Allen–Cahn equation

Summary

The Allen–Cahn equation is the reaction–diffusion equation $u_t = \Delta u - \varepsilon^{-2}(u^3 - u)$. It models phase separation in a binary alloy, where $u = \pm 1$ are the two pure phases. It is the gradient flow in $L^2$ of the Ginzburg–Landau energy. As $\varepsilon \to 0$, the thin interfaces between the phases move approximately by mean curvature flow.