In this article, we derive a large deviation principle for a 2D Allen–Cahn–Navier–Stokes model under random influences. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn equation for the order (phase) parameter. The proof relies on the weak convergence method that was introduced in (Ann. Inst. Henri Poincaré Probab. Stat.47 (2011) 725–747) and based on a variational representation on infinite-dimensional Brownian motion.
It is well known that the strong law of large numbers and the central limit theorem are fundamental theorems in probability theory. Basically, these limit theorems state that averages taken over large samples converge to expected values. Unfortunately, these results say little or nothing about the rate of convergence which is important for many applications of probability theory, e.g., statistical mechanics, [14]. One way to address this issue is the theory of large deviations. A clear definition of this theory was introduced by Varadhan in [39], where he formulated the large deviation principle (LDP) for a collection of random variables on a Polish space , a complete separable metric space.
Let be a family of random variables defined on a probability space taking values in a Polish space . The theory of large deviation is concerned with events O for which the probability converges to zero exponentially fast as . The exponential decay rate of the probability is usually expressed in terms of a “rate function” I. Let the Borel σ-field of . We first recall some definitions.
(Rate Function).
A function is called a rate function if for each , the level set is a compact subset of . For , we define .
(Large Deviation Principle).
Let I be a rate function on . The sequence satisfies a large deviation principle with rate function I if the following conditions are satisfied.
Large deviation upper bound. For each closed subset of ,
Large deviation lower bound. For each open subset of ,
In the literature of large deviations for stochastic differential equations (SDEs), is usually the space of solutions and is the solution, when the intensity of noise is multiplied by ϵ. To establish the LDP for solutions of SDEs, the following lemma plays a key role.
(Varadhan’s Contraction Principle).
Letbe a family of random variables on the Polish spacewhich satisfies the LDP with the rate function, and letbe a continuous transformation. Then the familysatisfies the LDP on the Polish spacewith the rate function
Here we use the usual convention.
In their pioneering book [21], Freidlin and Wentzell used a time discretization argument to freeze the diffusion term in the case of SDE’s with multiplicative noise. This method is summarized in [14] as follows. In each step of the time discretization process, the solution is a continuous function of the noise and therefore satisfies the LDP by the Varadhan’s contraction principle. Finally, one must prove that the equation with the frozen drift is a good exponential estimate of the original solution. To apply this method in infinite dimensions, after freezing the diffusion, one must project it to a finite dimensional system. Then, after establishing the LDP in finite dimension, one must further prove that the LDP remains valid as one approaches the infinite dimensional system. The fundamental ideas of this method for infinite dimensional spaces can be found in [1,27,30]. We also refer the reader to [20] for a first work in LDP for stochastic reaction–diffusion equations. Thereafter, many papers in different approaches to SPDE have been written. In [29], the author used these fundamental ideas to establish LDP for semi-linear stochastic evolution equations with Lipschitz nonlinearity and multiplicative noise. In [32], the author studied the LDP problem for stochastic reaction–diffusion equations with diffusion coefficients bounded on the unit circle. In [9], the authors investigated the large deviation property for stochastic reaction–diffusion systems with unbounded diffusion terms and locally Lipschitz reaction terms which satisfy a polynomial growth.
In this paper, we use the weak convergence approach to establish the LDP. In this approach, which has been initiated in [4,5], we consider an equivalent formulation of the LDP, called the Laplace principle. We refer the reader to [18, Section 1.2] for a proof of equivalence between Laplace principle and large deviation principle.
(Laplace principle).
The family of random variables on the Polish space is said to satisfy the Laplace principle with the rate function I if for all bounded continuous functions ,
An important tool for studying LDP is the weak convergent approach, [3–6]. Briefly, the basic setup in this approach is to associate with the given LDP problem a family of minimal cost functions which gives a variational representation for these integrals. Then the asymptotics of the minimal cost functions will be determined by the weak convergence methods. In [18], the authors present a comprehensive reference of weak convergence methods. For more illustrations of this method, we refer to [8,10–12,28,31–33,40,42].
In this article, we establish a LDP for an Allen–Cahn–Navier–Stokes equations (AC-NSE) driven by a multiplicative noise of Gaussian type. We recall that the incompressible Navier–Stokes equations (NSE) govern 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, [22,23]. 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 phase-field system, [7,22–24]. In the isothermal compressible case, the existence of a global weak solution is proved in [19]. 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, [22]. The associated initial and boundary value problem was studied in [22] in which the authors proved that the system generated a strongly continuous semigroup on a suitable phase space which possesses a global attractor. 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. [25,26]). 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.
