Well-posedness of stochastic Cahn-Hilliard-Brinkman system with regular potential

We consider the stochastic Cahn-Hilliard-Brinkman system with regular potential in a bounded domain D subset of & Ropf;3 driven by a multiplicative noise. Since the noise term and coupling terms create some difficulties in the process of showing the existence of weak solutions, we will first prove the existence of weak solutions by the Faedo-Galerkin methods and stochastic compactness method when the initial data satisfies some "regular" condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of "regular" initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences.