The nonlocal Stefan problem via a Martingale transport

• 成果类型: Article; Early Access
• 署名作者: Chu, Raymond; Kim, Inwon; Kim, Young-Heon; Nam, Kyeongsik
• 署名单位: University of California System; University of California Los Angeles; University of British Columbia; Korea Advanced Institute of Science & Technology (KAIST)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01374-9
• 发表日期: 2025
• 关键词: distributional solutions; free-boundary; REGULARITY

摘要

We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced in Kim and Kim (The Stefan problem and free targets of optimal Brownian martingale transport, 2021. arXiv preprint arXiv:2110.03831), we construct global-time weak solutions and give a probabilistic interpretation for the solutions. An important ingredient in our analysis is a probabilistic interpretation of the enthalpy and temperature variables in terms of a particle system. Our approach in particular establishes the connection between the parabolic obstacle problem and the Stefan problem for the nonlocal diffusions. For the melting problem, we show that our temperature-based solution coincides with the enthalpy-based ones studied in Athanasopoulos and Caffarelli (Adv Math 224(1):293-315, 2010), del Teso et al. (C R Math 355(11):1154-1160, 2017, Adv Math 305:78-143, 2017, Appl Sci 31(01):83-131, 2021), and obtain a new exponential convergence result.