[{"date_created":"2026-05-20T08:25:25Z","type":"journal_article","ddc":["510"],"file":[{"file_id":"21906","file_name":"2026_AnnalsProbability_Cornalba.pdf","checksum":"3e60c0e25a1c96342029a7d2b031505f","success":1,"relation":"main_file","file_size":865745,"access_level":"open_access","content_type":"application/pdf","date_updated":"2026-05-21T07:11:27Z","date_created":"2026-05-21T07:11:27Z","creator":"dernst"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"abstract":[{"lang":"eng","text":"The Dean–Kawasaki equation—one of the most fundamental SPDEs of\r\nfluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly\r\nsingular and fails to be renormalizable, even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is\r\nnatural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields\r\nwell-posed equations that should still precisely approximate the law of the\r\nparticle density fluctuations.\r\nIn the present work, we prove that a regularization in the form of a formal\r\ndiscretization of the Dean–Kawasaki equation indeed accurately describes\r\ndensity fluctuations in systems of weakly interacting diffusing particles: We\r\nshow that, in suitable weak metrics, the law of fluctuations as predicted by\r\nthe discretized Dean–Kawasaki SPDE approximates the law of fluctuations\r\nof the original particle system, up to an error that is of arbitrarily high order in\r\nthe inverse particle number and a discretization error. In particular, the Dean–\r\nKawasaki equation provides a means for efficient and accurate simulations of\r\ndensity fluctuations in weakly interacting particle systems."}],"publication_status":"published","article_type":"original","publication":"The Annals of Probability","keyword":["Weakly interacting particle systems","fluctuating hydrodynamics","Dean-Kawasaki equation","stochastic PDEs","numerical approximation"],"volume":54,"date_updated":"2026-05-21T07:21:25Z","_id":"21894","APC_amount":"1352,08 EUR","citation":{"ama":"Cornalba F, Fischer JL, Ingmanns J, Raithel C. Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. The Annals of Probability. 2026;54(1):155-215. doi:10.1214/25-aop1763","chicago":"Cornalba, Federico, Julian L Fischer, Jonas Ingmanns, and Claudia Raithel. “Density Fluctuations in Weakly Interacting Particle Systems via the Dean–Kawasaki Equation.” The Annals of Probability. Institute of Mathematical Statistics, 2026. https://doi.org/10.1214/25-aop1763.","apa":"Cornalba, F., Fischer, J. L., Ingmanns, J., & Raithel, C. (2026). Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. The Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/25-aop1763","ieee":"F. Cornalba, J. L. Fischer, J. Ingmanns, and C. Raithel, “Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation,” The Annals of Probability, vol. 54, no. 1. Institute of Mathematical Statistics, pp. 155–215, 2026.","short":"F. Cornalba, J.L. Fischer, J. Ingmanns, C. Raithel, The Annals of Probability 54 (2026) 155–215.","ista":"Cornalba F, Fischer JL, Ingmanns J, Raithel C. 2026. Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. The Annals of Probability. 54(1), 155–215.","mla":"Cornalba, Federico, et al. “Density Fluctuations in Weakly Interacting Particle Systems via the Dean–Kawasaki Equation.” The Annals of Probability, vol. 54, no. 1, Institute of Mathematical Statistics, 2026, pp. 155–215, doi:10.1214/25-aop1763."},"publication_identifier":{"eissn":["2168-894X"],"issn":["0091-1798"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Institute of Mathematical Statistics","title":"Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation","ec_funded":1,"license":"https://creativecommons.org/licenses/by/4.0/","year":"2026","intvolume":" 54","author":[{"first_name":"Federico","last_name":"Cornalba","full_name":"Cornalba, Federico"},{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0479-558X","last_name":"Fischer","full_name":"Fischer, Julian L","first_name":"Julian L"},{"id":"71523d30-15b2-11ec-abd3-f80aa909d6b0","first_name":"Jonas","orcid":"0009-0008-1310-7946","last_name":"Ingmanns","full_name":"Ingmanns, Jonas"},{"last_name":"Raithel","full_name":"Raithel, Claudia","first_name":"Claudia"}],"corr_author":"1","acknowledgement":"All authors gratefully acknowledge funding from the Austrian Science Fund (FWF) through the project F65. CR gratefully acknowledges support from the Austrian Science Fund (FWF), grants P30000, P33010, W1245. FC gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411.","page":"155-215","arxiv":1,"oa_version":"Published Version","department":[{"_id":"JuFi"}],"quality_controlled":"1","status":"public","external_id":{"arxiv":["2303.00429"]},"project":[{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"},{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"}],"date_published":"2026-01-01T00:00:00Z","oa":1,"OA_type":"hybrid","has_accepted_license":"1","issue":"1","OA_place":"publisher","article_processing_charge":"Yes (in subscription journal)","language":[{"iso":"eng"}],"scopus_import":"1","day":"01","file_date_updated":"2026-05-21T07:11:27Z","PlanS_conform":"1","month":"01","doi":"10.1214/25-aop1763"},{"abstract":[{"text":"Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular.","lang":"eng"}],"article_type":"original","publication_status":"published","publication":"SIAM Journal on Numerical Analysis","volume":63,"date_created":"2025-02-16T23:02:34Z","type":"journal_article","file":[{"access_level":"open_access","file_size":2435019,"relation":"main_file","success":1,"checksum":"53505647e848ed50f7e0d00c369b14e7","file_name":"2025_SIAMNumerAnaly_Cornalba.pdf","file_id":"19029","creator":"dernst","date_updated":"2025-02-17T08:32:23Z","date_created":"2025-02-17T08:32:23Z","content_type":"application/pdf"}],"ddc":["510"],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"publisher":"Society for Industrial and Applied Mathematics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics","year":"2025","date_updated":"2025-09-30T10:30:31Z","_id":"19027","citation":{"chicago":"Cornalba, Federico, and Julian L Fischer. “Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics, 2025. https://doi.org/10.1137/23M1617345.","apa":"Cornalba, F., & Fischer, J. L. (2025). Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/23M1617345","short":"F. Cornalba, J.L. Fischer, SIAM Journal on Numerical Analysis 63 (2025) 262–287.","ieee":"F. Cornalba and J. L. Fischer, “Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics,” SIAM Journal on Numerical Analysis, vol. 63, no. 1. Society for Industrial and Applied Mathematics, pp. 262–287, 2025.","ista":"Cornalba F, Fischer JL. 2025. Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. 63(1), 262–287.","mla":"Cornalba, Federico, and Julian L. Fischer. “Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” SIAM Journal on Numerical Analysis, vol. 63, no. 1, Society for Industrial and Applied Mathematics, 2025, pp. 262–87, doi:10.1137/23M1617345.","ama":"Cornalba F, Fischer JL. Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. 2025;63(1):262-287. doi:10.1137/23M1617345"},"publication_identifier":{"eissn":["1095-7170"],"issn":["0036-1429"]},"department":[{"_id":"JuFi"}],"oa_version":"Published Version","quality_controlled":"1","status":"public","external_id":{"isi":["001447583400011"],"arxiv":["2311.08872"]},"project":[{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"date_published":"2025-02-01T00:00:00Z","OA_type":"hybrid","oa":1,"isi":1,"intvolume":" 63","author":[{"first_name":"Federico","orcid":"0000-0002-6269-5149","full_name":"Cornalba, Federico","last_name":"Cornalba","id":"2CEB641C-A400-11E9-A717-D712E6697425"},{"last_name":"Fischer","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"}],"acknowledgement":"The work of the authors was supported by the Austrian Science Fund (FWF) projectF65.","corr_author":"1","arxiv":1,"page":"262-287","file_date_updated":"2025-02-17T08:32:23Z","month":"02","doi":"10.1137/23M1617345","has_accepted_license":"1","issue":"1","OA_place":"publisher","language":[{"iso":"eng"}],"article_processing_charge":"Yes (in subscription journal)","scopus_import":"1","day":"01"},{"month":"09","file_date_updated":"2024-09-09T08:43:32Z","doi":"10.1007/s00205-024-02020-9","issue":"5","has_accepted_license":"1","day":"03","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","scopus_import":"1","oa_version":"Published Version","department":[{"_id":"JuFi"}],"status":"public","quality_controlled":"1","oa":1,"project":[{"_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","name":"Bridging Scales in Random Materials","call_identifier":"H2020","grant_number":"948819"}],"external_id":{"isi":["001305530600001"],"pmid":["39239088"],"arxiv":["2311.02997"]},"date_published":"2024-09-03T00:00:00Z","isi":1,"intvolume":" 248","arxiv":1,"author":[{"last_name":"Abels","full_name":"Abels, Helmut","first_name":"Helmut"},{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X"},{"id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","first_name":"Maximilian","full_name":"Moser, Maximilian","last_name":"Moser"}],"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.","publisher":"Springer Nature","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","ec_funded":1,"title":"Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system","year":"2024","_id":"17887","date_updated":"2025-09-08T09:11:41Z","pmid":1,"publication_identifier":{"issn":["0003-9527"],"eissn":["1432-0673"]},"citation":{"apa":"Abels, H., Fischer, J. L., & 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. Springer Nature. https://doi.org/10.1007/s00205-024-02020-9","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.” Archive for Rational Mechanics and Analysis. Springer Nature, 2024. https://doi.org/10.1007/s00205-024-02020-9.","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.","mla":"Abels, Helmut, et al. “Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System.” Archive for Rational Mechanics and Analysis, vol. 248, no. 5, 77, Springer Nature, 2024, doi:10.1007/s00205-024-02020-9.","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,” Archive for Rational Mechanics and Analysis, vol. 248, no. 5. Springer Nature, 2024.","short":"H. Abels, J.L. Fischer, M. Moser, Archive for Rational Mechanics and Analysis 248 (2024).","ama":"Abels H, Fischer JL, Moser M. Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. Archive for Rational Mechanics and Analysis. 2024;248(5). doi:10.1007/s00205-024-02020-9"},"publication_status":"published","article_type":"original","article_number":"77","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."}],"volume":248,"publication":"Archive for Rational Mechanics and Analysis","type":"journal_article","date_created":"2024-09-08T22:01:10Z","file":[{"content_type":"application/pdf","date_updated":"2024-09-09T08:43:32Z","date_created":"2024-09-09T08:43:32Z","creator":"dernst","file_id":"17938","file_name":"2024_ArchiveRatAnalysis_Abels.pdf","checksum":"98493a05b84e4513b6394dfad4851ddf","relation":"main_file","success":1,"file_size":811131,"access_level":"open_access"}],"ddc":["510"],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"}},{"has_accepted_license":"1","issue":"5","article_processing_charge":"Yes","language":[{"iso":"eng"}],"scopus_import":"1","day":"24","file_date_updated":"2024-09-09T07:46:42Z","month":"01","doi":"10.4171/AIHPC/109","isi":1,"intvolume":" 41","author":[{"full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Marveggio, Alice","last_name":"Marveggio","first_name":"Alice","id":"25647992-AA84-11E9-9D75-8427E6697425"}],"corr_author":"1","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.","page":"1117-1178","oa_version":"Published Version","department":[{"_id":"JuFi"}],"status":"public","quality_controlled":"1","project":[{"grant_number":"948819","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020","name":"Bridging Scales in Random Materials"}],"external_id":{"isi":["001293853900003"]},"date_published":"2024-01-24T00:00:00Z","oa":1,"date_updated":"2026-08-12T09:15:49Z","_id":"17481","citation":{"ama":"Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. Annales de l’Institut Henri Poincaré C. 2024;41(5):1117-1178. doi:10.4171/AIHPC/109","apa":"Fischer, J. L., & Marveggio, A. (2024). Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. Annales de l’Institut Henri Poincaré C. EMS Press. https://doi.org/10.4171/AIHPC/109","chicago":"Fischer, Julian L, and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen–Cahn Equation towards Multiphase Mean Curvature Flow.” Annales de l’Institut Henri Poincaré C. EMS Press, 2024. https://doi.org/10.4171/AIHPC/109.","mla":"Fischer, Julian L., and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen–Cahn Equation towards Multiphase Mean Curvature Flow.” Annales de l’Institut Henri Poincaré C, vol. 41, no. 5, EMS Press, 2024, pp. 1117–78, doi:10.4171/AIHPC/109.","ista":"Fischer JL, Marveggio A. 2024. Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. Annales de l’Institut Henri Poincaré C. 41(5), 1117–1178.","short":"J.L. Fischer, A. Marveggio, Annales de l’Institut Henri Poincaré C 41 (2024) 1117–1178.","ieee":"J. L. Fischer and A. Marveggio, “Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow,” Annales de l’Institut Henri Poincaré C, vol. 41, no. 5. EMS Press, pp. 1117–1178, 2024."},"publication_identifier":{"issn":["0294-1449"],"eissn":["1873-1430"]},"publisher":"EMS Press","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ec_funded":1,"title":"Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow","year":"2024","date_created":"2024-09-01T22:01:09Z","type":"journal_article","ddc":["510"],"file":[{"creator":"dernst","date_updated":"2024-09-09T07:46:42Z","date_created":"2024-09-09T07:46:42Z","content_type":"application/pdf","access_level":"open_access","relation":"main_file","success":1,"file_size":1348896,"file_name":"2024_AnnInstHPoincare_Fischer.pdf","checksum":"b5ad02d9abd5b4701269cd1ad0a1cc8f","file_id":"17923"}],"related_material":{"record":[{"relation":"earlier_version","id":"14597","status":"public"}]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"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."}],"article_type":"original","publication_status":"published","publication":"Annales de l'Institut Henri Poincaré C","volume":41},{"author":[{"first_name":"Marcello","full_name":"Carioni, Marcello","last_name":"Carioni"},{"first_name":"Julian L","full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Schlömerkemper, Anja","last_name":"Schlömerkemper","first_name":"Anja"}],"corr_author":"1","page":"217-247","arxiv":1,"isi":1,"intvolume":" 30","external_id":{"isi":["001115503400013"],"arxiv":["1811.09857"]},"date_published":"2023-01-01T00:00:00Z","oa":1,"oa_version":"Preprint","department":[{"_id":"JuFi"}],"status":"public","quality_controlled":"1","article_processing_charge":"No","language":[{"iso":"eng"}],"scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1811.09857","open_access":"1"}],"day":"01","issue":"1","month":"01","date_created":"2023-12-10T23:00:59Z","type":"journal_article","publication":"Journal of Convex Analysis","volume":30,"abstract":[{"text":"This paper is concerned with equilibrium configurations of one-dimensional particle systems with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a Γ-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates and thus may model dead as well as live loads. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order Γ-limit in the presence of external forces: instead of comparing a configuration of n atoms to a global minimizer of the Γ-limit, we compare the configuration to a minimizer in some subclass of functions which in some sense are \"close to\" the configuration. The paper is complemented with the study of the minimizers of the Γ-limit.","lang":"eng"}],"publication_status":"published","article_type":"original","citation":{"mla":"Carioni, Marcello, et al. “External Forces in the Continuum Limit of Discrete Systems with Non-Convex Interaction Potentials: Compactness for a Γ-Development.” Journal of Convex Analysis, vol. 30, no. 1, Heldermann Verlag, 2023, pp. 217–47.","ista":"Carioni M, Fischer JL, Schlömerkemper A. 2023. External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development. Journal of Convex Analysis. 30(1), 217–247.","short":"M. Carioni, J.L. Fischer, A. Schlömerkemper, Journal of Convex Analysis 30 (2023) 217–247.","ieee":"M. Carioni, J. L. Fischer, and A. Schlömerkemper, “External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development,” Journal of Convex Analysis, vol. 30, no. 1. Heldermann Verlag, pp. 217–247, 2023.","apa":"Carioni, M., Fischer, J. L., & Schlömerkemper, A. (2023). External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development. Journal of Convex Analysis. Heldermann Verlag.","chicago":"Carioni, Marcello, Julian L Fischer, and Anja Schlömerkemper. “External Forces in the Continuum Limit of Discrete Systems with Non-Convex Interaction Potentials: Compactness for a Γ-Development.” Journal of Convex Analysis. Heldermann Verlag, 2023.","ama":"Carioni M, Fischer JL, Schlömerkemper A. External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development. Journal of Convex Analysis. 2023;30(1):217-247."},"publication_identifier":{"eissn":["2363-6394"],"issn":["0944-6532"]},"date_updated":"2024-10-09T21:07:35Z","_id":"14661","year":"2023","publisher":"Heldermann Verlag","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development"},{"type":"journal_article","date_created":"2021-12-16T12:15:35Z","ddc":["510"],"file":[{"file_name":"2023_JourNonlinearScience_Fellner.pdf","checksum":"f3f0f0886098e31c81116cff8183750b","file_id":"13149","access_level":"open_access","relation":"main_file","success":1,"file_size":742315,"date_created":"2023-06-19T07:33:53Z","date_updated":"2023-06-19T07:33:53Z","content_type":"application/pdf","creator":"dernst"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"publication_status":"published","article_type":"original","article_number":"66","abstract":[{"text":"The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.","lang":"eng"}],"volume":33,"publication":"Journal of Nonlinear Science","_id":"10550","date_updated":"2023-08-01T14:40:33Z","publication_identifier":{"issn":["0938-8974"],"eissn":["1432-1467"]},"citation":{"ama":"Fellner K, Fischer JL, Kniely M, Tang BQ. Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. Journal of Nonlinear Science. 2023;33. doi:10.1007/s00332-023-09926-w","apa":"Fellner, K., Fischer, J. L., Kniely, M., & Tang, B. Q. (2023). Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. Journal of Nonlinear Science. Springer Nature. https://doi.org/10.1007/s00332-023-09926-w","chicago":"Fellner, Klemens, Julian L Fischer, Michael Kniely, and Bao Quoc Tang. “Global Renormalised Solutions and Equilibration of Reaction-Diffusion Systems with Non-Linear Diffusion.” Journal of Nonlinear Science. Springer Nature, 2023. https://doi.org/10.1007/s00332-023-09926-w.","mla":"Fellner, Klemens, et al. “Global Renormalised Solutions and Equilibration of Reaction-Diffusion Systems with Non-Linear Diffusion.” Journal of Nonlinear Science, vol. 33, 66, Springer Nature, 2023, doi:10.1007/s00332-023-09926-w.","ista":"Fellner K, Fischer JL, Kniely M, Tang BQ. 2023. Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. Journal of Nonlinear Science. 33, 66.","ieee":"K. Fellner, J. L. Fischer, M. Kniely, and B. Q. Tang, “Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion,” Journal of Nonlinear Science, vol. 33. Springer Nature, 2023.","short":"K. Fellner, J.L. Fischer, M. Kniely, B.Q. Tang, Journal of Nonlinear Science 33 (2023)."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publisher":"Springer Nature","title":"Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion","year":"2023","isi":1,"intvolume":" 33","arxiv":1,"author":[{"first_name":"Klemens","full_name":"Fellner, Klemens","last_name":"Fellner"},{"first_name":"Julian L","orcid":"0000-0002-0479-558X","last_name":"Fischer","full_name":"Fischer, Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"id":"2CA2C08C-F248-11E8-B48F-1D18A9856A87","first_name":"Michael","full_name":"Kniely, Michael","last_name":"Kniely","orcid":"0000-0001-5645-4333"},{"first_name":"Bao Quoc","full_name":"Tang, Bao Quoc","last_name":"Tang"}],"acknowledgement":"We thank the referees for their valuable comments and suggestions. A major part of this work was carried out when B. Q. Tang visited the Institute of Science and Technology Austria (ISTA). The hospitality of ISTA is greatly acknowledged. This work was partially supported by NAWI Graz.\r\nOpen access funding provided by University of Graz.","department":[{"_id":"JuFi"}],"oa_version":"Published Version","quality_controlled":"1","status":"public","oa":1,"external_id":{"isi":["001002343400002"],"arxiv":["2109.12019"]},"date_published":"2023-06-07T00:00:00Z","has_accepted_license":"1","day":"07","article_processing_charge":"No","language":[{"iso":"eng"}],"scopus_import":"1","month":"06","file_date_updated":"2023-06-19T07:33:53Z","doi":"10.1007/s00332-023-09926-w"},{"year":"2023","ec_funded":1,"title":"The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles","publisher":"Springer Nature","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"eissn":["1432-0673"],"issn":["0003-9527"]},"citation":{"ieee":"F. Cornalba and J. L. Fischer, “The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles,” Archive for Rational Mechanics and Analysis, vol. 247, no. 5. Springer Nature, 2023.","short":"F. Cornalba, J.L. Fischer, Archive for Rational Mechanics and Analysis 247 (2023).","ista":"Cornalba F, Fischer JL. 2023. The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. Archive for Rational Mechanics and Analysis. 247(5), 76.","mla":"Cornalba, Federico, and Julian L. Fischer. “The Dean-Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles.” Archive for Rational Mechanics and Analysis, vol. 247, no. 5, 76, Springer Nature, 2023, doi:10.1007/s00205-023-01903-7.","chicago":"Cornalba, Federico, and Julian L Fischer. “The Dean-Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles.” Archive for Rational Mechanics and Analysis. Springer Nature, 2023. https://doi.org/10.1007/s00205-023-01903-7.","apa":"Cornalba, F., & Fischer, J. L. (2023). The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. Archive for Rational Mechanics and Analysis. Springer Nature. https://doi.org/10.1007/s00205-023-01903-7","ama":"Cornalba F, Fischer JL. The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. Archive for Rational Mechanics and Analysis. 2023;247(5). doi:10.1007/s00205-023-01903-7"},"_id":"10551","pmid":1,"date_updated":"2025-04-23T13:06:01Z","volume":247,"publication":"Archive for Rational Mechanics and Analysis","article_number":"76","publication_status":"published","article_type":"original","abstract":[{"text":"The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N≫1. The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in N−1 (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.","lang":"eng"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"access_level":"open_access","success":1,"relation":"main_file","file_size":1851185,"file_name":"2023_ArchiveRationalMech_Cornalba.pdf","checksum":"4529eeff170b6745a461d397ee611b5a","file_id":"14904","creator":"dernst","date_created":"2024-01-30T12:09:34Z","date_updated":"2024-01-30T12:09:34Z","content_type":"application/pdf"}],"ddc":["510"],"type":"journal_article","date_created":"2021-12-16T12:16:03Z","doi":"10.1007/s00205-023-01903-7","month":"08","file_date_updated":"2024-01-30T12:09:34Z","day":"04","scopus_import":"1","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"issue":"5","has_accepted_license":"1","oa":1,"date_published":"2023-08-04T00:00:00Z","project":[{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships"},{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"}],"external_id":{"arxiv":["2109.06500"],"isi":["001043086800001"],"pmid":["37547904"]},"quality_controlled":"1","status":"public","department":[{"_id":"JuFi"}],"oa_version":"Published Version","arxiv":1,"acknowledgement":"We thank the anonymous referee for his/her careful reading of the manuscript and valuable suggestions. FC gratefully acknowledges funding from the Austrian Science Fund (FWF) through the project F65, and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411.\r\nOpen access funding provided by Austrian Science Fund (FWF).","corr_author":"1","author":[{"last_name":"Cornalba","full_name":"Cornalba, Federico","orcid":"0000-0002-6269-5149","first_name":"Federico","id":"2CEB641C-A400-11E9-A717-D712E6697425"},{"full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"}],"intvolume":" 247","isi":1},{"page":"1394-1434","arxiv":1,"author":[{"last_name":"De Nitti","full_name":"De Nitti, Nicola","first_name":"Nicola"},{"first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"}],"acknowledgement":"N. De Nitti acknowledges the kind hospitality of IST Austria within the framework of the ISTernship Summer Program 2018, during which most of the present article was written. N. DeNitti has received funding by The Austrian Agency for International Cooperation in Education &Research (OeAD-GmbH) via its financial support of the ISTernship Summer Program 2018. N.De Nitti would also like to thank Giuseppe Coclite, Giuseppe Devillanova, Giuseppe Florio, Sebastian Hensel, and Francesco Maddalena for several helpful conversations on topics related to this work.","corr_author":"1","isi":1,"intvolume":" 47","oa":1,"external_id":{"arxiv":["1907.05342"],"isi":["000805689800001"]},"date_published":"2022-07-01T00:00:00Z","department":[{"_id":"JuFi"}],"oa_version":"Preprint","status":"public","quality_controlled":"1","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.1907.05342"}],"day":"01","article_processing_charge":"No","language":[{"iso":"eng"}],"scopus_import":"1","issue":"7","doi":"10.1080/03605302.2022.2056702","month":"07","type":"journal_article","date_created":"2023-01-16T10:06:50Z","volume":47,"publication":"Communications in Partial Differential Equations","keyword":["Applied Mathematics","Analysis"],"publication_status":"published","article_type":"original","abstract":[{"text":"We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.","lang":"eng"}],"publication_identifier":{"eissn":["1532-4133"],"issn":["0360-5302"]},"citation":{"ista":"De Nitti N, Fischer JL. 2022. Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation. Communications in Partial Differential Equations. 47(7), 1394–1434.","mla":"De Nitti, Nicola, and Julian L. Fischer. “Sharp Criteria for the Waiting Time Phenomenon in Solutions to the Thin-Film Equation.” Communications in Partial Differential Equations, vol. 47, no. 7, Taylor & Francis, 2022, pp. 1394–434, doi:10.1080/03605302.2022.2056702.","ieee":"N. De Nitti and J. L. Fischer, “Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation,” Communications in Partial Differential Equations, vol. 47, no. 7. Taylor & Francis, pp. 1394–1434, 2022.","short":"N. De Nitti, J.L. Fischer, Communications in Partial Differential Equations 47 (2022) 1394–1434.","apa":"De Nitti, N., & Fischer, J. L. (2022). Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation. Communications in Partial Differential Equations. Taylor & Francis. https://doi.org/10.1080/03605302.2022.2056702","chicago":"De Nitti, Nicola, and Julian L Fischer. “Sharp Criteria for the Waiting Time Phenomenon in Solutions to the Thin-Film Equation.” Communications in Partial Differential Equations. Taylor & Francis, 2022. https://doi.org/10.1080/03605302.2022.2056702.","ama":"De Nitti N, Fischer JL. Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation. Communications in Partial Differential Equations. 2022;47(7):1394-1434. doi:10.1080/03605302.2022.2056702"},"_id":"12304","date_updated":"2024-10-09T21:03:57Z","year":"2022","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publisher":"Taylor & Francis","title":"Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation"},{"title":"Global existence analysis of energy-reaction-diffusion systems","publisher":"Society for Industrial and Applied Mathematics","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","year":"2022","date_updated":"2023-08-02T13:37:03Z","_id":"10547","citation":{"ieee":"J. L. Fischer, K. Hopf, M. Kniely, and A. Mielke, “Global existence analysis of energy-reaction-diffusion systems,” SIAM Journal on Mathematical Analysis, vol. 54, no. 1. Society for Industrial and Applied Mathematics, pp. 220–267, 2022.","short":"J.L. Fischer, K. Hopf, M. Kniely, A. Mielke, SIAM Journal on Mathematical Analysis 54 (2022) 220–267.","mla":"Fischer, Julian L., et al. “Global Existence Analysis of Energy-Reaction-Diffusion Systems.” SIAM Journal on Mathematical Analysis, vol. 54, no. 1, Society for Industrial and Applied Mathematics, 2022, pp. 220–67, doi:10.1137/20M1387237.","ista":"Fischer JL, Hopf K, Kniely M, Mielke A. 2022. Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. 54(1), 220–267.","chicago":"Fischer, Julian L, Katharina Hopf, Michael Kniely, and Alexander Mielke. “Global Existence Analysis of Energy-Reaction-Diffusion Systems.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/20M1387237.","apa":"Fischer, J. L., Hopf, K., Kniely, M., & Mielke, A. (2022). Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/20M1387237","ama":"Fischer JL, Hopf K, Kniely M, Mielke A. Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. 2022;54(1):220-267. doi:10.1137/20M1387237"},"publication_identifier":{"issn":["0036-1410"]},"abstract":[{"text":"We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities,\r\nwhile at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.","lang":"eng"}],"article_type":"original","publication_status":"published","keyword":["Energy-Reaction-Diffusion Systems","Cross Diffusion","Global-In-Time Existence of Weak/Renormalised Solutions","Entropy Method","Onsager System","Soret/Dufour Effect"],"publication":"SIAM Journal on Mathematical Analysis","volume":54,"date_created":"2021-12-16T12:08:56Z","type":"journal_article","month":"01","doi":"10.1137/20M1387237","issue":"1","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","day":"04","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2012.03792"}],"status":"public","quality_controlled":"1","department":[{"_id":"JuFi"}],"oa_version":"Preprint","date_published":"2022-01-04T00:00:00Z","external_id":{"isi":["000762768000006"],"arxiv":["2012.03792 "]},"oa":1,"intvolume":" 54","isi":1,"acknowledgement":"M.K. gratefully acknowledges the hospitality of WIAS Berlin, where a major part of the project was carried out. The research stay of M.K. at WIAS Berlin was funded by the Austrian Federal Ministry of Education, Science and Research through a research fellowship for graduates of a promotio sub auspiciis. The research of A.M. has been partially supported by Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 1114 “Scaling Cascades in Complex Systems” (Project no. 235221301), Subproject C05 “Effective models for materials and interfaces with multiple scales”. J.F. and A.M. are grateful for the hospitality of the Erwin Schrödinger Institute in Vienna, where some ideas for this work have been developed. The authors are grateful to two anonymous referees for several helpful comments, in particular for the short proof of estimate (2.7).","author":[{"orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Katharina","full_name":"Hopf, Katharina","last_name":"Hopf"},{"id":"2CA2C08C-F248-11E8-B48F-1D18A9856A87","first_name":"Michael","full_name":"Kniely, Michael","last_name":"Kniely","orcid":"0000-0001-5645-4333"},{"full_name":"Mielke, Alexander","last_name":"Mielke","first_name":"Alexander"}],"arxiv":1,"page":"220-267"},{"month":"04","doi":"10.1214/21-AAP1705","issue":"2","article_processing_charge":"No","language":[{"iso":"eng"}],"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.04088"}],"day":"28","department":[{"_id":"JuFi"}],"oa_version":"Preprint","quality_controlled":"1","status":"public","external_id":{"arxiv":["1910.04088"],"isi":["000791003700011"]},"date_published":"2022-04-28T00:00:00Z","oa":1,"isi":1,"intvolume":" 