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.
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Marcio Gameiro was partially supported by the JSPS Postdoctoral Fellowship No. P08016 and by the JSPS Grant-in-Aid for Scientific Research No. 2008016, Ministry of Education, Science, Technology, Culture and Sports, Japan. Jean-Philippe Lessard was partially supported by NSF grant DMS-0511115, by DARPA, and by DOE grant DE-FG02-05ER25711.
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Gameiro, M., Lessard, JP. Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation. Numer. Math. 117, 753–778 (2011). https://doi.org/10.1007/s00211-010-0350-3
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DOI: https://doi.org/10.1007/s00211-010-0350-3