Global weak solutions to the Cahn-Hilliard equation with degenerate mobility and singular diffusion

Abstract

We study the initial-boundary value problem for the Cahn-Hilliard equation with degenerate mobility and singular diffusion at pure phases. This model describes the dynamics of phase separation in polymer blends with associated Flory-Huggins-de Gennes free energy. We prove the existence of suitable global weak solutions in three-dimensional bounded, smooth, and convex domains, assuming that the initial datum has finite energy. A key novelty of our analysis is the derivation of $L^4(0,T; H^2(\Omega))$ estimates for both the solution $u$ and the function $\phi(u)=\arcsin(u)$, obtained without relying on the classical entropy-based approach.