[{"title":"Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model","type":"journal_article","arxiv":1,"citation":{"ista":"Hurm C, Moser M. 2025. Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model. GAMM-Mitteilungen. 48(2), e70003.","ama":"Hurm C, Moser M. Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model. <i>GAMM-Mitteilungen</i>. 2025;48(2). doi:<a href=\"https://doi.org/10.1002/gamm.70003\">10.1002/gamm.70003</a>","chicago":"Hurm, Christoph, and Maximilian Moser. “Nonlocal‐to‐local Convergence for a Cahn–Hilliard Tumor Growth Model.” <i>GAMM-Mitteilungen</i>. Wiley, 2025. <a href=\"https://doi.org/10.1002/gamm.70003\">https://doi.org/10.1002/gamm.70003</a>.","ieee":"C. Hurm and M. Moser, “Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model,” <i>GAMM-Mitteilungen</i>, vol. 48, no. 2. Wiley, 2025.","mla":"Hurm, Christoph, and Maximilian Moser. “Nonlocal‐to‐local Convergence for a Cahn–Hilliard Tumor Growth Model.” <i>GAMM-Mitteilungen</i>, vol. 48, no. 2, e70003, Wiley, 2025, doi:<a href=\"https://doi.org/10.1002/gamm.70003\">10.1002/gamm.70003</a>.","apa":"Hurm, C., &#38; Moser, M. (2025). Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model. <i>GAMM-Mitteilungen</i>. Wiley. <a href=\"https://doi.org/10.1002/gamm.70003\">https://doi.org/10.1002/gamm.70003</a>","short":"C. Hurm, M. Moser, GAMM-Mitteilungen 48 (2025)."},"OA_place":"publisher","publisher":"Wiley","abstract":[{"lang":"eng","text":"We consider a local Cahn–Hilliard‐type model for tumor growth as well as a nonlocal model where, compared to the local system, the Laplacian in the equation for the chemical potential is replaced by a nonlocal operator. The latter is defined as a convolution integral with suitable kernels parametrized by a small parameter. For sufficiently smooth bounded domains in three dimensions, we prove convergence of weak solutions of the nonlocal model toward strong solutions of the local model together with convergence rates with respect to the small parameter. The proof is done via a Gronwall‐type argument and a convergence result with rates for the nonlocal integral operator toward the Laplacian due to Abels and Hurm."}],"quality_controlled":"1","department":[{"_id":"JuFi"}],"has_accepted_license":"1","date_published":"2025-06-01T00:00:00Z","month":"06","oa_version":"Published Version","acknowledgement":"C. Hurm was partially supported by the Graduiertenkolleg 2339 IntComSin of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project-ID 321821685. M. Moser has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 948819). The support is gratefully acknowledged. Finally, we thank Daniel Böhme and Jonas Stange for careful proofreading. Open Access funding enabled and organized by Projekt DEAL.","issue":"2","language":[{"iso":"eng"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_type":"original","date_updated":"2025-06-03T09:14:17Z","volume":48,"date_created":"2025-06-03T08:58:01Z","status":"public","doi":"10.1002/gamm.70003","_id":"19783","day":"01","author":[{"full_name":"Hurm, Christoph","last_name":"Hurm","first_name":"Christoph"},{"first_name":"Maximilian","last_name":"Moser","full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c"}],"ec_funded":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ddc":["510"],"publication_identifier":{"eissn":["1522-2608"],"issn":["0936-7195"]},"project":[{"name":"Bridging Scales in Random Materials","grant_number":"948819","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020"}],"license":"https://creativecommons.org/licenses/by/4.0/","file":[{"file_name":"2025_GAMM_Hurm.pdf","creator":"dernst","date_created":"2025-06-03T09:12:22Z","date_updated":"2025-06-03T09:12:22Z","access_level":"open_access","file_id":"19786","checksum":"6bac9d3e566b68519ae80ac8b0f41f20","file_size":513741,"success":1,"relation":"main_file","content_type":"application/pdf"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"arxiv":["2402.13790"]},"publication":"GAMM-Mitteilungen","scopus_import":"1","file_date_updated":"2025-06-03T09:12:22Z","OA_type":"hybrid","article_number":"e70003","intvolume":"        48","year":"2025","publication_status":"published"},{"external_id":{"pmid":["39239088"],"isi":["001305530600001"],"arxiv":["2311.02997"]},"article_processing_charge":"Yes (via OA deal)","file":[{"checksum":"98493a05b84e4513b6394dfad4851ddf","file_size":811131,"success":1,"relation":"main_file","content_type":"application/pdf","creator":"dernst","file_name":"2024_ArchiveRatAnalysis_Abels.pdf","date_created":"2024-09-09T08:43:32Z","date_updated":"2024-09-09T08:43:32Z","access_level":"open_access","file_id":"17938"}],"scopus_import":"1","publication":"Archive for Rational Mechanics and Analysis","article_number":"77","file_date_updated":"2024-09-09T08:43:32Z","publication_status":"published","year":"2024","intvolume":"       248","author":[{"full_name":"Abels, Helmut","first_name":"Helmut","last_name":"Abels"},{"first_name":"Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Moser","first_name":"Maximilian","full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c"}],"pmid":1,"day":"03","ec_funded":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ddc":["510"],"project":[{"grant_number":"948819","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020","name":"Bridging Scales in Random Materials"}],"publication_identifier":{"eissn":["1432-0673"],"issn":["0003-9527"]},"month":"09","issue":"5","acknowledgement":"J. Fischer and M. Moser have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819).\r\nOpen Access funding enabled and organized by Projekt DEAL.","oa_version":"Published Version","article_type":"original","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"status":"public","isi":1,"doi":"10.1007/s00205-024-02020-9","_id":"17887","volume":248,"date_updated":"2025-09-08T09:11:41Z","date_created":"2024-09-08T22:01:10Z","title":"Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system","type":"journal_article","arxiv":1,"citation":{"ama":"Abels H, Fischer JL, Moser M. Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. <i>Archive for Rational Mechanics and Analysis</i>. 2024;248(5). doi:<a href=\"https://doi.org/10.1007/s00205-024-02020-9\">10.1007/s00205-024-02020-9</a>","ista":"Abels H, Fischer JL, Moser M. 2024. Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. Archive for Rational Mechanics and Analysis. 248(5), 77.","ieee":"H. Abels, J. L. Fischer, and M. Moser, “Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system,” <i>Archive for Rational Mechanics and Analysis</i>, vol. 248, no. 5. Springer Nature, 2024.","apa":"Abels, H., Fischer, J. L., &#38; Moser, M. (2024). Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. <i>Archive for Rational Mechanics and Analysis</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00205-024-02020-9\">https://doi.org/10.1007/s00205-024-02020-9</a>","short":"H. Abels, J.L. Fischer, M. Moser, Archive for Rational Mechanics and Analysis 248 (2024).","mla":"Abels, Helmut, et al. “Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System.” <i>Archive for Rational Mechanics and Analysis</i>, vol. 248, no. 5, 77, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s00205-024-02020-9\">10.1007/s00205-024-02020-9</a>.","chicago":"Abels, Helmut, Julian L Fischer, and Maximilian Moser. “Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System.” <i>Archive for Rational Mechanics and Analysis</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00205-024-02020-9\">https://doi.org/10.1007/s00205-024-02020-9</a>."},"publisher":"Springer Nature","abstract":[{"lang":"eng","text":"We show convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility  mε>0  in the Allen-Cahn equation tends to zero in a subcritical way, i.e.,  mε=m0εβ  for some  β∈(0,2)  and  m0>0 . The proof proceeds by showing via a relative entropy argument that the solution to the Navier-Stokes/Allen-Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term  mεHΓt  in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow."}],"has_accepted_license":"1","date_published":"2024-09-03T00:00:00Z","quality_controlled":"1","department":[{"_id":"JuFi"}]},{"ddc":["510"],"project":[{"_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020","grant_number":"948819","name":"Bridging Scales in Random Materials"}],"publication_identifier":{"issn":["0944-2669"],"eissn":["1432-0835"]},"ec_funded":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"author":[{"full_name":"Abels, Helmut","first_name":"Helmut","last_name":"Abels"},{"full_name":"Fei, Mingwen","last_name":"Fei","first_name":"Mingwen"},{"full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","first_name":"Maximilian","last_name":"Moser"}],"day":"01","year":"2024","publication_status":"published","intvolume":"        63","article_number":"94","file_date_updated":"2024-04-23T07:30:48Z","scopus_import":"1","publication":"Calculus of Variations and Partial Differential Equations","external_id":{"arxiv":["2304.12096"],"isi":["001199418100002"]},"article_processing_charge":"Yes (via OA deal)","file":[{"access_level":"open_access","file_id":"15343","creator":"dernst","file_name":"2024_CalculusEquations_Abels.pdf","date_created":"2024-04-23T07:30:48Z","date_updated":"2024-04-23T07:30:48Z","relation":"main_file","content_type":"application/pdf","checksum":"b1095fad4cae596f52cc616a973bdde2","file_size":975186,"success":1}],"has_accepted_license":"1","date_published":"2024-05-01T00:00:00Z","quality_controlled":"1","department":[{"_id":"JuFi"}],"publisher":"Springer Nature","abstract":[{"text":"We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility mε=ε√, where the small parameter ε>0 related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable ε-scaled and coupled model problem. Moreover, we apply the novel idea of introducing ε-dependent coordinates.","lang":"eng"}],"arxiv":1,"citation":{"short":"H. Abels, M. Fei, M. Moser, Calculus of Variations and Partial Differential Equations 63 (2024).","mla":"Abels, Helmut, et al. “Sharp Interface Limit for a Navier–Stokes/Allen–Cahn System in the Case of a Vanishing Mobility.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 63, no. 4, 94, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s00526-024-02715-7\">10.1007/s00526-024-02715-7</a>.","ieee":"H. Abels, M. Fei, and M. Moser, “Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 63, no. 4. Springer Nature, 2024.","apa":"Abels, H., Fei, M., &#38; Moser, M. (2024). Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-024-02715-7\">https://doi.org/10.1007/s00526-024-02715-7</a>","chicago":"Abels, Helmut, Mingwen Fei, and Maximilian Moser. “Sharp Interface Limit for a Navier–Stokes/Allen–Cahn System in the Case of a Vanishing Mobility.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00526-024-02715-7\">https://doi.org/10.1007/s00526-024-02715-7</a>.","ama":"Abels H, Fei M, Moser M. Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility. <i>Calculus of Variations and Partial Differential Equations</i>. 2024;63(4). doi:<a href=\"https://doi.org/10.1007/s00526-024-02715-7\">10.1007/s00526-024-02715-7</a>","ista":"Abels H, Fei M, Moser M. 2024. Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility. Calculus of Variations and Partial Differential Equations. 63(4), 94."},"title":"Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility","type":"journal_article","status":"public","isi":1,"_id":"15334","doi":"10.1007/s00526-024-02715-7","volume":63,"date_updated":"2025-09-04T13:45:40Z","date_created":"2024-04-21T22:00:52Z","article_type":"original","language":[{"iso":"eng"}],"oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","issue":"4","acknowledgement":"Open Access funding enabled and organized by Projekt DEAL.\r\nM. Fei was partially supported by NSF of China under Grant No. 12271004 and Anhui Provincial Funding Project under Grant Nos. gxbjZD2022009 and 2308085J10. Moreover, M. Moser has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 948819).","oa_version":"Published Version","month":"05"},{"day":"02","author":[{"full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","last_name":"Moser","first_name":"Maximilian"}],"publication_identifier":{"eissn":["1875-8576"],"issn":["0921-7134"]},"article_processing_charge":"No","external_id":{"arxiv":["2105.07100"],"isi":["000927801300001"]},"publication":"Asymptotic Analysis","scopus_import":"1","corr_author":"1","intvolume":"       131","year":"2023","publication_status":"published","type":"journal_article","title":"Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result","citation":{"ama":"Moser M. Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result. <i>Asymptotic Analysis</i>. 2023;131(3-4):297-383. doi:<a href=\"https://doi.org/10.3233/asy-221775\">10.3233/asy-221775</a>","ista":"Moser M. 2023. Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result. Asymptotic Analysis. 131(3–4), 297–383.","apa":"Moser, M. (2023). Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result. <i>Asymptotic Analysis</i>. IOS Press. <a href=\"https://doi.org/10.3233/asy-221775\">https://doi.org/10.3233/asy-221775</a>","mla":"Moser, Maximilian. “Convergence of the Scalar- and Vector-Valued Allen–Cahn Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, Part I: Convergence Result.” <i>Asymptotic Analysis</i>, vol. 131, no. 3–4, IOS Press, 2023, pp. 297–383, doi:<a href=\"https://doi.org/10.3233/asy-221775\">10.3233/asy-221775</a>.","short":"M. Moser, Asymptotic Analysis 131 (2023) 297–383.","ieee":"M. Moser, “Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result,” <i>Asymptotic Analysis</i>, vol. 131, no. 3–4. IOS Press, pp. 297–383, 2023.","chicago":"Moser, Maximilian. “Convergence of the Scalar- and Vector-Valued Allen–Cahn Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, Part I: Convergence Result.” <i>Asymptotic Analysis</i>. IOS Press, 2023. <a href=\"https://doi.org/10.3233/asy-221775\">https://doi.org/10.3233/asy-221775</a>."},"arxiv":1,"abstract":[{"text":"We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse interface has developed and intersects the boundary ∂ Ω. The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [ 0 , T ] for some time T &gt; 0. Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).","lang":"eng"}],"publisher":"IOS Press","department":[{"_id":"JuFi"}],"quality_controlled":"1","date_published":"2023-02-02T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2105.07100"}],"page":"297-383","month":"02","acknowledgement":"The author gratefully acknowledges support through DFG, GRK 1692 “Curvature,\r\nCycles and Cohomology” during parts of the work.","oa_version":"Preprint","issue":"3-4","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"language":[{"iso":"eng"}],"keyword":["General Mathematics"],"article_type":"original","date_created":"2024-01-08T13:13:28Z","date_updated":"2025-09-09T14:14:55Z","volume":131,"_id":"14755","doi":"10.3233/asy-221775","isi":1,"status":"public"},{"citation":{"mla":"Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn Equation with Boundary Contact Energy: The Non-Perturbative Regime.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 6, 201, Springer Nature, 2022, doi:<a href=\"https://doi.org/10.1007/s00526-022-02307-3\">10.1007/s00526-022-02307-3</a>.","apa":"Hensel, S., &#38; Moser, M. (2022). Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-022-02307-3\">https://doi.org/10.1007/s00526-022-02307-3</a>","short":"S. Hensel, M. Moser, Calculus of Variations and Partial Differential Equations 61 (2022).","ieee":"S. Hensel and M. Moser, “Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 6. Springer Nature, 2022.","chicago":"Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn Equation with Boundary Contact Energy: The Non-Perturbative Regime.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00526-022-02307-3\">https://doi.org/10.1007/s00526-022-02307-3</a>.","ama":"Hensel S, Moser M. Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. <i>Calculus of Variations and Partial Differential Equations</i>. 2022;61(6). doi:<a href=\"https://doi.org/10.1007/s00526-022-02307-3\">10.1007/s00526-022-02307-3</a>","ista":"Hensel S, Moser M. 2022. Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. Calculus of Variations and Partial Differential Equations. 61(6), 201."},"type":"journal_article","title":"Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime","date_published":"2022-08-24T00:00:00Z","has_accepted_license":"1","department":[{"_id":"JuFi"}],"quality_controlled":"1","abstract":[{"text":"We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90∘. We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order ε12 for general contact angles α∈(0,π). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order ε; again for general contact angles α∈(0,π). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).","lang":"eng"}],"publisher":"Springer Nature","issue":"6","acknowledgement":"This Project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 948819)  , and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2047/1 - 390685813.