Wellposedness and regularity estimate for stochastic Cahn--Hilliard equation with unbounded noise diffusion

TL;DR

This paper establishes the well-posedness and regularity of solutions for a stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise with unbounded diffusion, using spectral Galerkin and semigroup methods.

Contribution

It introduces a spectral Galerkin approach to prove existence, uniqueness, and regularity of solutions for the stochastic Cahn--Hilliard equation with unbounded noise diffusion.

Findings

Proved well-posedness of the approximated finite-dimensional equations.

Established strong convergence of the approximation process.

Demonstrated global existence and regularity of the solution.

Abstract

In this article, we consider the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. By introducing the spectral Galerkin method, we first obtain the well-posedness of the approximated equation in finite dimension. Then with the help of the semigroup theory and the factorization method, the approximation processes is shown to possess many desirable properties. Further, we show that the approximation process is strongly convergent in certain Banach space via the interpolation inequality and variational approach. Finally, the global existence and regularity estimate of the unique solution process are proven by means of the strong convergence of the approximation process.