There are few works available on stochastic two phase flow models. In [16], the authors considered the stochastic 3D globally modified Cahn–Hilliard–Navier–Stokes (CH-NSE) equations with multiplicative Gaussian noise. They proved the existence and uniqueness of strong solution (in the sense of partial differential equations and stochastic analysis). Moreover, they studied the asymptotic behavior of the unique solution and obtained the existence of a probabilistic weak solution for the stochastic 3D CH-NSE. In [15], they also considered the asymptotic stability of the unique strong solution for the 3D globally modified CH-NSE. In [34], the author proved the existence and uniqueness of the probabilistic strong solution for the stochastic 2D CH-NSE with multiplicative noise. We recall that the presence of noise can lead to new and important phenomena. For example, the 2D Navier–Stokes equations with sufficiently degenerate noise have a unique invariant measure and hence exhibit ergodic behavior in the sense that the time average of a solution is equal to the average over all possible initial data. Despite continuous efforts in the last thirty years, such a property has so far not been found for the deterministic counterpart of these equations. This property could lead to profound understanding of the nature of turbulence. The aforementioned Navier–Stokes Equations (NSE) are now a widely accepted model of fluid motion, see for instance the well known monograph [37,38].
We consider a stochastic version of the AC-NSE driven by random forces of the form , where is a small parameter. Denoting by the solution to the stochastic AC-NSE, our main result (see Theorem 4.4) is a Wentzell–Freidlin type LDP as , which describes the exponential rate of convergence of the solution to the solution to the deterministic AC-NSE. More precisely, we prove that the solution satisfies a LDP in the space with the good rate function defined in (4.58).
We recall that the LDP for partial differential equations such as the Navier–Stokes equations are investigated in several articles, [12,31,40–42]. A LDP for stochastic partial differential equations (SPDE) with Gaussian noise has been investigated for instance in [8,10,11,28,32]. In these papers, the LDP is usually obtained using the weak convergence approach. This is the method adopted in this article. Let us recall that the coupling of the Navier–Stokes system and the Allen–Cahn equation yields a nonlinear term that makes the analysis of the model more involved.
The article is organized as follows. In the next section we present a stochastic AC-NSE and its mathematical setting. The well posedness and some a priori estimates are given in Sect. 3. The main result appears in Sect. 4, where we prove the LDP for a stochastic AC-NSE driven with a small multiplicative noise of Gaussian type. The method of proof is based on the weak convergence method.
A stochastic AC-NSE and its mathematical setting
Governing equations
We assume that the domain of the fluid is a bounded domain in with a smooth boundary (e.g., of class ). Then, we consider the system
In (2.1), the unknown functions are the velocity , of the fluid, its pressure and the order (phase) parameter ϕ. The term represents the random external forces that eventually depend on , W is a cylindrical Wiener process and is a small parameter. Precise assumptions on the data are given below. The model (2.1) describes the motion of a binary fluid excited by random forces.
The quantity is the variational derivative of the following free energy functional
where, e.g., . Here, the constants and correspond to the kinematic viscosity of the fluid and the capillarity (stress) coefficient respectively. Here 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. A typical example of potential F is that of logarithmic type. However, this potential is often replaced by a polynomial approximation of the type , , being positive constants.
Regarding the boundary conditions for these models, we assume that the boundary conditions for are the natural no-flux condition
where is the boundary of and η is the outward normal to . These conditions ensure the mass conservation. Note that (2.3) implies that
Concerning the boundary condition for , we assume the Dirichlet (no-slip) boundary condition
Therefore we assume that there is no relative motion at the fluid-solid interface.
The initial condition is given by
Mathematical setting
We first recall from [23] a weak formulation of (2.1), (2.5)–(2.6). We assume that satisfies
where is some positive constant and is fixed. It follows from (2.7) that
Note that the derivative of the typical double-well potential f satisfies conditions similar to (2.7).
If X is a real Hilbert space with inner product , we will denote the induced norm by , while will indicate its dual. We will also denote by the set of right continuous functions with left limits from into X.
