by A : V 1 → V * 1 the variational realization of the −∆ with homogeneous Neumann boundary condition, namely For any Banach space E, we use the symbol E for the product space E d . We also need to define the following solenoidal vector-valued spaces The space H σ is endowed with the norm · Hσ of H and its respective scalar product (·, ·) H σ . By means of the Poincaré inequality, on the space V σ we can use the norm v V σ := ∇v L 2 (O) , v ∈ V σ , induced by the scalar product (·, ·) V σ . The d-dimensional realisation of the −∆ with homogeneous Dirichlet boundary condition L : Furthermore, we also point out that for any u ∈ (H 1 0 (Ω)) * and v ∈ [

The Stokes operator
Employing the spectral properties of the operator A, as customary, we also define the family of operators A s for any s ∈ R. In particular, if {β k , e k } k∈N + ⊂ R × V σ denote the eigencouples of A, where {e k } k∈N + is an orthonormal basis of H σ and an orthogonal basis of V σ , we introduce for any s ≥ 0 the family of Hilbert spaces and we set D(A −s ) = D(A s ) * . Next, for all s ≥ 0, we define the operators Hereafter, we recall a number of standard facts: In light of the previous considerations, using H σ as pivot space, we also have the general variational structure for any s > t > 0, with dense and compact embeddings in two and three spatial dimensions. Finally, we remind that, owing to the Korn inequality, we have where Dv = 1 2 (∇v + (∇v) t ) denotes the symmetric gradient. Furthermore, we define the usual Stokes and the associated bilinear form B : Let us recall that b(u, v, w) = −b(u, w, v) for all u, v, w ∈ V σ , from which it follows in particular that b(u, v, v) = 0 for all u, v ∈ V σ . Moreover, we point out that thanks to the usual functional embeddings it holds that B : V σ × V σ → L 6 5 (O), hence, in particular, that B : V σ × V σ → V * 1 . We now report for the reader's convenience a basic embedding result and its proof. This will be useful in the forthcoming analysis.
Proof. The embedding holds trivially for every α ∈ (0, 1) if p = r. The same follows in the case 1 < p < r from the chain of embeddings Let now p > r. If α ∈ (0, 1), q ∈ [1, +∞] satisfy then the fractional Gagliardo-Nirenberg inequality (see [17,Theorem 1]) entails that for any f ∈ W 1,r (0, T ). Taking into account the embedding valid for every r > 1, we infer that the right hand side of the inequality is finite for every q ∈ [1, +∞]. Moreover, we also get If we set q = +∞, then we have α = r p .
If u ∈ W 1,r (0, T ; X) the claim follows applying the proved inequality to t → f (t) = u(t) X .
Finally, we shall make precise the rigorous interpretation of the stochastic terms (see (1.7) and (1.9)). As a cylindrical process on U i , i = 1, 2, W i admits the following representation where {b k } k∈N is a family of real and independent Brownian motions. However, it is well known that (2.1) does not converge in U i , in general. That being said, it always exists a larger Hilbert space U i 0 , such that U i ֒→ U i 0 with Hilbert-Schmidt embedding ι i , such that we can identify W i as a Q 0 i -Wiener process on U i 0 , for some trace-class operator Q 0 i (see [62,Subsections 2.5.1]). Actually, it holds that Q 0 i = ι i • ι * i . In the following, we may implicitly assume this extension by simply saying that W i is a cylindrical process on U i . This holds also for stochastic integration with respect to W i . The symbol   Furthermore, there exists c F > 0 such that (A3) Setting B as the closed unit ball in L ∞ (O), the operator G 2 : B → L 2 (U 2 , H) satisfies In particular, note that this implies that G 2 : B → L 2 (U 2 , H) is L 2 -Lipschitz-continuous with respect to the H-metric on B, and also G 2 (B ∩ V 1 ) ⊂ L 2 (U 2 , V 1 ). With a slight abuse of notation, we will use the symbol to indicate the operator , then linear boundedness is directly implied by Lipschitz continuity.

Main results.
We first introduce suitable notions of solution for problem (1.7)-(1.11). The first is a martingale solution, the second is a probabilistically-strong solution.
Definition 2.4. Let p ≥ 1 and let (u 0 , ϕ 0 ) satisfy A martingale solution to problem (1.7)-(1.11) with respect to the initial datum (u 0 , ϕ 0 ) is a family is a filtered probability space satisfying the usual conditions; W 1 , W 2 are two independent cylindrical Wiener processes on U 1 and U 2 , respectively; the pair of processes ( u, ϕ) satisfies and, for every v ∈ V σ and ψ ∈ V 1 , it holds that and Remark 2.6. Note that in Definitions 2.4 and 2.5 one has in particular that u ∈ C 0 w ([0, T ]; H σ ) P-almost surely and u ∈ C 0 w ([0, T ]; H σ ) P-almost surely, respectively. Here, the subscript "w" stands for weak continuity in time. Thus the initial conditions (2.8) and (2.15) make sense.
