2012/05/22 by Colli, Pierluigi, Gilardi, Gianni, Podio-Guidugli, Paolo +1
#35A05 #35K61 #74A15 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1205.4979
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 lattice; in the balance equations of microforces and microenergy, the two unknowns are the order parameter and the chemical potential. A simpler version of the same system has recently been discussed in arXiv:1103.4585v1. In this paper, a fairly more general phase-field equation is coupled with a genuinely nonlinear diffusion equation. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of a constant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.