32","author":[{"first_name":"Mitia","last_name":"Duerinckx","full_name":"Duerinckx, Mitia"},{"last_name":"Fischer","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Gloria","full_name":"Gloria, Antoine","first_name":"Antoine"}],"acknowledgement":"The authors thank Ivan Nourdin and Felix Otto for inspiring discussions. The work of MD is financially supported by the CNRS-Momentum program. Financial support of AG is acknowledged from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).","corr_author":"1","page":"1179-1209","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Institute of Mathematical Statistics","title":"Scaling limit of the homogenization commutator for Gaussian coefficient fields","year":"2022","date_updated":"2026-08-12T09:22:31Z","_id":"10548","citation":{"chicago":"Duerinckx, Mitia, Julian L Fischer, and Antoine Gloria. “Scaling Limit of the Homogenization Commutator for Gaussian Coefficient Fields.” Annals of Applied Probability. Institute of Mathematical Statistics, 2022. https://doi.org/10.1214/21-AAP1705.","apa":"Duerinckx, M., Fischer, J. L., & Gloria, A. (2022). Scaling limit of the homogenization commutator for Gaussian coefficient fields. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/21-AAP1705","short":"M. Duerinckx, J.L. Fischer, A. Gloria, Annals of Applied Probability 32 (2022) 1179–1209.","ieee":"M. Duerinckx, J. L. Fischer, and A. Gloria, “Scaling limit of the homogenization commutator for Gaussian coefficient fields,” Annals of Applied Probability, vol. 32, no. 2. Institute of Mathematical Statistics, pp. 1179–1209, 2022.","mla":"Duerinckx, Mitia, et al. “Scaling Limit of the Homogenization Commutator for Gaussian Coefficient Fields.” Annals of Applied Probability, vol. 32, no. 2, Institute of Mathematical Statistics, 2022, pp. 1179–209, doi:10.1214/21-AAP1705.","ista":"Duerinckx M, Fischer JL, Gloria A. 2022. Scaling limit of the homogenization commutator for Gaussian coefficient fields. Annals of Applied Probability. 32(2), 1179–1209.","ama":"Duerinckx M, Fischer JL, Gloria A. Scaling limit of the homogenization commutator for Gaussian coefficient fields. Annals of Applied Probability. 2022;32(2):1179-1209. doi:10.1214/21-AAP1705"},"publication_identifier":{"issn":["1050-5164"]},"abstract":[{"text":"Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution operator displays fluctuations around its expectation. The recently developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting\r\nfrom a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we\r\ninvestigate the (non)degeneracy of the limit. This extends to general dimension $d\\ge1$ previous results so far limited to dimension $d=1$, and to the continuum setting with strong correlations recent results in the discrete iid case.","lang":"eng"}],"publication_status":"published","article_type":"original","publication":"Annals of Applied Probability","volume":32,"date_created":"2021-12-16T12:10:16Z","type":"journal_article"},{"arxiv":1,"related_material":{"record":[{"relation":"dissertation_contains","id":"14587","status":"public"},{"status":"public","relation":"later_version","id":"17481"}]},"author":[{"first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Marveggio, Alice","last_name":"Marveggio","first_name":"Alice","id":"25647992-AA84-11E9-9D75-8427E6697425"}],"corr_author":"1","type":"preprint","date_created":"2023-11-23T09:30:02Z","oa":1,"external_id":{"arxiv":["2203.17143"]},"project":[{"grant_number":"948819","name":"Bridging Scales in Random Materials","call_identifier":"H2020","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d"}],"publication":"arXiv","date_published":"2022-03-31T00:00:00Z","publication_status":"draft","article_number":"2203.17143","department":[{"_id":"JuFi"}],"oa_version":"Preprint","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."}],"status":"public","main_file_link":[{"url":"https://arxiv.org/abs/2203.17143","open_access":"1"}],"day":"31","article_processing_charge":"No","language":[{"iso":"eng"}],"citation":{"ieee":"J. L. Fischer and A. Marveggio, “Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow,” arXiv. .","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.” ArXiv, 2203.17143, doi:10.48550/ARXIV.2203.17143.","ista":"Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. arXiv, 2203.17143.","chicago":"Fischer, Julian L, and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen-Cahn Equation towards Multiphase Mean Curvature Flow.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2203.17143.","apa":"Fischer, J. L., & Marveggio, A. (n.d.). Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. arXiv. https://doi.org/10.48550/ARXIV.2203.17143","ama":"Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. arXiv. doi:10.48550/ARXIV.2203.17143"},"_id":"14597","date_updated":"2026-08-12T09:15:49Z","doi":"10.48550/ARXIV.2203.17143","year":"2022","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"03","ec_funded":1,"title":"Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow"},{"page":"343-452","arxiv":1,"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). SN acknowledges partial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 405009441.","author":[{"orcid":"0000-0002-0479-558X","last_name":"Fischer","full_name":"Fischer, Julian L","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Stefan","last_name":"Neukamm","full_name":"Neukamm, Stefan"}],"intvolume":" 242","isi":1,"oa":1,"date_published":"2021-06-30T00:00:00Z","external_id":{"isi":["000668431200001"],"arxiv":["1908.02273"]},"status":"public","quality_controlled":"1","department":[{"_id":"JuFi"}],"oa_version":"Published Version","day":"30","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","issue":"1","has_accepted_license":"1","doi":"10.1007/s00205-021-01686-9","month":"06","file_date_updated":"2021-12-16T14:58:08Z","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ddc":["530"],"file":[{"creator":"cchlebak","date_created":"2021-12-16T14:58:08Z","date_updated":"2021-12-16T14:58:08Z","content_type":"application/pdf","access_level":"open_access","file_size":1640121,"relation":"main_file","success":1,"checksum":"cc830b739aed83ca2e32c4e0ce266a4c","file_name":"2021_ArchRatMechAnalysis_Fischer.pdf","file_id":"10558"}],"type":"journal_article","date_created":"2021-12-16T12:12:33Z","volume":242,"keyword":["Mechanical Engineering","Mathematics (miscellaneous)","Analysis"],"publication":"Archive for Rational Mechanics and Analysis","publication_status":"published","article_type":"original","abstract":[{"lang":"eng","text":"We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on \\mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale \\varepsilon >0, we establish homogenization error estimates of the order \\varepsilon in case d\\geqq 3, and of the order \\varepsilon |\\log \\varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence \\varepsilon ^\\delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/\\varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,\\alpha } regularity theory is available."