The first main result is the existence of a martingale solution.
Theorem 2.7. Assume (A1)-(A3) and let p > 2. Then, for every initial datum (u 0 , ϕ 0 ) satisfying for every t ∈ [0, T ], P-almost surely. Here |O| stands for the Lebesgue measure of O. Furthermore, there Finally, the following estimate holds: Remark 2.8. The above result still holds if the viscosity depends on ϕ in a smooth way and it is bounded from below by a positive constant. Moreover, we recall that, in [33], the energy inequality is written P-a.s. in a distributional sense.
The second is a stronger result in dimension two, namely, the existence of a (unique) probabilisticallystrong solution.
Remark 2.10. Referring to [33], we observe that a more general G 1 (u, ϕ) can be considered. Instead, considering G 2 (u, ϕ) would require appropriate assumptions on account of (A3). For instance, in place of g k (ψ) we could suppose to have g k (ψ)h k (u) for a suitable {h k } k∈N + .

Proof of Theorem 2.7
Here we prove the existence of martingale solutions to problem (1.7)-(1.12). For the sake of clarity, the proof is split into several steps.
3.1. Regularization of the singular potential. First of all, note that assumption (A1) implies that the function γ : , can be identified with a maximal monotone graph in R × R. Consequently, one can consider, for every λ ∈ (0, 1), the resolvent operator and the Yosida approximation of β, defined as follows For notation and general properties of monotone operators we refer the reader to [8]. For every λ ∈ (0, 1), we define an approximation of F as follows Thus it holds In order to preserve the scaling of the Yosida-approximation on F ′ , we analogously define the λ-approximation of G 2 by setting Notice that, by assumption (A3) and the non-expansivity of J λ , the operator G 2,λ is L 2 -Lipschitzcontinuous (therefore uniformly in λ), and converges pointwise to G 2 as λ → 0 + . Now, we consider the λ-approximated (formal) problem 3.2. Faedo-Galerkin approximation. A discretization scheme is now applied to problem (3.4)-(3.9). Let us consider the (countably many) eigencouples of the negative Laplace operator with homogeneous Neumann boundary condition, namely the couples Analogously, we also consider the (countably many) eigencouples of the Stokes operator, namely the couples It is well known that, up to a renormalization, the set {e j } j∈N + (resp. {e k } k∈N + ) is an orthonormal system in H (resp. H σ ) and an orthogonal system in V 1 (resp. V σ ). Let n ∈ N + and consider the finite-dimensional spaces Z n := span{e 1 , . . . , e n } and Z n := span{e 1 , . . . , e n }, both endowed with the L 2 -norm. In order to approximate the stochastic perturbation, we define the operators G 1,n and G 2,λ,n as H) and such that for any k ∈ N, v ∈ Z n and v ∈ Z n . Notice that, fixed any v ∈ Z n and v ∈ Z n , G 1,n (v) and G 2,λ,n (v) are actually well defined as elements of L 2 (U 1 , Z n ) and L 2 (U 2 , Z n ), respectively. Indeed, for instance, Hσ) . (3.10) Moreover, since G 1 is L 1 -Lipschitz continuous in the sense of assumption (A2) and the orthogonal projection on Z n is non-expansive as an operator from V σ to itself, we can deduce by the same argument that G 1,n is also L 1 -Lipschitz continuous as an operator from Y to L 2 (U 1 , Y ). Similar considerations also apply to G 2,λ,n . More precisely, we have Proposition 3.1. Let λ ∈ (0, 1) and n ∈ N + . The operators are well defined and uniformly Lipschitz continuous with respect to n and λ. In particular, Next, we define suitable projections (orthogonal with respect to the L 2 -inner products) of initial data (evaluated at some point in Ω) on the discrete spaces Z n and Z n , namely, for all n ∈ N + , we set It is now possible to formulate the discretized problem, which reads du λ,n + [−∆u λ,n + (u λ,n · ∇)u λ,n + ∇p λ,n − µ λ,n ∇ϕ λ,n ] dt = G 1,n (u λ,n ) dW 1 The variational formulation of problem (3.11)-(3.16) is given by for every v ∈ Z n and ψ ∈ Z n . Fixed any λ ∈ (0, 1) and n ∈ N + , we search for a weak solution to (3.17)-(3.18) of the form where a λ,n = (a 1 λ,n , a 2 λ,n , ..., a n λ,n ) : are suitable stochastic processes. Inserting (3.19) into (3.17)-(3.18) and