Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity

Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $\rho$ and the chemical potential $\mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $\rho$ is coupled with a genuinely nonlinear diffusion equation for $\mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables