{"file_date_updated":"2024-09-09T07:46:42Z","file":[{"success":1,"creator":"dernst","file_id":"17923","date_created":"2024-09-09T07:46:42Z","relation":"main_file","checksum":"b5ad02d9abd5b4701269cd1ad0a1cc8f","date_updated":"2024-09-09T07:46:42Z","access_level":"open_access","file_name":"2024_AnnInstHPoincare_Fischer.pdf","content_type":"application/pdf","file_size":1348896}],"publication":"Annales de l'Institut Henri Poincare C","license":"https://creativecommons.org/licenses/by/4.0/","oa_version":"Published Version","scopus_import":"1","month":"01","intvolume":"        41","page":"1117-1178","ddc":["510"],"volume":41,"article_processing_charge":"Yes","doi":"10.4171/AIHPC/109","ec_funded":1,"language":[{"iso":"eng"}],"quality_controlled":"1","day":"24","has_accepted_license":"1","date_created":"2024-09-01T22:01:09Z","_id":"17481","year":"2024","date_published":"2024-01-24T00:00:00Z","oa":1,"publisher":"EMS Press","article_type":"original","publication_status":"published","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"14597"}]},"issue":"5","author":[{"orcid":"0000-0002-0479-558X","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","full_name":"Fischer, Julian L","first_name":"Julian L","last_name":"Fischer"},{"last_name":"Marveggio","first_name":"Alice","full_name":"Marveggio, Alice","id":"25647992-AA84-11E9-9D75-8427E6697425"}],"department":[{"_id":"JuFi"}],"citation":{"mla":"Fischer, Julian L., and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen–Cahn Equation towards Multiphase Mean Curvature Flow.” <i>Annales de l’Institut Henri Poincare C</i>, vol. 41, no. 5, EMS Press, 2024, pp. 1117–78, doi:<a href=\"https://doi.org/10.4171/AIHPC/109\">10.4171/AIHPC/109</a>.","ista":"Fischer JL, Marveggio A. 2024. Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. Annales de l’Institut Henri Poincare C. 41(5), 1117–1178.","apa":"Fischer, J. L., &#38; Marveggio, A. (2024). Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. <i>Annales de l’Institut Henri Poincare C</i>. EMS Press. <a href=\"https://doi.org/10.4171/AIHPC/109\">https://doi.org/10.4171/AIHPC/109</a>","ieee":"J. L. Fischer and A. Marveggio, “Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow,” <i>Annales de l’Institut Henri Poincare C</i>, vol. 41, no. 5. EMS Press, pp. 1117–1178, 2024.","short":"J.L. Fischer, A. Marveggio, Annales de l’Institut Henri Poincare C 41 (2024) 1117–1178.","chicago":"Fischer, Julian L, and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen–Cahn Equation towards Multiphase Mean Curvature Flow.” <i>Annales de l’Institut Henri Poincare C</i>. EMS Press, 2024. <a href=\"https://doi.org/10.4171/AIHPC/109\">https://doi.org/10.4171/AIHPC/109</a>.","ama":"Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. <i>Annales de l’Institut Henri Poincare C</i>. 2024;41(5):1117-1178. doi:<a href=\"https://doi.org/10.4171/AIHPC/109\">10.4171/AIHPC/109</a>"},"publication_identifier":{"eissn":["1873-1430"],"issn":["0294-1449"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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. In 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(ε \r\n1/2\r\n ). Our 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."}],"title":"Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow","date_updated":"2024-09-09T07:48:43Z","project":[{"call_identifier":"H2020","name":"Bridging Scales in Random Materials","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","grant_number":"948819"}],"acknowledgement":"The authors thank Sebastian Hensel for useful and helpful commentson the first draft of this work.\r\nThis project has received funding from the European Research Council (ERC)\r\nunder the European Union’s Horizon 2020 research and innovation programme (grant\r\nagreement no. 948819.","corr_author":"1","status":"public","type":"journal_article"}