The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

In this article, we study the stability of weak solutions to a stochastic version of a coupled Cahn-Hilliard-Navier-Stokes model with multiplicative noise. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.


1.
Introduction. 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, [17]. For instance, this approach is used in [4] to describe cavitation phenomena in a flowing liquid. The model consists of the NS equation coupled with a phase-field system, [8,17,16,18]. 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 or a Cahn-Hilliard equation, [17,16]. The associated initial and boundary value problem was studied in [17,16], in which the authors proved that the system generated a strongly continuous semigroup on a suitable phase space.
The long-time behavior of flows is a very interesting and important problem in the theory of fluid dynamic. As the vast literature [27,15,3,21,23,25,20,19,11,10] shows, the problem has been receiving very much attention over the last three decades. Another interesting question is to analyze the effects produced on a deterministic system by some stochastic or random disturbances appearing in the model. This problem has been studied for the NS model, [12,13,26]. In [12], the authors studied the stability of the stationary solutions of the stochastic 2D NS equations. In particular, they proved that the weak solutions converge 1118 T. TACHIM MEDJO exponentially in the mean square and almost surely exponentially to the stationary solutions under some restrictions on the viscosity and the forcing terms. In [13], the authors generalized to the results of [12] to a class of dissipative nonlinear systems that include the 3D lagrangian average NS equations.
Our work is motivated by the above references. We study the stability of weak solutions to the stochastic 2D CH-NS model with multiplicative noise. One of our goals is to investigate the exponential convergence in mean square and almost surely of the weak probabilistic solutions to stationary solutions. Using techniques similar to that of [12,13], we prove that the exponential convergence in mean square and almost surely of the weak probabilistic solutions to stationary solutions hold, provided some reasonable restrictions on the forcing terms g 1 , g 2 , as well as some physical parameters such as the viscosity ν and the physical constant > 0, which is related to the thickness of the interface separating the two fluids. Under similar restrictions on the noise term g 2 and the physical parameters ν and , we also show that there is a stabilization effect of the stochastic perturbation. It is worth mentioning that our model does not fall into the general framework considered in [13], since the coupling between the Navier-Stokes and the Cahn-Hilliard systems introduces in the model a highly nonlinear term that makes the analysis of the problem more involved.
The article is divided as follows. In the next section, we introduce the stochastic 2D CH-NS model and its mathematical setting. The third section studies the stability of weak solutions. As in [12], applying the Itô formula, we study the stability of stationary solutions to the stochastic 2D CH-NS model. We also prove in the fourth section a result related to the stabilization of these equations.

2.
The stochastic CH-NS model and its mathematical setting.
2.1. Governing equations. In this article, we consider a coupled CH-NS model with multiplicative noise. More precisely, we assume that the domain M of the fluid is a bounded domain in 2 . Then, we consider the system In (2.1), the unknown functions are the velocity v = (v 1 , · · · v d ) of the fluid, its pressure p and the order (phase) parameter φ. The external volume force t ,Ẇ 2 t ) denotes the time derivative of a cylindrical Wiener process. The quantity µ is the variational derivative of the following free energy functional where, e.g., F (x) = x 0 f (ζ)dζ. Here, the constants ν > 0 and K > 0 correspond to the kinematic viscosity of the fluid and the capillarity (stress) coefficient respectively. Here , α > 0 are two physical parameters describing the interaction between the two phases. In particular, is related with the thickness of the interface separating the two fluids. Hereafter, as in [17] we assume that ≤ α.
A typical example of potential F is that of logarithmic type. However, this potential is often replaced by a polynomial approximation of the type F (x) = γ 1 x 4 − γ 2 x 2 , γ 1 , γ 2 being positive constants. As noted in [16], (2.1) 1 can be replaced by The stress tensor ∇φ ⊗ ∇φ is considered the main contribution modeling capillary forces due to surface tension at the interface between the two phases of the fluid. Regarding the boundary conditions for the model, as in [16] we assume that the boundary conditions for φ are the natural no-flux condition where ∂M is the boundary of M and η is the outward normal to ∂M. These conditions ensure the mass conservation. In fact, from ∂ η µ = 0 on ∂M × [0, T ], we have the conservation of the following quantity where |M| stands for the Lebesgue measure of M. More precisely, we have Concerning the boundary condition for v, we assume the Dirichlet (no-slip) boundary condition v = 0, on ∂M × (0, ∞).
(2.7) Therefore we assume that there is no relative motion at the fluid-solid interface.
(2.10) If X is a real Hilbert space with inner product (·, ·) X , we will denote the induced norm by | · | X , while X * will indicate its dual. We set We denote by H 1 and V 1 the closure of V 1 in (L 2 (M)) 2 and (H 1 0 (M)) 2 respectively. The scalar product in H 1 is denoted by (·, ·) L 2 and the associated norm by | · | L 2 . Moreover, the space V 1 is endowed with the scalar product

