Averaging principles for multiscale stochastic Cahn-Hilliard system

In this paper, we will establish averaging principles for the multiscale stochastic Cahn-Hilliard system. The stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. Under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale system can be reduced to a single stochastic Cahn-Hilliard equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic Cahn-Hilliard system towards to the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).