Equilibrium Validation in Models for Pattern Formation Based on Sobolev Embeddings

TL;DR

This paper develops a computer-assisted proof method to rigorously validate equilibrium branches in PDE models, specifically applied to the Ohta-Kawasaki diblock copolymer model across multiple dimensions.

Contribution

It extends previous techniques to validate equilibrium solutions for the Ohta-Kawasaki model, providing a detailed analytical framework applicable to other PDEs.

Findings

Validated existence and isolation of equilibrium branches in the Ohta-Kawasaki model

Applicable to 1D, 2D, and 3D cases

Method can be generalized to other parabolic PDEs

Abstract

In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.