{"status":"public","abstract":[{"lang":"eng","text":"Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N≥3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow.\r\nIn the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε↘0. We even establish the rate of convergence O(ε1/2).\r\nOur approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of \"gradient flow calibrations\" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times."}],"_id":"14597","day":"31","main_file_link":[{"url":"https://arxiv.org/abs/2203.17143","open_access":"1"}],"publication":"arXiv","author":[{"orcid":"0000-0002-0479-558X","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","last_name":"Fischer","full_name":"Fischer, Julian L","first_name":"Julian L"},{"id":"25647992-AA84-11E9-9D75-8427E6697425","first_name":"Alice","last_name":"Marveggio","full_name":"Marveggio, Alice"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","type":"preprint","department":[{"_id":"JuFi"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"14587"}]},"date_created":"2023-11-23T09:30:02Z","date_updated":"2023-11-30T13:25:02Z","title":"Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow","oa":1,"year":"2022","language":[{"iso":"eng"}],"oa_version":"Preprint","publication_status":"submitted","external_id":{"arxiv":["2203.17143"]},"date_published":"2022-03-31T00:00:00Z","project":[{"name":"Bridging Scales in Random Materials","call_identifier":"H2020","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","grant_number":"948819"}],"article_processing_charge":"No","citation":{"apa":"Fischer, J. L., &#38; Marveggio, A. (n.d.). Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2203.17143\">https://doi.org/10.48550/ARXIV.2203.17143</a>","ista":"Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. arXiv, <a href=\"https://doi.org/10.48550/ARXIV.2203.17143\">10.48550/ARXIV.2203.17143</a>.","short":"J.L. Fischer, A. Marveggio, ArXiv (n.d.).","mla":"Fischer, Julian L., and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen-Cahn Equation towards Multiphase Mean Curvature Flow.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/ARXIV.2203.17143\">10.48550/ARXIV.2203.17143</a>.","chicago":"Fischer, Julian L, and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen-Cahn Equation towards Multiphase Mean Curvature Flow.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2203.17143\">https://doi.org/10.48550/ARXIV.2203.17143</a>.","ama":"Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2203.17143\">10.48550/ARXIV.2203.17143</a>","ieee":"J. L. Fischer and A. Marveggio, “Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow,” <i>arXiv</i>. ."},"ec_funded":1,"doi":"10.48550/ARXIV.2203.17143","month":"03"}