2002/09/17 by Th. Gallay, Gallay, Th., A. Mielke +2
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #35Q99 (Primary) 35C05 #74H40 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Solidification and crystal growth phenomena #Theoretical and Computational Physics #math.AP #msc:35C05 #msc:35Q99 #msc:74H40
paper · pdf · doi:10.48550/arxiv.math/0209208
34 pages, 2 figures
arxiv created 2002/09/17 · openalex publication_date 2002/09/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a coarsening model describing the dynamics of interfaces in the one-dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in constantly eliminating the shortest interval of the partition by merging it with its two neighbors. We show that the mean-field equation for the time-dependent distribution of interval lengths can be explicitly solved using a global linearization transformation. This allows us to derive rigorous results on the long-time asymptotics of the solutions. If the average length of the intervals is finite, we prove that all distributions approach a uniquely determined self-similar solution. We also obtain global stability results for the family of self-similar profiles which correspond to distributions with infinite expectation.