In [1], two mathematical models for phase segregation and diffusion of an order parameter are derived, within one and the same continuum mechanical framework. These models are, respectively, of the Allen-Cahn type and of the Cahn-Hilliard type, but differ from those in that they are based on a system of two evolution equations, rather than one. I plan to concentrate on the deduction of those models. In the last part of my talk, I shall restrict attention to the first, which consists in a system of a partial and an ordinary differential equation, and indicate briefly how this system is reduced to an Allen-Cahn equation with a memory term [2] and how global existence and uniqueness of a smooth solution are proven.

[1] P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice. Ricerche di Matematica, 55 (1) (2006), 105-118.

[2] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Existence and uniqueness of a global-in-time solution to a phase segregation problem of the Allen-Cahn type. Mathematical Models and Methods in Applied Sciences, (2009) to appear.$. One consequence is that $w(A) = c(A)$ for the Frobenius norm too, and another is the perhaps surprising result that the minimal distance is attained by a defective matrix in all cases. Our results suggest a new computational approach to approximating the nearest defective matrix.