We establish a large deviation principle for the 2D stochastic Cahn–Hilliard–Navier–Stokes equations driven by multiplicative noise in a bounded domain. This system consists of the Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard model for the order parameter. The proof is completed using a weak convergence approach based on the variational representational of functional of infinite-dimensional Brownian motion. In particular, we directly prove the existence and uniqueness of the solution of the stochastic controlled equations instead of using the Girsanov transformation.

中文翻译：

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二维随机Cahn–Hilliard–Navier–Stokes方程的大偏差原理

我们为有限域中的乘性噪声驱动的二维随机Cahn–Hilliard–Navier–Stokes方程建立了大偏差原理。该系统由用于速度的Navier–Stokes方程以及用于阶数参数的Cahn–Hilliard模型组成。使用基于无穷维布朗运动函数的变分表示的弱收敛方法来完成证明。特别是，我们直接证明了随机控制方程解的存在性和唯一性，而不是使用Girsanov变换。