}],"publication_identifier":{"issn":["0003-9527"],"eissn":["1432-0673"]},"citation":{"ama":"Fischer JL, Neukamm S. Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems. Archive for Rational Mechanics and Analysis. 2021;242(1):343-452. doi:10.1007/s00205-021-01686-9","chicago":"Fischer, Julian L, and Stefan Neukamm. “Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems.” Archive for Rational Mechanics and Analysis. Springer Nature, 2021. https://doi.org/10.1007/s00205-021-01686-9.","apa":"Fischer, J. L., & Neukamm, S. (2021). Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems. Archive for Rational Mechanics and Analysis. Springer Nature. https://doi.org/10.1007/s00205-021-01686-9","short":"J.L. Fischer, S. Neukamm, Archive for Rational Mechanics and Analysis 242 (2021) 343–452.","ieee":"J. L. Fischer and S. Neukamm, “Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems,” Archive for Rational Mechanics and Analysis, vol. 242, no. 1. Springer Nature, pp. 343–452, 2021.","mla":"Fischer, Julian L., and Stefan Neukamm. “Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems.” Archive for Rational Mechanics and Analysis, vol. 242, no. 1, Springer Nature, 2021, pp. 343–452, doi:10.1007/s00205-021-01686-9.","ista":"Fischer JL, Neukamm S. 2021. Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems. Archive for Rational Mechanics and Analysis. 242(1), 343–452."},"_id":"10549","date_updated":"2023-08-17T06:23:21Z","year":"2021","title":"Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems","publisher":"Springer Nature","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8"},{"publication_identifier":{"issn":["0036-1429"]},"citation":{"ista":"Fischer JL, Matthes D. 2021. The waiting time phenomenon in spatially discretized porous medium and thin film equations. SIAM Journal on Numerical Analysis. 59(1), 60–87.","mla":"Fischer, Julian L., and Daniel Matthes. “The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations.” SIAM Journal on Numerical Analysis, vol. 59, no. 1, Society for Industrial and Applied Mathematics, 2021, pp. 60–87, doi:10.1137/19M1300017.","short":"J.L. Fischer, D. Matthes, SIAM Journal on Numerical Analysis 59 (2021) 60–87.","ieee":"J. L. Fischer and D. Matthes, “The waiting time phenomenon in spatially discretized porous medium and thin film equations,” SIAM Journal on Numerical Analysis, vol. 59, no. 1. Society for Industrial and Applied Mathematics, pp. 60–87, 2021.","apa":"Fischer, J. L., & Matthes, D. (2021). The waiting time phenomenon in spatially discretized porous medium and thin film equations. SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/19M1300017","chicago":"Fischer, Julian L, and Daniel Matthes. “The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations.” SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics, 2021. https://doi.org/10.1137/19M1300017.","ama":"Fischer JL, Matthes D. The waiting time phenomenon in spatially discretized porous medium and thin film equations. SIAM Journal on Numerical Analysis. 2021;59(1):60-87. doi:10.1137/19M1300017"},"_id":"9335","date_updated":"2023-08-08T13:10:40Z","year":"2021","publisher":"Society for Industrial and Applied Mathematics","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"The waiting time phenomenon in spatially discretized porous medium and thin film equations","type":"journal_article","date_created":"2021-04-18T22:01:42Z","volume":59,"publication":"SIAM Journal on Numerical Analysis","publication_status":"published","article_type":"original","abstract":[{"text":"Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin film equations, and we obtain sufficient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. For the spatially discrete solution, the waiting time phenomenon refers to a deviation of the edge of support from its original position by a quantity comparable to the mesh width, over a mesh-independent time interval. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique à la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1911.04185"}],"day":"01","language":[{"iso":"eng"}],"article_processing_charge":"No","scopus_import":"1","issue":"1","doi":"10.1137/19M1300017","month":"01","arxiv":1,"page":"60-87","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer"},{"last_name":"Matthes","full_name":"Matthes, Daniel","first_name":"Daniel"}],"acknowledgement":"This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”.","isi":1,"intvolume":" 59","oa":1,"external_id":{"isi":["000625044600003"],"arxiv":["1911.04185"]},"date_published":"2021-01-01T00:00:00Z","department":[{"_id":"JuFi"}],"oa_version":"Preprint","quality_controlled":"1","status":"public"},{"date_created":"2021-04-25T22:01:31Z","type":"journal_article","abstract":[{"text":"This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\\varepsilon/H)^{d/2}$, $\\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.","lang":"eng"}],"publication_status":"published","article_type":"original","publication":"SIAM Journal on Numerical Analysis","volume":59,"date_updated":"2023-08-08T13:13:37Z","_id":"9352","citation":{"apa":"Fischer, J. L., Gallistl, D., & Peterseim, D. (2021). A priori error analysis of a numerical stochastic homogenization method. SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/19M1308992","chicago":"Fischer, Julian L, Dietmar Gallistl, and Dietmar Peterseim. “A Priori Error Analysis of a Numerical Stochastic Homogenization Method.” SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics, 2021. https://doi.org/10.1137/19M1308992.","ista":"Fischer JL, Gallistl D, Peterseim D. 2021. A priori error analysis of a numerical stochastic homogenization method. SIAM Journal on Numerical Analysis. 