Abstract
The aim of this paper is to study relaxation rates for the Cahn–Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish differential and algebraic relationships between the energy, the dissipation, and the squared \(\dot{H}^{-1}\) distance to a kink. This leads to a scale separation of the dynamics into two different stages: a first fast phase of the order \(t^{-\frac{1}{2}}\) where one sees convergence to some kink, followed by a slow relaxation phase with rate \(t^{-\frac{1}{4}}\) where convergence to the centered kink is observed.
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Acknowledgements
We warmly thank J. F. Babadjian, F. Cacciafesta and G. De Philippis for very useful discussions related to the local existence result Theorem 5.1. The hospitality of the Université Paris-Diderot and the Università di Roma “La Sapienza” where part of this research was done is gratefully acknowledged. In the early stage of this work, LDL was funded by the DFG Collaborative Research Center CRC 109 “Discretization in Geometry and Dynamics”. Part of this work was carried out while LDL was visiting Università di Roma “La Sapienza” and Università di Milano “Bicocca”, thanks to the program “Global Challenges for Women in Math Science”. MG was partially supported by the PGMO project COCA. MS was supported in the early stage of this work by the INdAM Fellowship in Mathematics for Experienced Researchers cofounded by Marie Curie actions. LDL and MS are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Communicated by Y. Giga.
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De Luca, L., Goldman, M. & Strani, M. A gradient flow approach to relaxation rates for the multi-dimensional Cahn–Hilliard equation. Math. Ann. 374, 2041–2081 (2019). https://doi.org/10.1007/s00208-018-1765-x
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DOI: https://doi.org/10.1007/s00208-018-1765-x