The Singular Limit Problem to the Extended Cahn–Hillard Equation

Abstract

A mathematical model with a small parameter, which describes the hardening process of the binary tin–lead alloy, is investigated on the basis of nonlinear asymptotic analysis. A singular limit problem, namely an extended Stefan problem in the case of short relaxation time in the phase transformation zone, is derived. We prove the existence of an asymptotic solution with any accuracy on the time interval where the solution to the singular limit problem exists. The phase-separation interface is determined uniquely by three leading approximations. We also show that the stability of the separation interface depends on the so-called dissipation condition obtained for the solutions of the interface problem. Nonsymmetry of the surface tension tensor leads to a situation where the limit values of concentration distributions are in dependence on the geometry of the interface. This provokes the dispersion of the interface problem solutions on the part of the interface that not is tangent to the main crystallographic axis.