Allen–Cahn equation

The Allen–Cahn equation (after John W. Cahn and Sam Allen) 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 $$\eta$$ on a domain $$\Omega$$ during a time interval $$\mathcal{T}$$, and is given by:


 * $${{\partial \eta}\over{\partial t}}=M_\eta[\operatorname{div}(\varepsilon^{2}_{\eta}\nabla\,\eta)-f'(\eta)]\quad \text{on } \Omega\times\mathcal{T},

\quad \eta=\bar\eta\quad\text{on }\partial_\eta\Omega\times\mathcal{T},$$
 * $$\quad -(\varepsilon^2_\eta\nabla\,\eta)\cdot m = q\quad\text{on } \partial_q \Omega \times \mathcal{T},

\quad \eta=\eta_o \quad\text{on } \Omega\times\{0\},$$ where $$M_{\eta}$$ is the mobility, $$f$$ is a double-well potential, $$\bar\eta$$ is the control on the state variable at the portion of the boundary $$\partial_\eta\Omega$$, $$q$$ is the source control at $$\partial_q\Omega$$,  $$\eta_o$$ is the initial condition, and $$m$$ is the outward normal to  $$\partial\Omega$$.

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
 * $$\Omega\subset \R^n$$ be an open set,
 * $$v_0(x)\in L^2(\Omega)$$ an arbitrary initial function,
 * $$\varepsilon>0$$ and $$T>0$$ two constants.

A function $$v(x,t):\Omega\times [0,T]\to \R$$ is a solution to the Allen–Cahn equation if it solves

\partial_t v-\Delta_x v = -\frac{1}{\varepsilon^2}f(v),\quad \Omega \times[0,T] $$ where
 * $$\Delta_x$$ is the Laplacian with respect to the space $$x$$,
 * $$f(v)=F'(v)$$ is the derivative of a non-negative $$F\in C^1(\R)$$ with two minima $$F(\pm 1)=0$$.

Usually, one has the following initial condition with the Neumann boundary condition
 * $$\begin{cases}

v(x,0) = v_0(x), & \Omega \times \{0\}\\ \partial_n v = 0, & \partial \Omega \times [0,T]\\ \end{cases} $$ where $$\partial_n v$$ is the outer normal derivative.

For $$F(v)$$ one popular candidate is
 * $$ F(v)=\frac{(v^2-1)^2}{4},\qquad f(v)=v^3-v.$$