We set
We denote by and the closure of in and respectively. The scalar product in is denoted by and the associated norm by . Moreover, the space is endowed with the scalar product
We now define the operator by
where is the Leray–Helmotz projector in onto . Then, is a self-adjoint positive unbounded operator in which is associated with the scalar product defined above. Furthermore, is a compact linear operator on and is a norm on that is equivalent to the -norm.
Note that from (2.7), we can find such that
We define the linear positive unbounded operator on by:
where
Note that is a compact linear operator on and is a norm on that is equivalent to the -norm.
We define the Hilbert spaces and by
endowed with the scalar products whose associated norms are respectively
We also set
The norm on is defined by:
We introduce the bilinear operators , (and their associated trilinear forms , ) as well as the coupling mapping , which are defined from into , into , and into , respectively. More precisely, we set
Note that
We recall from [23] that , and satisfy the following estimates
We set
and observe that still satisfies (2.8) with γ in place of since . Also its primitive is bounded from below.
Hereafter, we assume that also satisfies
for some constant .
Note that (2.18) is satisfies if there exists a constant such that
Using the notations above, we rewrite (2.1), (2.3), (2.5)–(2.6) as follows:
In the formulation (2.20), the term is replaced by . This is justified since is the gradient and can be incorporated into the pressure gradient, see [23] for details.
We recall from [36] an equivalent formulation to (2.20).
We first introduce the trilinear form defined by:
The proofs of the following propositions is given in [36].
There exists a constantsuch that
There exists a bilinear operatordefined onwith values insuch thatMoreover, there exists a constantsuch that
For anyand, we have
We recall from [22] that
It is clear that for , . Therefore
It follows from (2.26)–(2.27) that
Using the notations above, we rewrite (2.20) as
We assume that the random force in (2.20) is driven by a Wiener process W with an intensity σ that may depend on the unknowns and . To be more precise, let Q be a linear positive operator in the Hilbert space , which belongs to the trace class, and hence compact. Let . Then is a Hilbert space with the scalar product
The associated norm is denoted and is defined by , . The embedding is Hilbert–Schimidt and therefore compact. Moreover, .
Let be the space of linear operators such that is a Hilbert–Schimdt operator from to . The norm in the space is defined by ; where denotes adjoint operator of S. We recall from [13] that the -norm can also be written as
where is any orthonormal basis in .
Let be a Wiener process defined on a filtered probability space , taking values in and with covariance Q. Therefore, W is Gaussian, has independent time increment and for , ,
Moreover, we also have the following representation
where , are standard (scalar) mutually independent Wiener processes, is an orthonormal basis in consisting on eigen-elements of Q, with .
As in [12 ,17], we assume that the intensity σ satisfies the following assumption.
There exist constants , such that
Well-posedness
In this section, our goal is to prove the existence and uniqueness of strong solution to (2.20).
Let be the class of -values -predictable stochastic process φ that satisfies , -a.s.
For , we define as follows:
On we define the following weak topology:
where is a complete orthonormal basis of , Then endowed with this topology is a Polish space, [6].
We define as follows:
For some technical reasons, we first consider a more general AC-NSE, which contains an extra forcing (control) term driven by elements of . We assume that the intensity of the control term satisfies the following assumptions.
There exist constants , such that
and .
We recall that
It follows that
We derive from (3.1) that
We also note that if , it follows from (2.18) that
Therefore, (3.1) and (3.3) imply that for all , we have
where and .
Hereafter, we set
For , we set
where is a constant large enough and independent on such that is nonnegative (note that F is bounded from below).
We can check that (see [22,23] for details) there exists a monotone non-decreasing function (independent on time and the initial condition) such that
Let and . For satisfying Assumptions A and
A
¯
respectively, we consider the following more general (control) system
Let, there existssuch that the following is true for. Let, withand, where k is given in (
2.7
). Letand. Then there exists a unique pathwise solutionto (
3.7
). Furthermore, the following estimate holds true:where.
We note that if , we deduce the existence and uniqueness of solution to the following “deterministic” control system
If , we can prove as in [22,23,34] that the solution satisfies the estimates
For the existence of solution to (3.7), we consider the following Galerkin scheme. We consider the family of eigenfunctions of the Stokes operator , and the family of eigenfunctions of the operator . We set . We denote by the orthogonal projection of onto . We assume that the -valued process W with covariance Q is such that , .