choosing as test functions ψ = e i and v = e i for each i ∈ {1, . . . , n}, we deduce that the three processes a λ,n , b λ,n and c λ,n satisfy the system of 3n ordinary stochastic differential equations Let us point out that, in order to derive (3.20)-(3.24), we exploited the fact that, for every choice of integers j and k between 1 and n, as well as the orthogonality in V σ of the basis {e j } j∈N . The stochastic integrals in (3.20)-(3.21) have to be regarded as G i 1,λ,n dW 1 and G i 2,λ,n dW 2 for every i = 1, . . . , n, where for every k ∈ N. By Lipschitz continuity of all the nonlinearities appearing in (3.20)-(3.24), the standard theory of abstract stochastic evolution equations applies. Therefore, we are able to infer that Proposition 3.2. For every λ ∈ (0, 1) and n ∈ N + , there exists a unique triplet of (F t ) t -adapted processes a λ,n , b λ,n , c λ,n satisfying problem (3.20)-(3.24). Furthermore, for every r ≥ 2, we have 3.3. Uniform estimates with respect to n. First of all, we prove some uniform estimates with respect to the Galerkin parameter n, keeping λ ∈ (0, 1) fixed. Hereafter, the symbol C (possibly numbered) denote positive constants whose special dependencies are explicitly pointed out when necessary. 'In some cases, in order to ease notation, we may use the same symbol to denote different constants throughout the same argument. In any case, such constants are always independent of n.
First estimate. We exploit the Itô formula for the H-norm of ϕ λ,n given in [62,Theorem 4.2.5]. This gives Let us now address the above equality term by term. First of all, recalling (3.2) and that F ′ λ (0) = 0, we find Next, owing to (3.10) and (A3), we have where C depends on p and also on c F , L 2 , |O|, T . The Burkholder-Davis-Gundy and Hölder inequalities jointly with (3.28) entail where C only depends on p. In turn, thanks to (3.30) and the Young inequality, we can refine the estimate and get The Gronwall lemma entails that there exists C 1 , independent of n and λ, but depending on p and the structural data of the problem, such that Next, we want to apply the standard Itô formula to the regularized energy functional E λ : However, notice that E λ exactly contains the kinetic energy contribution linked to the fluid velocity field which we just handled in (3.32). Thus, it is sufficient to apply the Itô formula only to the portion of the energy linked to the order parameter ϕ λ,n . Let us stress that this is only possible since no coupling energy terms are present. We set It has already been shown in [70, Subsection 3.2] that Ψ λ is twice Fréchet-differentiable. Thus it is possible to apply the Itô formula in its classical version [23,Theorem 4.32]. This yields where we recall that DΨ λ (ϕ λ,n ) = µ λ,n . Adding (3.32) and (3.33) together, we find F ′′ λ (ϕ λ,n (τ ))|g k (J λ (ϕ λ,n (τ )))| 2 dτ + t 0 (u λ,n (τ ), G 1 (u λ,n (τ )) dW 1 (τ )) Hσ + t 0 (µ λ,n (τ ), G 2,λ,n (ϕ λ,n (τ )) dW 2 (τ )) H .
In the following, we assume that, given p ≥ 2, the exponents β = β(p) and γ = γ(p) are fixed. Notice that if p > 2, then trivially β and γ are both greater than 1.
3.4. Passage to the limit as n → +∞. Owing to the previously proven uniform estimates, we now pass to the limit as n → +∞ keeping λ ∈ (0, 1) fixed. Let p > 2.
Proof. To prove the claims, we follow a standard argument (refer, for instance, to [ are compact (the intersection spaces are endowed with their canonical norm), since β, γ > 1 p , p > 2. Here, δ ∈ (0, 1 2 ). Let us prove the first one, the other three cases being similar. For any R > 0, let B R denote the closed ball of radius R in L ∞ (0, T ; V 1 ) ∩ W β,p (0, T ; V * 1 ). Then the Markov inequality, jointly with estimates (3.45) and (3.58), implies so that the first claim is proven. The remaining claims can be proven analogously, replacing the spaces accordingly and exploiting the corresponding estimates.
We now set With a little modification in the proof of Lemma 3.5, we can also prove Lemma 3.6. The family of laws of (G 1,n (u λ,n ) · W 1 ) n∈N is tight in the space C 0 ([0, T ]; V * σ ). The family of laws of (G 2,λ,n (ϕ λ,n ) · W 2 ) n∈N is tight in the space C 0 ([0, T ]; H).
