On the diffuse interface models for high codimension dispersed inclusions

Diffuse interface models are widely used to describe evolution of multi-phase system of different nature. Usually dispersed inclusions are three dimensional objects. When describing elastic fractute evolution, elements of the dispersed phase are effectively 2d objects. An example of the model which governs evolution of effectively 1d dispersed inclusions is phase field model for electric breakdown in solids. Phase field model is defined by appropriate free energy functional, which depends on phase field and its derivatives. In this work we show that codimension of the dispersed 'inclusion' significantly restrict the functional dependency of system energy on the derivatives of the problem state variables. It is shown that free energy of any phase field model suitable to describe 1d diffuse objects necessary depends on higher order derivatives of the phase field or need an additional smoothness of the solution — it should have first derivatives integrated with power greater then two. To support theoretical discussion, some numerical experience are presented.