For , we consider the following Faedo–Galerkin approximation to (3.7):
As in [34], using the properties of the operators , , and , we can prove the existence of a maximal solution, i.e a stopping time such that (3.11) holds for and as , .
We will need the following version of the Gronwall lemma (see [17])
Let,and I be non-decreasing, non-negative processes, φ be a non-negative process andbe a non-negative integrable random variable. We assume thatalmost surely and that there exist positive constantsandsuch that for,Then if, we have for
For any integer, there existsuch that for, the following result holds. Letand. Thena.s. and (
3.11
) has a unique solutionthat satisfieswhereis independent of n.
Let be the maximal solution to (3.11) described above. For every , we set
Then, almost surely, on . The following estimates will show that a.s.
To simplify the notations, we set , and . Let be the projection operator defined by
where is the orthonormal basis of made of eigen-elements of Q and used in (2.30).
Applying Ito’s formula to and using (3.11)1, we derive that
Now we multiply (3.11)2 by and using (3.11)3 to derive that
Integrating (3.16) and adding the result to (3.15) gives
Note that in (3.17), we have used the fact that
Now applying Ito’s formula to and using (3.17), we derive that
where
As in [12 ,17], we have
We also have
Using the Burkholder–Davies–Gundy inequality, (A1) and the Schwarz inequality, we derive that
We define the following property for any integer i, by:
where is independent of .
Let us prove by induction that is true. It is clear that is true since we can take and . Let us assume that hold true. Let us prove that holds true. We set
It follows that
Let
It follows from (3.18)–(3.22) that the assumptions for Lemma 3.2 hold true since . It then follows from Lemma 3.2 that is true.
As , on , . Therefore, and for almost all ω, for large enough, and . By the Lebesgue convergence theorem, we complete the proof. □
Letand. Letbe given in Proposition
3.3
with. Then there exists a constantsuch that for, we have
From (3.4), we have
Therefore (3.23) follows Proposition 3.3. □
We assume that , where k is given in (2.7)2. We also assume that . Let be given by Proposition 3.3. Then there exists a constant independent of n and of such that for , we have
From (3.11)3, we have
Therefore, (3.24) follows from (3.14) and (3.25). □
Letbe-measurable;be an adapted process such that for almost every ω the map. For, we set. For, letsatisfies Assumption (A1),andsatisfies Assumption
A
¯
. Letand,be the solution corresponding toWe set. Then, for every, we havewhere
Let us set . Then satisfies
Applying Ito’s formula to gives
where is given in (3.28).
We also have
Adding (3.30) and (3.31) give
Note that
As in [22 ,23], we also have
Let us note that
It follows from (3.30)–(3.37) that
which proves (3.27). □
In this part, we give a sketch of the proof of Theorem 3.1. A detailed proof of the existence and uniqueness of strong solution to a stochastic AC-NSE driven by a Gaussian noise is given in [34].
Let be endowed with the product measure on . Let , be defined by Proposition 3.3 with and respectively. We set .
We set
There exist a subsequence ofstill denoted the same and processes,,,and a random variablesuch thatMoreover,is a strong solution to (
3.7
).
In fact, (3.40)1–(3.40)6 follows from the estimates (3.14), (3.23), (3.24) as well as the uniqueness of the limit for an appropriate choice of φ.
We also note that for , we have
which gives
We also have
From (2.28) in Remark 2.2 and (2.24), we have
which gives (see (3.14) with )
This gives
From (3.41)–(3.47), we deduce (3.40)7.
As in [12 ,17], we also have
where is independent of n. This proves (3.40)8.
For (3.40)9, using Assumption
A
¯
1, Holder’s inequality and (3.23), we derive that for , we have
where is independent of n. This proves (3.40)9.
As in [34], we can check that satisfies
As in [34 ,35], we can check that , , , , where
This proves that is a strong solution to (3.7). □
The proof of the existence and unique of strong solution to a stochastic AC-NSE driven by a Gaussian noise was given in [34]. Since the result involves some restrictions on the parameter ϵ, we will give below a detail proof.
Let be another solution to (3.7). Let
Since and are a.s. bounded on , we have a.s. as . Let us set . We also set , , , , , .
