Allen–Cahn equation


The Allen–Cahn equation is a reaction–diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions.
The equation describes the time evolution of a scalar-valued state variable on a domain during a time interval, and is given by:
where is the mobility, is a double-well potential, is the control on the state variable at the portion of the boundary, is the source control at, is the initial condition, and is the outward normal to .
It is the L2 gradient flow of the Ginzburg–Landau free energy functional. It is closely related to the Cahn–Hilliard equation.

Mathematical description

Let
A function is a solution to the Allen–Cahn equation if it solves
where
Usually, one has the following initial condition with the Neumann boundary condition
where is the outer normal derivative.
For one popular candidate is