Integro-differential equations in materials science
This thesis deals with nonlocal models for solid-solid phase transitions, such as ferromagnetic phase transition or phase separation in binary alloys. We discuss here, among others, nonlocal versions of the Allen-Cahn and Cahn-Hilliard equations, as well as a nonlocal version of the viscous Cahn-Hilliard equation. The analysis of these models can be motivated by the fact that their local analogues fail to be applicable when the wavelength of microstructure is very small, e. g. at the nanometre scale. Though the solutions of these nonlocal equations and those of the local versions share some common properties, we find many differences between them, which are mainly due to the lack of compactness of the semigroups generated by nonlocal equations. Directly from microscopic considerations, we derive and analyse two new types of equations. One of the equations approximately represents the dynamic Ising model with vacancy-driven dynamics, and the other one is the vacancy-driven model obtained using the Vineyard formalism. These new equations are being put forward as possible improvements of the local and nonlocal Cahn-Hilliard models, as well as of the mean-field model for the Ising model with Kawasaki dynamics.