Let , where is defined in (3.28). We also set
From Lemma 3.5, we have
But from Assumption
A
¯
3 and (3.2), we have
It follows from (3.51)–(3.52) that
Using the Burkholder’s inequality and Assumption (A3), we derive that for all and , we have
where
Note that
Therefore, Lemma 3.2 implies that for and small enough such that , we have
Since a.s., we conclude that a.s. on . □
Large deviation
In this section, using the weak convergence method introduced in [4,5], we study the LDP for the AC-NSE (2.20). We recall that this method is based on variational approximations of infinite dimensional Wiener processes. The solution to (2.20) will be denoted for a Borel measurable function , where endowed with the norm (3.5).
In order to establish the LDP, we impose the following stronger assumption on σ and .
There exist positive constants L and K such that
To prove the LDP, as in [12,17] we will need the following lemma, which studies time increments of the solution to the stochastic control system (3.7). For every integer , and , we set and .
For and , , ϵ small enough, let be the solution to (3.7) given by Theorem 3.1. We set
Letand σ,satisfy (A1), (A4) and (A5). Letbe-measurable random variable andbe a solution to (
3.7
). There exists a constantsuch that,, we have
Let , . From Itô’s formula, we have
Let us set
It follows that
where
We also note that
Let us set
Then
where
where
It is clear that
We also note that for all . It follows that on , for .
As in [12 ,17], using the Burkholder–Davis–Gundy inequality as well as (A4), we derive that
where .
For , using property (A4), we also have
where .
Using Schwarz’s inequality, Fubini’s theorem as well as (A4) give
where .
For , we note that
It follows that
where
Since on we have
it follows that
We now consider and . Note that
It follows that , where
Note that
We also have
As in (4.17)–(4.18), we derive from (4.20)–(4.21) that
It is clear that can be handle in the same way as . This gives
For , we note that
for some non-decreasing polynomial that depends only on . It follows that
The estimate (4.1) follows from (4.12), (4.2)–(4.13) and (4.15)–(4.23). □
Let be defined as in Theorem 3.1 and let be a family of random elements taking values in . Let be the solution to
It is clear that (due to the uniqueness of solution) we have
For all , let be the solution to the control system
Let
For every , define by:
(Weak convergence).
We assume that σ is time-independent and satisfies (A1), (A4) and (A5). Letbe-measurable random variable such that. We assume thatconverges to h in distribution as random elements taking values in. Then, as,converges in distribution toin. That is
Since is a Polish space, by the Skorokhod representation theorem, we can construct a process such that the joint distribution of is the same as that of , the distribution of coincide with that of h and , a.s. in the (weak) topology of . Hence. a.s. for every , we have
Let . Then satisfies
Let be defined as in Theorem 3.1. We set , , , , , . Then , satisfy (3.26). Therefore, from Lemma 3.5, (A4) and (A5), we derive that for , we have
where
Our goal is to prove that as ,
in probability, which will imply that in distribution in .
Let fixed. For , we set
Claim 1. For any , we have
In fact, for , , it follows from (3.8) and the Markov inequality that
where is a constant. This proves Claim 1.
Claim 2. For fixed and such that in the weak topology of a.s. as , we have
In fact, from (4.28) and the Gronwall lemma, it follows that on , we have
Note that
and
We derive from (4.33) that
where C is independent of ϵ. Therefore, using again (4.33), we derive that
Since is decreasing, we have
where
Note that the scalar-valued random variables in as . In fact, using the Burkholder–Davis–Gundy inequality and (A4), we have
We recall that for every integer , and , we have set and . Then, for any , we have
where
As in [12 ,17], using the Schwarz inequality, Lemma 4.1 and (A4), we derive that there exists a constant such that
Using (A5), Lemma 4.1 and similar computations, we derive that
For , Using Schwarz’s inequality and (A4), we deduce that
We note that from the weak convergence of to h, we deduce that for any , , as , we have in the weak topology of . Therefore, since for , the operator is compact from to , we deduce that
Hence, a.s., for n fixed, as , . Moreover, and hence, from the dominated convergence theorem, we conclude that for any n fixed, as . Thus, for fixed, we can choose large enough such that for all . Then, for , let such that for , we have . From (4.42)–(4.44), we obtain that for ,
Claim 2 follows from inequalities (4.36), (4.39) and (4.44). To complete the proof of Proposition 4.2, let , . We set
From the Markov inequality, we have
From Claim 1, we choose N large enough such that
For N fixed, using Claim 2, it follows that there exists ϵ small enough such that
which concludes the proof of the proposition. □
The next result will show that the rate of function of the LDP satisfied by the solution to (4.25) is a good function.
