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.

翻译：相场模型是计算物理学中描述多相物质复杂动力学的常用选择，并广泛应用于各类场景。本文提出了包含非光滑双阱势垒势的Cahn-Hilliard型和Allen-Cahn型非等温非局域相场模型。在数学上，该模型以弱形式转化为耦合温度演化方程的变分不等式系统。我们证明，在特定条件下并通过谨慎选择非局域算子，可构建出解中允许存在随时间演变的尖锐界面的模型——这是许多应用场景所期望的特性。与之对比，弥散界面局部模型无法解析尖锐界面。本文给出了模型的适定性分析，讨论了合理的数值离散格式，并通过多项数值实验补充验证了我们的结论。