Proof. By [36, Theorem 2.2], since βp > 1 and γp > 1, we have that the embeddings The argument of the proof of Lemma 3.5, recalling estimates (3.51) and (3.52), is enough to conclude the claims.
Next, we consider the constant sequences of cylindrical Wiener processes Lemma 3.7. The family of laws of (W 1,n ) n∈N is tight in C 0 ([0, T ]; U 0 1 ). The family of laws of (W 2,n ) n∈N is tight in C 0 ([0, T ]; U 0 2 ).
Proof. It directly follows from the fact that every measure on a complete separable metric space is tight.
Finally, we consider the sequences of approximated initial conditions. Lemma 3.8. The family of laws of (u 0,n ) n∈N is tight in V * σ . The family of laws of (ϕ 0,n ) n∈N is tight in H.
Proof. It is a third iteration of the proof of Lemma 3.5, exploting the compact embeddings and the Markov inequality on closed balls of H σ and V 1 , respectively.
Let us notice that it is possible to take the probability space ( Ω, F , P) large enough so that it does not depend on λ. Taking into account the previous considerations and further straightforward weak convergences, the limit processes fulfill the following regularity properties: From this starting point, we now address several issues.
Next, we address the Korteweg term representing the capillary force. Let us prove that and both terms tend to zero as n → +∞ by the above convergences (note that ∇ ϕ λ ·w ∈ L 2 ( Ω×(0, T )×O)).
Thus, by the Hölder inequality, it holds that The stochastic integrals. Let us now identify I λ and J λ . The procedure is standard, for instance see [23,Section 8.4]. We introduce a family of filtrations on ( Ω, F , P), namely we set F λ,n,t := σ u λ,n (s), ϕ λ,n (s), I λ,n (s), J λ,n (s), W 1,n (s), W 2,n (s), u 0,n , ϕ 0,n , s ∈ [0, t] , for any t ≥ 0, n ∈ N and λ ∈ (0, 1), in such a way that both W 1,n and W 2,n are adapted. In particular, by preservation of laws and the definitions of Wiener process and stochastic integral, we readily have that W i,n is a Q 0 i -Wiener process on U 0 i and are respectively a V * σ -valued and an H-valued martingale. Let us iterate the same procedure on the limit processes: we define It is easy to infer, by the proven convergences, that both W 1 (0) and W 2 (0) are zero. Let now t > 0, s ∈ [0, t] and set Let ψ : X s → R be a bounded and continuous function. By definition of martingale, we have E W i,n (t) − W i,n (s) ψ ϕ λ,n , u λ,n , I λ,n , J λ,n , W 1,n , W 2,n , u 0,n , ϕ 0,n = 0 (3.59) for i = 1, 2. Here, the arguments of ψ are intended to be restricted over [0, s] when necessary and E denotes the expectation with respect to P. Letting n → +∞ in (3.59), an application of the dominated convergence theorem, owing to the proven convergences and the properties of ψ, entails which expresses the fact that W i is a U 0 i -valued ( F λ,t ) t -martingale for i = 1, 2. The characterization of Q-Wiener processes given in [23,Theorem 4.6] leads us to compute the quadratic variation of W i . To this end, notice that (3.60) means that, for every v, w ∈ U 0 ψ ϕ λ,n , u λ,n , I λ,n , J λ,n , W 1,n , W 2,n , u 0,n , ϕ 0,n = 0, and using once more the dominated convergence theorem, we get which is enough to conclude that W i is a Q 0 i -Wiener process, adapted to ( F λ,t ) t , owing to [23, Theorem 4.6]. We are now in a position to study the stochastic integrals. Arguing exactly as in (3.59)-(3.60), we find that I λ (resp. J λ ) is a V * σ -valued (resp. an H-valued) martingale. As far as the quadratic variations are concerned, an application of [23,Theorem 4.27] yields ⟪ I λ,n ⟫ (t) = t 0 G 1,n ( u λ,n (τ ))•G 1,n ( u λ,n (τ )) * dτ, ⟪ J λ,n ⟫ (t) = t 0 G 2,λ,n ( ϕ λ,n (τ ))•G 2,λ,n ( ϕ λ,n (τ )) * dτ, for every t ∈ [0, T ]. Let us outline the argument for the first sequence (similar considerations hold for the second one). Once again, fixing v, w ∈ V σ , we have Hσ dτ ψ ϕ λ,n , u λ,n , I λ,n , J λ,n , W 1,n , W 2,n , u 0,n , ϕ 0,n = 0, and, as n → +∞, the dominated convergence theorem implies that Notice that in the above equality the dualities are necessary. The quadratic variation of I λ is therefore Let us identify I λ with the martingale which is a V * σ -valued ( F λ,t ) t -martingale having the same quadratic variation of I λ . By [66, Theorem 3.2], we can write (3.61) Thus, we now compute the cross quadratic variation appearing on the right hand side in (3.61). To this end, notice that by [66, Theorem 3.2], we have where we also used the fact that Q 0 1 = ι 1 • ι * 1 , where ι 1 : U 1 → U 0 1 is the classical Hilbert-Schmidt embedding. This implies that ⟪ W 1,n , I λ,n ⟫ = · 0 ι 1 • G 1,n ( u λ,n (τ )) * dτ.