\r\nOpen Access funding enabled and organized by Projekt DEAL.","oa_version":"Published Version","month":"08","doi":"10.1007/s00526-022-02307-3","_id":"12079","isi":1,"status":"public","date_created":"2022-09-11T22:01:54Z","volume":61,"date_updated":"2025-04-14T07:53:59Z","article_type":"original","oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","language":[{"iso":"eng"}],"author":[{"first_name":"Sebastian","orcid":"0000-0001-7252-8072","last_name":"Hensel","id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87","full_name":"Hensel, Sebastian"},{"id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","full_name":"Moser, Maximilian","last_name":"Moser","first_name":"Maximilian"}],"day":"24","publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"project":[{"name":"Bridging Scales in Random Materials","call_identifier":"H2020","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","grant_number":"948819"}],"ddc":["510"],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ec_funded":1,"scopus_import":"1","publication":"Calculus of Variations and Partial Differential Equations","article_processing_charge":"No","external_id":{"isi":["000844247300008"]},"file":[{"date_created":"2023-01-20T08:56:01Z","date_updated":"2023-01-20T08:56:01Z","creator":"dernst","file_name":"2022_Calculus_Hensel.pdf","file_id":"12320","access_level":"open_access","file_size":1278493,"success":1,"checksum":"b2da020ce50440080feedabeab5b09c4","content_type":"application/pdf","relation":"main_file"}],"year":"2022","publication_status":"published","intvolume":"        61","article_number":"201","file_date_updated":"2023-01-20T08:56:01Z"},{"year":"2022","publication_status":"published","intvolume":"        54","corr_author":"1","scopus_import":"1","publication":"SIAM Journal on Mathematical Analysis","external_id":{"isi":["000762768000004"],"arxiv":["2105.08434"]},"article_processing_charge":"No","publication_identifier":{"issn":["0036-1410"],"eissn":["1095-7154"]},"author":[{"first_name":"Helmut","last_name":"Abels","full_name":"Abels, Helmut"},{"full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","last_name":"Moser","first_name":"Maximilian"}],"day":"04","doi":"10.1137/21m1424925","_id":"12305","isi":1,"status":"public","date_created":"2023-01-16T10:07:00Z","date_updated":"2024-10-09T21:03:58Z","volume":54,"keyword":["Applied Mathematics","Computational Mathematics","Analysis"],"article_type":"original","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"language":[{"iso":"eng"}],"issue":"1","oa_version":"Preprint","month":"01","page":"114-172","date_published":"2022-01-04T00:00:00Z","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2105.08434","open_access":"1"}],"department":[{"_id":"JuFi"}],"quality_controlled":"1","abstract":[{"text":"This paper is concerned with the sharp interface limit for the Allen--Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Ω⊂\\R2. We assume that a diffuse interface already has developed and that it is in contact with the boundary ∂Ω. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant α-contact angle. For α close to 90° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen--Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.","lang":"eng"}],"publisher":"Society for Industrial and Applied Mathematics","citation":{"ista":"Abels H, Moser M. 2022. Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°. SIAM Journal on Mathematical Analysis. 54(1), 114–172.","ama":"Abels H, Moser M. Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°. <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(1):114-172. doi:<a href=\"https://doi.org/10.1137/21m1424925\">10.1137/21m1424925</a>","chicago":"Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2022. <a href=\"https://doi.org/10.1137/21m1424925\">https://doi.org/10.1137/21m1424925</a>.","short":"H. Abels, M. Moser, SIAM Journal on Mathematical Analysis 54 (2022) 114–172.","mla":"Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 1, Society for Industrial and Applied Mathematics, 2022, pp. 114–72, doi:<a href=\"https://doi.org/10.1137/21m1424925\">10.1137/21m1424925</a>.","apa":"Abels, H., &#38; Moser, M. (2022). Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/21m1424925\">https://doi.org/10.1137/21m1424925</a>","ieee":"H. Abels and M. Moser, “Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 1. Society for Industrial and Applied Mathematics, pp. 114–172, 2022."},"arxiv":1,"type":"journal_article","title":"Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°"}]