Computer-assisted equilibrium validation for the diblock copolymer model

Abstract

The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.