Identification of the limit solution.
We are now left to prove that the limiting processes solve the regularized Allen-Cahn-Navier-Stokes system (3.4)-(3.9). Testing (3.11) by some v ∈ V σ and integrating the obtained identity with respect to time yield Letting n → +∞, owing to above convergences and using the dominated convergence theorem, we obtain Next, we identify the limit chemical potential. Testing (3.14) by some v ∈ V 1 , passing to the limit as n → +∞ yields and exploiting the proven convergences entail almost everywhere in [0, T ] and P-almost surely. Finally, consider the approximating Allen-Cahn equation. Testing (3.13) by some v ∈ V 1 and passing to the limit as n → +∞, we get Therefore, system (3.4)-(3.9) is satisfied (in the weak sense) once we identify (the law of) the initial state. By the properties of X n , we know that ϕ 0,n L = ϕ 0,n , u 0,n L = u 0,n for any n ∈ N, and by uniqueness of the distributional limit (jointly with the above convergences) we conclude The initial conditions are therefore attained in law.

3.5.
Uniform estimates with respect to λ. Here, we prove further uniform estimates, now independent of the Yosida parameter λ. The symbol K (possibly numbered) denotes a positive constant, always independent of λ, which may change from line to line.
First estimate. Notice that the constant C 1 in (3.31) does not depend on λ. By lower semicontinuity and preservation of laws of X n , we infer ϕ λ L p P (Ω;C 0 ([0,T ];H)) + ϕ λ L p P (Ω;L 2 (0,T ;V 1 )) ≤ K 1 . (3.65) Second estimate. Let us collect, in (3.35), all controls which are already uniform with respect to λ, that is, the bounds on the diffusion coefficients (3.36) and (3.37), the bounds on the stochastic terms (3.39) and (3.40), and the initial data bounds given in (3.41). This can be summarized as follows (we can express the result in the new variables since X n preserves laws) where C depends on p but is independent of λ. Next, we would like to take the limit as n → +∞ in (3.66).
Identification of the limit solution. Again, a multiple application of the dominated convergence theorem allows us to infer that the limit processes form a martingale solution of the original problem. The existence of a martingale solution is proved. 3.7. The energy inequality. We are left to prove the energy inequality. To this end, we simply pass to the limit in a suitable approximating energy inequality. Let us add (3.32) and (3.33) together and take expectations. Recalling that stochastic integrals are martingales, we obtain the identity Thank to (3.36) and (3.37), from (3.78) we infer Exploiting the preservation of laws by X n , and letting n → +∞, we find Here we have used the lower semicontinuity of the norms and the dominated convergence theorem. A second passage to the limit entails the claimed inequality. Indeed, exploiting preservation of laws by Y λ in (3.80) as well as (A3), and letting λ → 0 + , we get Observe that, passing in the limit in the third term on the left hand side of (3.80) is possible by lower semicontinuity since recalling that it follows J λ ϕ λ → ϕ almost everywhere in Ω × O × (0, T ). Fixed any t > 0, the energy inequality follows taking the supremum over [0, t] in both sides of (3.81).
for i = 1, 2. Then, using the Hölder, Young and Ladyzhenskaya inequalities, together with the definition of ζ n , we find Here, we also used the well-known fact that Au H σ is an equivalent norm in H 2 (O) ∩ V σ . Next, we address the coupling term. We make use of the customary formula for i = 1, 2. The above makes sense in V * σ , since the chemical potential is not regular enough. Therefore, integrating by parts, we recover the identities On the other hand, by Hölder, Young and Ladyzhenskaya inequalities, we obtain By Assumption (A2) we also get (recall that Y = V * σ ), s 0 u(τ ), A −1 G 1 ( u 1 (τ )) − G 1 ( u 2 (τ )) d W 1 (τ ) Hσ dτ. (4.6)