Motion by curvature in generalized Cahn-Allen models
The authors generalize the Cahn-Allen model for the motion of phase-antiphase boundaries by allowing the kinetic coefficient and the coefficient of the gradient free to depend on the order parameter \(u\). The resulting equation is \[ \varepsilon^2 u_t= \alpha(u) \{\varepsilon^2\sqrt{\kappa(u)} \nabla\cdot (\sqrt{\kappa(u)} \nabla u)- f(u)\}, \] where \(\alpha\) and \(\kappa\) are physical quantities and \(f\) is bistable. This reduces to the classical Cahn-Allen equation if \(\alpha\) and \(\kappa\) are constants. Grain boundaries in a solid material, away from triple junctions, are modelled by solutions of the above equation with internal layers. The authors show that the classical motion by mean curvature law for internal layers remains valid, the only difference being the value of the coefficient entering into the relation. They also extend the analysis to the case that \(\alpha\) is infinite for some values of \(u\).
scientific article; zbMATH DE number 843814