(Compactness).
Letbe a fixed number andbe deterministic. We setwhereis the unique solution to the deterministic control problemWe assume that σ is independent of time t and satisfies (A1), (A4) and (A5). Thenis a compact subset of.
Let be a sequence in corresponding to solution to (4.48) with controls in :
Since is bounded in , there exists a subsequence (still) denoted such that converges weakly to a limit . Note that is closed. Let be the solution to (4.48) corresponding to h. Our goal is to prove that converges to in the space . We set . Then satisfies
We apply Lemma 3.5 with , , , , , , , . Then , satisfy (3.26). Let us set
It follows from Lemma 3.5 that
For and , we set . For , , let . Note that (3.10) implies that there exists a constant independent of n such that
We derive from the Gronwall lemma that
where is independent of N and
Using Schwarz’s inequality, (A4), (A5) and Lemma 4.1, we derive that
It follows that for fixed, we can choose large enough such that
Then, for N fixed, and , as , from the weak convergence of to h, we conclude that
Since is a compact operator, we conclude that for k fixed, the sequence
converges strongly to 0 in as . Since , we obtain that
Therefore, as
From this convergence and (4.53), we conclude that
This proves that is compact in . □
With the results proved above, we now deduce the following large deviation theorem.
We assume that σ does not depend on time and satisfies (A1), (A4) and (A5). Letbe the solution to (
2.20
). Thensatisfies the LDP inwith the good rate functionwhere.
Here, the infimum of an empty set is taken as ∞.
From Proposition 4.2 and Proposition 4.3, we deduce that satisfies the Laplace principle (which is equivalent to the LDP) in with the good rate function (4.58); see Theorem 4.4 in [12] as well as Theorem 5 in [17]. □
Footnotes
Acknowledgements
The first author is supported by the Fulbright Scholar Program Advanced Research and the Florida International University, 2019.
References
1.
R.Azencott, Grandes deviations et applications, in: Ecole d’ Eté de Probabilités de Saint-Flour VII, Lecture Notes in Math, Springer, New York, 1980, pp. 1–176.
2.
T.Blesgen, A generalization of the Navier–Stokes equation to two-phase flow, Pysica D (Applied Physics)32 (1999), 1119–1123. doi:10.1088/0022-3727/32/10/307.
3.
A.Budhiraja, J.Chen and P.Dupuis, Large deviations for stochastic partial differential equations driven by a Poisson random measure, Stochastic Process. Appl.123(2) (2013), 523–560. doi:10.1016/j.spa.2012.09.010.
4.
A.Budhiraja and P.Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist.20(1) (2000), 39–61.
5.
A.Budhiraja, P.Dupuis and V.Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab.36(4) (2008), 1390–1420. doi:10.1214/07-AOP362.
6.
A.Budhiraja, P.Dupuis and V.Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat.47(3) (2011), 725–747. doi:10.1214/10-AIHP382.
7.
G.Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal.92(3) (1986), 205–245. doi:10.1007/BF00254827.
8.
C.Cardon-Weber, Large deviations for a Burgers’-type SPDE, Stochastic Process. Appl.84(1) (1999), 53–70. doi:10.1016/S0304-4149(99)00047-2.
9.
S.Cerrai and M.Rǒckner, Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab.32(18) (2004), 1100–1139. doi:10.1214/aop/1079021473.
10.
F.Chenal and A.Millet, Uniform large deviations for parabolic SPDEs and applications, Stochastic Process. Appl.72(2) (1997), 161–186. doi:10.1016/S0304-4149(97)00091-4.
11.
P.L.Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math.45(1) (1992), 97–120. doi:10.1002/cpa.3160450105.
12.
I.Chueshov and A.Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim.61 (2010), 379–420. doi:10.1007/s00245-009-9091-z.
13.
G.Da Prato and J.Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd edn, Encyclopedia of Mathematics and Its Applications, Vol. 152, Cambridge University Press, Cambridge, 2014.
14.
H.Dadashi-Arani and B.Zangeneh, Large deviation principle for semilinear stochastic evolution equations with monotone nonlinearity and multiplicative noise, Differential Integral Equations23.
15.
