Equilibrium validation in models for pattern formation based on Sobolev embeddings

71. Evelyn Sander, Thomas Wanner:
    Equilibrium validation in models for pattern formation based on Sobolev embeddings
    Discrete and Continuous Dynamical Systems, Series B 26(1), pp. 603-632, 2021.

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.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/2005.14224, while the published version of the paper can be found at https://doi.org/10.3934/dcdsb.2020260.

Bibtex

@article{sander:wanner:21a,
   author = {Evelyn Sander and Thomas Wanner},
   title = {Equilibrium validation in models for pattern formation based
            on {S}obolev embeddings},
   journal = {Discrete and Continuous Dynamical Systems, Series B},
   volume = {26},
   number = {1},
   year = {2021},
   pages = {603--632},
   doi = {10.3934/dcdsb.2020260}
   }