Convergence of the One-Dimensional Cahn--Hilliard Equation
Convergence of the One-Dimensional Cahn--Hilliard Equation
We consider the Cahn--Hilliard equation in one space dimension with scaling parameter $\varepsilon$, i.e., $u_t = (W'(u)-\varepsilon^2 u_{xx})_{xx}$, where $W$ is a nonconvex potential. In the limit $\varepsilon\downarrow 0$, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof …