Non-isothermal nonlocal phase-field models with a double-obstacle potential

Abstract

Phase-field models are a popular choice in computational physics to describe complex dynamics of substances with multiple phases and are widely used in various applications. We present nonlocal non-isothermal phase-field models of Cahn-Hilliard and Allen-Cahn types involving a nonsmooth double-well obstacle potential. Mathematically, in a weak form, the model translates to a system of variational inequalities coupled to a temperature evolution equation. We demonstrate that under certain conditions and with a careful choice of the nonlocal operator one can obtain a model that allows for sharp interfaces in the solution that evolve in time, which is a desirable property in many applications. This can be contrasted to the diffuse-interface local models that can not resolve sharp interfaces. We present the well-posedness analysis of the models, discuss an appropriate numerical discretization scheme, and supplement our findings with several numerical experiments.