The stochastic viscous Cahn-Hilliard equation: well-posedness, regularity and vanishing viscosity limit

TL;DR

This paper establishes well-posedness, regularity, and the vanishing viscosity limit for the stochastic viscous Cahn-Hilliard equation with broad potential growth conditions and multiplicative noise.

Contribution

It proves well-posedness for the stochastic viscous Cahn-Hilliard equation with general potential growth and demonstrates convergence to the pure Cahn-Hilliard equation as viscosity vanishes.

Findings

Proved well-posedness under broad potential growth conditions.

Established regularity results for viscous and non-viscous cases.

Demonstrated convergence of viscous solutions to the pure Cahn-Hilliard solutions.

Abstract

Well-posedness is proved for the stochastic viscous Cahn-Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular, also super-polynomial) provided that it is everywhere defined on the real line. A vanishing viscosity argument is carried out and the convergence of the solutions to the ones of the pure Cahn-Hilliard equation is shown. Some refined regularity results are also deduced for both the viscous and the non-viscous case.