Numerische Mathematik

, Volume 117, Issue 4, pp 753–778 | Cite as

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

Article

Abstract

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.

Mathematics Subject Classification (2010)

65N15 37M20 35K55 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.BCAM, Basque Center for Applied MathematicsBizkaia Technology ParkDerio, BizkaiaSpain
  3. 3.Department of MathematicsRutgers UniversityPiscatawayUSA

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