Large deviation for a 2D Cahn-Hilliard-Navier-Stokes model under random influences

Abstract

In this article, we derive a large deviation principle for a 2D Cahn-Hilliard-Navier-Stokes model under random influences. The model consists of the Navier-Stokes equations for the velocity, coupled with a Cahn-Hilliard equation for the order (phase) parameter. The proof relies on the weak convergence method that was introduced in [3], [4], [5] and based on a variational representation on infinite-dimensional Brownian motion.

Introduction

Large deviation theory is concerned with the study of the precise asymptotic behavior governing the decay rate of probabilities of rare events. This is an important research topic in probability theory. In recent years, this topic has received considerable attention, [29], [27], [10], [23], [8], [9], [7], [24], [20]. Large deviation principle (LDP) has applications in many different areas such as thermodynamics, statistics, information theory, risk management, etc., [10], [13]. LDP for stochastic differential equations such as the Navier-Stokes system has been investigated in [29], [27], [10], [23]. An important tool for studying LDP is the weak convergent approach developed in [2], [5], [4], [3]. The key idea in this method is to prove some variational representation formula about the Laplace transform of bounded continuous functionals.

In this article, we establish a LDP for Cahn-Hilliard-Navier-Stokes equations (CH-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, [16], [15]. For instance, this approach is used in [1] to describe cavitation phenomena in a flowing liquid. The model consists of the NS equation coupled with the phase-field system, [6], [16], [15], [17]. 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, [15]. The associated initial and boundary value problem was studied in [15] 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. [19], [18]). 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 [12], 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 [11], they also considered the asymptotic stability of the unique strong solution for the 3D globally modified CH-NSE. In [21], 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 [26], [25].

The purpose of this work is to establish a large deviation principle (LDP) for a stochastic CH-NSE driven by a multiplicative Gaussian noise. The LDP for partial differential equations such as the Navier-Stokes equations are investigated in several articles, [29], [27], [10], [23], [28]. A LDP for stochastic partial differential equations (SPDE) with Gaussian noise has been investigated in several articles such as [8], [9], [7], [24], [20]. In these papers, the LDP is usually obtained using the weak convergence approach introduced in [5], [4], [3]. This is the approach adopted in this article. Let us recall that the coupling of the Navier-Stokes system and the Cahn-Hilliard equation yields a nonlinear term that makes the analysis of the model more involve.

The article is organized as follows. In the next section we present a stochastic CH-NSE and its mathematical setting. The well posedness and some a priori estimates are given in Section 3. The main result appears in Section 4, where we prove the LDP for a stochastic CH-NSE driven with a small multiplicative noise of Gaussian type. The method of proof is based on the weak convergence method introduced in [2].