Nonconvex and random effects in complex energy landscapes
Final Report Abstract
With the help of this grant support, we focused on developing and applying new tools for the analysis of complex energy landscapes, both in terms of quantifying the shape of these landscapes and in deducing quantitative information about the dynamics of the associated gradient flow. Work in progress (with Grafke, Scholtes, Vanden-Eijnden, and Wagner) combines numerical and analytical tools to further analyze saddle points of the Cahn-Hilliard energy for mean value close to one. This is part of a body of work in which we combine information about Γ-limits and mountain-pass type arguments to establish existence and fine properties of critical points. The application of the String Method serves both to confirm our intuition and guide the way to new results. Optimal relaxation rates for gradient flows with respect to nonconvex energies is a largely open field. We show how the HED framework developed can be applied to the Mullins-Sekerka evolution in R2. As far as we know, this is the first result of its kind. The framework has also recently been applied to derive new results for the thin-film and Cahn-Hilliard equations. The relaxation framework developed is also interesting in order to better understand metastability. Our work shows how the information gleaned from the tools introduced can be combined to establish metastability of the 1-d Cahn-Hilliard equation on compact intervals. Multiple timescales of the evolution are identified and quantified. A multi-pronged approach of this sort should not be necessary for the 1-d Allen-Cahn equation, even on the line. Confirming this fact requires extending some of the tools; this is carried out. Despite the usefulness of the HED framework, it turns out that an L1-type relaxation framework is necessary and possible to achieve. This new tool is developed and applied there for a new result on the 1-d Cahn-Hilliard equation on the line. Work in progress studies this framework for a different kind of long-time limit, which is a necessary next step for an optimal metastability result. In addition, work in progress (partially supported by this grant but not yet complete) extends the L1-type framework to the Stefan problem in R2.
Publications
- Metastability of the Cahn-Hilliard equation in one space dimension, J. Differential Equations 265 (2018), no. 4, 1528–1575
Sebastian Scholtes and Maria G. Westdickenberg
(See online at https://doi.org/10.1016/j.jde.2018.04.012) - Optimal L1-type relaxation rates for the Cahn-Hilliard equation on the line, SIAM J. Math. Anal. 51 (2019), no. 6, 4645–4682
Felix Otto, Sebastian Scholtes, and Maria G. Westdickenberg
(See online at https://doi.org/10.1137/18M1192640) - Relaxation to a planar interface in the Mullins-Sekerka problem, Interfaces Free Bound. 21 (2019), no. 1, 21–40
Olga Chugreeva, Felix Otto, and Maria G. Westdickenberg
(See online at https://doi.org/10.4171/IFB/415) - On the metastability of the 1-d Allen-Cahn equation, Journal of Dynamics and Differential Equations, posted on 2020
Maria G. Westdickenberg
(See online at https://doi.org/10.1007/s10884-020-09874-z)
