Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion

In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation.