Front migration in the nonlinear Cahn-Hilliard equation.

*(English)*Zbl 0701.35159The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in \(N>1\) space dimensions. The results support the notion that the nonlinear Cahn- Hilliard equation gives a qualitatively correct description of both early and late stages of phase separation by spinodal decomposition. The expansions are formally valid when the thickness of internal transition layers separating phases is small compared with their radii of curvature and the distance separating layers. On the dominant (slowest) time scale, interface velocity is determined by the mean curvature of the interface through a non-local relation, which is identical to that is a well-known quasi-static model of solidification which exhibits the Mullins-Sekerka shape instability. On a faster time scale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem. Similarity solutions of the two phase Stefan problem approximately describe the development of boundary layers in the Cahn-Hilliard equation. Existence and uniqueness is proved rigorously in an appendix for such similarity solutions which admit metastable states at infinity.

Reviewer: R.L.Pego

##### MSC:

35R35 | Free boundary problems for PDEs |

80A17 | Thermodynamics of continua |

35K25 | Higher-order parabolic equations |

35C20 | Asymptotic expansions of solutions to PDEs |