On the Diffuse Interface Models for High Codimension Dispersed Inclusions

Diffuse interface models are widely used to describe the evolution of multi-phase systems various natures. Dispersed inclusions described by these usually three-dimensional (3D) objects characterized phase field distribution. When employed elastic fracture evolution, dispersed elements effectively two-dimensional (2D) objects. An example model with one-dimensional (1D) is a for electric breakdown in solids. Any diffuse defined an appropriate free energy functional, which depends on and its derivatives. In this work we show that codimension significantly restricts functional dependency derivatives problem state variables. It shown 2 objects, necessarily higher order or needs additional smoothness solution, i.e., first should be integrable power greater than two. Numerical experiments presented support our theoretical discussion.