T. TACHIM MEDJO
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.
Hereafter, we set Then we introduce the linear nonnegative unbounded operator on L 2 (M) and we endow D(A 1 ) with the norm |A 1 · | L 2 + | · | L 2 , which is equivalent to the H 2 −norm. Also we define the linear positive unbounded operator on the Hilbert space L 2 0 (M) of the L 2 − functions with null mean Note that B −1 n is a compact linear operator on L 2 0 (M). More generally, we can define B s n , for any s ∈ , noting that |B 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(A 0 ) × D(A 0 ) into H, D(A 0 ) × D(A 1 ) into L 2 (M), and L 2 (M) × (D(A 1 ) ∩ H 3 (M)) into H 1 , respectively. More precisely, we set (2.14) Note that R 0 (µ, φ) = Pµ∇φ.
We recall that B 0 , B 1 and R 0 satisfy the following estimates We recall that (due to the mass conservation) we have Thus, up to a shift of the order parameter field, we can always assume that the mean of φ is zero a the initial time and, therefore it will remain zero for all positive times. Hereafter, we assume that We set The space H is a complete metric space with respect to the norm We define the Hilbert space U by endowed with the scalar products whose associated norm is We will denote by λ 1 > 0 a positive constant such that We will also denote by c a generic positive constant that depends on the domain M. To simplify the notations, we set (without loss of generality) K = 1.
Let (Ω, P, J ) be a probability space on which an increasing and right continuous family {J t } t∈[0,∞) of complete sub σ− algebra of J is defined. Let β n (t) (n = 1, 2, 3, · · · ) be a sequence of real valued one-dimensional standard Brownian motions mutually independent on (Ω, P, J ). We seṫ where λ n (n = 1, 2, 3, · · · ) are nonnegative real numbers such that ∞ n=1 λ n < ∞, and {e n } (n = 1, 2, 3, · · · ) is a complete orthogonal basis in the real and separable Hilbert space K. Let Q ∈ L(K, K) be the operator defined by Qe n = λ n e n . The above K− valued stochastic process W (t) is called a Q−Wiener process.
Thus, we consider the stochastic coupled CH-NS model written in the following abstract mathematical setting: Remark 2.1. In the formulation (2.26) or (2.27), the term µ∇φ is replaced by A 1 ∇φ. This is justified since f (φ)∇φ is the gradient F (φ) and can be incorporated into the pressure gradient, see [16] for details.
In the deterministic case, the weak formulation of (2.26) was proposed and studied in [7,5,6,17,16] (see also [2,1,9]), where the existence and uniqueness results for weak and strong solutions were proved in the deterministic case.
In [22], the author studied a stochastic CH-NS model in a two or three-dimensional domain. Using a Galerkin approximation, he proved the existence of weak solutions. We will use the notation (2.29) We assume that for any in Ω × (0, T ) and where L 1 > 0, L 2 > 0 are fixed. Moreover, we also assume that for all T > 0, we have The following result is proved in [22,24].
Proof. The proof of the existence of weak solution is given in [22], using a Galerkin approximation as well as some compactness results. The proof of the uniqueness is based on the pathwise uniqueness, which implies the uniqueness of weak solution, is given in [24].
In this article, we study the exponential stability of the weak solutions to (2.26).
Hereafter, we assume that f satisfies the additional condition We also set 3. The exponential stability of solutions. In this section we discuss the moment exponential stability and almost sure exponential stability of weak solutions to (2.26) assuming that they exist. We discuss the long-time behavior of the weak solutions (v, φ)(t) under some conditions. As in [12], applying the Itô formula, we study the stability of stationary solutions to (3.1) below. Let us note that the coupling of the Cahan-Hilliard equation and the Navier-Stokes system introduces in the model a highly nonlinear term that makes the analysis of the problem more involved than the Navier-Stokes system studied for instance in [12]. The existence of weak solutions to the stochastic version of the coupled CH-NS model is proved in [22]. We now consider the following stationary equation (3.1)
Proof. If (v m , φ m ) ∈ U is a solution to (3.2), we can easily check that and (3.6) It follows that which gives where α 1 > 0 is given by (2.33).
Proof. Recall that (v m , φ m ) satisfies the a priori estimates (3.3). Let K 0 > 0 such that K 0 (α 1 − α 0 L 1 ) ≥ g 1 (0, 0) U * . Then from (3.8) we can check that Then ∧ m is a compact and convex subset of U m . It is also clear that T n maps ∧ m into itself. To prove the existence of solution, we apply the Brouwer fixed point theorem to the restriction of T m to ∧ m . Therefore it remains to check that T m is continuous.
For the continuity of T m , we proceed as follows.
(3.12) Multiplying (3.11) 1 and (3.11) 2 by w and A 1 ψ respectively and using (3.12), we derive that 14) and the continuity of the mapping T m follows. It follows that there exists a fixed point (v m , φ m ) of T m in ∧ m . Therefore we can extract a subsequence (still) denoted (v m , φ m ) that converges to (v * , φ * ) strongly in U. Using the same argument as in [17], we can prove that (v * , φ * ) is a weak solution to (3.1).

3.2.
Some a priori estimates on (v * , φ * ). We derive some explicit a priori estimates in the U−norm.
Step 4: Therefore, we have is satisfied for ν > 0, > 0 large enough, since K 1 > 0 is assumed to be a constant.
3.3. Stability of the steady state solutions. We study in this section the stability of the steady state solutions. We assume that ν and are large enough so that (3.1) has a unique solution (v * , φ * ), See Remark above. We first recall from [14] some preliminary definitions.
Definition 3.1. We say that a weak solution (v, φ)(t) to (2.26) converges to (v * , φ * ) ∈ H exponentially in the mean square if there exists η > 0 and M 0 = M 0 ((v, φ)(0)) > 0 such that is a solution to (3.1), we say that (v * , φ * ) is exponentially stable in the mean square provided that every weak solution to (2.26) converges to (v * , φ * ) exponentially in the mean square with the same exponential order η > 0.
Proof. Let N be a positive integer. By the Itô formula, for any t ≥ N we have By the Burkholder-Davis-Gundy lemma, we have where η 1 > 0, η 2 > 0 are some constants. Therefore as in (3.33)-(3.38), we obtain that it follows from Theorem 3.4 that there exist 3.47) and the proof of the theorem follows from the Borel-Cantelli lemma as in [12] (see also [13]).
Theorem 6. Let (v * , φ * ) ∈ U be the unique solution to (3.1). Furthermore, we assume that then any weak solution to (2.26) converges to (v * , φ * ) exponentially in the mean square. That is, there exists η > 0 such that Moreover, the path-wise exponential stability with probability one of (v * , φ * ) also holds true.
It follows from (3.53) that and the proof of the first part of the theorem follows as that of Theorem 3.4. The rest of the theorem is proved using a similar method to the one in the proof of Theorem 5.
Proof. Let η ∈ (0, ρ) be such that By the Gronwall lemma, we obtain that any weak solution to (2.26) converges to zero exponentially in the mean square. We can then finish the proof using the same method as in the proof of Theorem 5.

4.
Stabilization of the 2D CH-NS model (2.26). Hereafter, we briefly discuss the stabilization of the 2D CH-NS model (2.26). As noted in [12,13], in order to produce a stabilization effect, it is enough to consider a one dimensional Wiener process for that purpose.
Hereafter, we suppose that for some σ ∈ . We also assume that Lemma 2. Let (v * , φ * ) ∈ U be the unique solution to (3.1). If g 1 satisfies (4.1) and where L 1 is the Lipschitz constant of g 1 given in (4.1), then the stationary solution (v * , φ * ) to (3.1) is exponentially stable.
If the Lipschitz constant L 1 of g 1 is sufficiently large such that κ 2 < 0, then we do not know if (v * , φ * ) is exponentially stable or not. However, the following result related to the stabilization of the 2D CH-NS systems holds true.