Rigorous numerics for the Cahn-Hilliard equation on the unit square

42. Stanislaus Maier-Paape, Ulrich Miller, Konstantin Mischaikow, Thomas Wanner:
    Rigorous numerics for the Cahn-Hilliard equation on the unit square
    Revista Matematica Complutense 21(2), pp. 351-426, 2008.

Abstract

While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computer-assisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Zgliczynski and Mischaikow (2001), and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.

Links

The published version of the paper can be found at https://doi.org/10.5209/rev_REMA.2008.v21.n2.16380.

Bibtex

@article{maier:etal:08a,
   author = {Stanislaus Maier-Paape and Ulrich Miller and
             Konstantin Mischaikow and Thomas Wanner},
   title = {Rigorous numerics for the {C}ahn-{H}illiard equation on
            the unit square},
   journal = {Revista Matematica Complutense},
   year = 2008,
   volume = 21,
   number = 2,
   pages = {351--426},
   doi = {10.5209/rev_REMA.2008.v21.n2.16380},
   url = {http://www.mat.ucm.es/serv/revmat/vol21-2/vol21-2d.html}
   }