Truncated gradient flows of the van der Waals free energy

Abstract

We employ the Pade approximation to derive a set of new partial differential equations, which can be put forward as possible models for phase transitions in solids. We start from a nonlocal free energy functional, we expand in Taylor series the interface part of this energy, and then consider gradient flows for truncations of the resulting expression. We shall discuss here issues related to the existence and uniqueness of solutions of the newly obtained equations, as well as the convergence of the solutions of these equations to the solution of a nonlocal version of the Allen-Cahn equation.

Keywords

• Gradient flow
• van der Waals energy
• integro-differential equation
• Pade approximants.