G.Deugoué and T.Tachim Medjo, The exponential behavior of a stochastic globally modified Cahn–Hilliard–Navier–Stokes model with multiplicative noise, J. Math. Anal. Appl.460(1) (2018), 140–163. doi:10.1016/j.jmaa.2017.11.050.
16.
G.Deugoué and T.Tachim Medjo, On the convergence for the 3D globally modified Cahn–Hilliard–Navier–Stokes equations, J. Differential Equations265(2) (2018), 545–592. doi:10.1016/j.jde.2018.03.002.
17.
J.Duan and A.Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl.119(6) (2009), 2052–2081. doi:10.1016/j.spa.2008.10.004.
18.
P.Dupuis and R.Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley, New York, 1997.
19.
E.Feireisl, H.Petzeltová, E.Rocca and G.Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci.20(7) (2010), 1129–1160. doi:10.1142/S0218202510004544.
20.
I.Freidlin, Random perturbations of reactiondiffusion equations: The quasi deterministic approximation, Trans. Amer. Math. Soc.305.
21.
M.I.Freidlin and A.D.Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984.
22.
C.Gal and M.Grasselli, Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire27(1) (2010), 401–436. doi:10.1016/j.anihpc.2009.11.013.
23.
C.Gal and M.Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst.28(1) (2010), 1–39. doi:10.3934/dcds.2010.28.1.
24.
C.Gal and M.Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B31(5) (2010), 655–678. doi:10.1007/s11401-010-0603-6.
25.
M.E.Gurtin, D.Polignone and J.Vinals, Two-phase binary fluid and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci.6 (1996), 8–15.
26.
P.C.Hohenberg and B.I.Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys.49 (1977), 435–479. doi:10.1103/RevModPhys.49.435.
27.
R.Leandre, in: Barcelona Seminar on Stochastic Analysis, D.Nualart and M.Sanz Solé, eds, Birkh, 1993, pp. 72–76. doi:10.1007/978-3-0348-8555-3_5.
28.
W.Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim.61(1) (2010), 27–56. doi:10.1007/s00245-009-9072-2.
29.
S.Peszat, Large deviation estimates for stochastic evolution equations, Probab. Theory Related Fields98.
30.
P.Priouret, Remarques sur les petites perturbations de systemes dynamiques, in: Séminaire de Probabilités XVI, Lecture Notes in Math, Vol. 920, Springer, New York, 1982, pp. 184–200.
31.
M.Röckner and T.Zhang, Stochastic 3D tamed Navier–Stokes equations: Existence, uniqueness and small time large deviation principles, J. Differential Equations252 (2012), 716–744. doi:10.1016/j.jde.2011.09.030.
32.
R.Sowers, Large deviations for the invariant measure of a reaction–diffusion equation with non-Gaussian perturbations, Probab. Theory Related Fields92(3) (1992), 393–421. doi:10.1007/BF01300562.
33.
S.S.Sritharan and P.Sundar, Large deviations for the two dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl.116.
34.
T.Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn–Hilliard–Navier–Stokes model, J. Differential Equations262 (2017), 1028–1054. doi:10.1016/j.jde.2017.03.008.
35.
T.Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Allen–Cahn–Navier–Stokes model, Stoch. Dyn.19(1), (2019). 1950007.
36.
T.Tachim Medjo, A note on the regularity of weak solutions to the coupled 2D Allen–Cahn–Navier–Stokes system, J. Appl. Anal.25(1), (2019), 111–117.
37.
R.Temam, Infinite Dynamical Systems in Mechanics and Physics, 2nd edn, Appl. Math. Sci., Vol. 68, Springer-Verlag, New York, 1997.
38.
R.Temam, Navier–Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001.
39.
S.R.S.Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math.19.
40.
T.Xu and T.Zhang, Large deviation principles for 2D stochastic Navier–Stokes equations driven by Lévy processes, Journal of Functional Analysis257(5) (2009), 1519–1545. doi:10.1016/j.jfa.2009.05.007.
41.
K.Yamazaki, Large deviation principle for the micropolar, magneto-micropolar fluid systems, Discrete Contin. Dyn. Syst. Ser. B23(2) (2018), 913–938.
42.
J.Zhai and T.Zhang, Large deviations for 2D stochastic Navier–Stokes equations driven by multiplicative Lévy noises, Bernoulli21(4) (2015), 2351–2392. doi:10.3150/14-BEJ647.
