A Cahn-Hilliard equation with singular diffusion

TL;DR

This paper studies a singular diffusion Cahn-Hilliard equation modeling phase separation in polymer mixtures, proving existence, uniqueness, and regularity of solutions while ensuring they stay away from singularities.

Contribution

It establishes the existence, uniqueness, and regularity of solutions for a singular diffusion Cahn-Hilliard equation, a simplified model of phase separation.

Findings

Unique energy weak solution exists for any final time T.

Solutions are classical and regular for positive times.

Solutions remain separated from singular phase values.

Abstract

In the present work, we address a class of Cahn-Hilliard equations characterized by a singular diffusion term. The problem is a simplified version with constant mobility of the Cahn-Hilliard-de Gennes model of phase separation in binary, incompressible, isothermal mixtures of polymer molecules. It is proved that, for any final time T, the problem admits a unique energy type weak solution, defined over (0,T). For any s > 0 such solution is classical in the sense of belonging to a suitable Hoelder class over (s,T), and enjoys the property of being separated from the singular values corresponding to pure phases.