Analysis and discretization of the Ohta-Kawasaki equation with forcing and degenerate mobility

The Ohta-Kawasaki equation models the mesoscopic phase separation of immiscible polymer chains that form diblock copolymers, with applications in directed self-assembly for lithography. We perform a mathematical analysis of this model under degenerate mobility and an external force, proving the existence of weak solutions via an approximation scheme for the mobility function. Additionally, we propose a fully discrete scheme for the system and demonstrate the existence and uniqueness of its discrete solution, showing that it inherits essential structural-preserving properties. Finally, we conduct numerical experiments to compare the Ohta-Kawasaki system with the classical Cahn-Hilliard model, highlighting the impact of the repulsion parameter on the phase separation dynamics.

💡 Research Summary

This paper investigates the Ohta‑Kawasaki equation, a non‑local variant of the Cahn‑Hilliard model that describes microphase separation in diblock copolymers. The authors focus on two challenging extensions: (1) a degenerate mobility function m(φ)=¼(1−φ²)², which vanishes in the pure phases φ=±1, and (2) an external forcing term f(φ) that breaks the usual mass‑conservation property. Starting from the free‑energy functional
E(φ)=∫Ω