A new formulation of the Cahn-Hilliard equation

摘要

We obtain in this article a new formulation of the generalizations of the Cahn-Hilliard equation based on constitutive equations proposed by Gurtin. This formulation, which can be seen as a compatibility equation, is obtained by making a suitable change of function. The interest is that it allows, once this equation is solved, to obtain the order parameter and the chemical potential by simple explicit expressions. Indeed, we can note that, in the original setting, once the order parameter is known (by solving the (generalized) Cahn-Hilliard equation), then, the expressions giving the chemical potential in terms of the order parameter are, for nonisotropic materials, intricate and not convenient (in view of numerical simulations for instance); in the classical Cahn-Hilliard theory, the chemical potential is given, constitutively, as a function of the order parameter. We then study the existence and uniqueness of solutions. (c) 2005 Elsevier Ltd. All rights reserved.