Consistent models of Cahn–Hilliard–Gurtin equations with Neumann boundary conditions

In this paper, we prove the equivalence between two formulations for generalizations of the Cahn–Hilliard equation (based on constitutive equations proposed by Gurtin) in a parallelepiped associated with mixed periodic-Neumann boundary conditions. It is important to note that, in the case of the classical Cahn–Hilliard equation, both formulations are useful for the study of the existence and uniqueness of solutions. Furthermore, the interest for taking Neumann boundary conditions appears in particular when one takes into account the deformations of the material. Indeed, in that case, the order parameter and the displacement are coupled through the boundary conditions, as it is generally expected. We then obtain some existence and uniqueness results.