Phase-field model

Definition

Let $\Omega \subset \R^n$ be a bounded domain. A phase-field model on $\Omega$ consists of the following data.

1. An order parameter, or phase field, $\phi \colon \Omega \times [0, T) \to \R$. Its distinct preferred values label the phases.
2. A free energy functional $$F[\phi] = \int_\Omega \left( f(\phi) + \frac{\varepsilon^2}{2} \abs{\nabla \phi}^2 \right) dx,$$ where $f$ is a bulk energy density with a local minimum at each preferred value. The small parameter $\varepsilon > 0$ sets the width of the interfaces.
3. An evolution law that is a gradient flow of $F$. It is either non-conserved, $$\partial_t \phi = -M \frac{\delta F}{\delta \phi} = M\left(\varepsilon^2 \Delta \phi - f'(\phi)\right),$$ or conserved, $$\partial_t \phi = \nabla \cdot \left( M \nabla \frac{\delta F}{\delta \phi} \right).$$ Here $M > 0$ is a mobility, and $\delta F / \delta \phi = f'(\phi) - \varepsilon^2 \Delta \phi$ is the variational derivative.