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 global weak solutions in three-dimensional bounded and smooth domains, assuming that the initial datum has finite energy. We show the classical regularity in $L^2(0,T;H^2(\Omega ))$ for both the solution $u$ and the function $\varphi (u)=\arcsin (u)$ through entropy estimates. In addition, when $\Omega$ is convex, a new elliptic-type argument yields the refined regularity $u,\varphi (u)\in L^4(0,T;H^2(\Omega ))$.

arXiv commentCompared to the first version, the present manuscript replaces one of the two proofs of a main result, after a gap was identified in the original argument. Furthermore, the main result is extended to the setting of general bounded smooth domains