Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity
- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 35K61 35A05 74A15
- phase field model, nonlinear laws, existence of solutions, new uniqueness proof
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 ; 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 . 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.
- J. Differential Equations, 254 (2013) pp. 4217--4244.