The Allen-Cahn equation with generic initial datum

• 成果类型: Article
• 署名作者: Hairer, Martin; Le, Khoa; Rosati, Tommaso
• 署名单位: Imperial College London; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Technical University of Berlin; University of Leeds; University of Warwick
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01198-5
• 发表日期: 2023
• 页码: 957-998
• 关键词: motion

摘要

We consider the Allen-Cahn equation ?(t)u - ?u = u - u(3) with a rapidly mixing Gaussian field as initial condition. We show that provided that the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to mean curvature flow.