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

TL;DR

This paper analyzes the Ohta-Kawasaki equation with external forcing and degenerate mobility, establishing existence of solutions, proposing a numerical scheme, and comparing phase separation dynamics with the Cahn-Hilliard model.

Contribution

It provides the first rigorous analysis of the Ohta-Kawasaki equation with degenerate mobility and external force, along with a structure-preserving numerical scheme.

Findings

Existence of weak solutions under degenerate mobility

A fully discrete, structure-preserving numerical scheme

Numerical comparison showing the effect of the repulsion parameter

Abstract

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.