59(2), 660–674.","mla":"Fischer, Julian L., et al. “A Priori Error Analysis of a Numerical Stochastic Homogenization Method.” SIAM Journal on Numerical Analysis, vol. 59, no. 2, Society for Industrial and Applied Mathematics, 2021, pp. 660–74, doi:10.1137/19M1308992.","ieee":"J. L. Fischer, D. Gallistl, and D. Peterseim, “A priori error analysis of a numerical stochastic homogenization method,” SIAM Journal on Numerical Analysis, vol. 59, no. 2. Society for Industrial and Applied Mathematics, pp. 660–674, 2021.","short":"J.L. Fischer, D. Gallistl, D. Peterseim, SIAM Journal on Numerical Analysis 59 (2021) 660–674.","ama":"Fischer JL, Gallistl D, Peterseim D. A priori error analysis of a numerical stochastic homogenization method. SIAM Journal on Numerical Analysis. 2021;59(2):660-674. doi:10.1137/19M1308992"},"publication_identifier":{"issn":["0036-1429"]},"publisher":"Society for Industrial and Applied Mathematics","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"A priori error analysis of a numerical stochastic homogenization method","year":"2021","isi":1,"intvolume":" 59","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer"},{"full_name":"Gallistl, Dietmar","last_name":"Gallistl","first_name":"Dietmar"},{"first_name":"Dietmar","full_name":"Peterseim, Dietmar","last_name":"Peterseim"}],"acknowledgement":"This work was initiated while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn during the trimester program Multiscale Problems: Algorithms, Numerical Analysis, and Computation. D. Peterseim would like to acknowledge the kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), where parts of this research were developed under the frame of the thematic program Numerical Analysis of Complex PDE Models in the Sciences.","page":"660-674","arxiv":1,"department":[{"_id":"JuFi"}],"oa_version":"Preprint","quality_controlled":"1","status":"public","external_id":{"isi":["000646030400003"],"arxiv":["1912.11646"]},"date_published":"2021-03-09T00:00:00Z","oa":1,"issue":"2","article_processing_charge":"No","language":[{"iso":"eng"}],"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1912.11646"}],"day":"09","month":"03","doi":"10.1137/19M1308992"},{"month":"03","title":"The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2020","doi":"10.48550/arXiv.2003.05478","_id":"10012","date_updated":"2026-04-08T07:01:01Z","day":"11","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2003.05478"}],"citation":{"apa":"Fischer, J. L., Hensel, S., Laux, T., & Simon, T. (n.d.). The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions. arXiv. https://doi.org/10.48550/arXiv.2003.05478","chicago":"Fischer, Julian L, Sebastian Hensel, Tim Laux, and Thilo Simon. “The Local Structure of the Energy Landscape in Multiphase Mean Curvature Flow: Weak-Strong Uniqueness and Stability of Evolutions.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2003.05478.","mla":"Fischer, Julian L., et al. “The Local Structure of the Energy Landscape in Multiphase Mean Curvature Flow: Weak-Strong Uniqueness and Stability of Evolutions.” ArXiv, 2003.05478, doi:10.48550/arXiv.2003.05478.","ista":"Fischer JL, Hensel S, Laux T, Simon T. The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions. arXiv, 2003.05478.","short":"J.L. Fischer, S. Hensel, T. Laux, T. Simon, ArXiv (n.d.).","ieee":"J. L. Fischer, S. Hensel, T. Laux, and T. Simon, “The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions,” arXiv. .","ama":"Fischer JL, Hensel S, Laux T, Simon T. The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions. arXiv. doi:10.48550/arXiv.2003.05478"},"article_processing_charge":"No","language":[{"iso":"eng"}],"article_number":"2003.05478","publication_status":"draft","status":"public","abstract":[{"lang":"eng","text":"We prove that in the absence of topological changes, the notion of BV solutions to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees that one has reached a global minimum in the energy landscape, the existence of a \"gradient flow calibration\" ensures that the route of steepest descent in the energy landscape is unique and stable."}],"department":[{"_id":"JuFi"}],"oa_version":"Preprint","oa":1,"publication":"arXiv","date_published":"2020-03-11T00:00:00Z","external_id":{"arxiv":["2003.05478"]},"project":[{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385"}],"type":"preprint","date_created":"2021-09-13T12:17:11Z","related_material":{"record":[{"status":"public","id":"10007","relation":"dissertation_contains"}]},"arxiv":1,"acknowledgement":"Parts of the paper were written during the visit of the authors to the Hausdorff Research Institute for Mathematics (HIM), University of Bonn, in the framework of the trimester program “Evolution of Interfaces”. The support and the hospitality of HIM are gratefully acknowledged. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 665385.","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0479-558X","last_name":"Fischer","full_name":"Fischer, Julian L","first_name":"Julian L"},{"first_name":"Sebastian","orcid":"0000-0001-7252-8072","last_name":"Hensel","full_name":"Hensel, Sebastian","id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Laux","full_name":"Laux, Tim","first_name":"Tim"},{"first_name":"Thilo","full_name":"Simon, Thilo","last_name":"Simon"}]},{"title":"Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publisher":"IOP Publishing","year":"2020","license":"https://creativecommons.org/licenses/by/3.0/","date_updated":"2026-04-02T14:31:34Z","_id":"8697","citation":{"ieee":"J. L. Fischer and M. Kniely, “Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model,” Nonlinearity, vol. 33, no. 11. IOP Publishing, pp. 5733–5772, 2020.","short":"J.L. Fischer, M. Kniely, Nonlinearity 33 (2020) 5733–5772.","mla":"Fischer, Julian L., and Michael Kniely. “Variance Reduction for Effective Energies of Random Lattices in the Thomas-Fermi-von Weizsäcker Model.” Nonlinearity, vol. 33, no. 11, IOP Publishing, 2020, pp. 5733–72, doi:10.1088/1361-6544/ab9728.","ista":"Fischer JL, Kniely M. 2020. Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model. Nonlinearity. 33(11), 5733–5772.","chicago":"Fischer, Julian L, and Michael Kniely. “Variance Reduction for Effective Energies of Random Lattices in the Thomas-Fermi-von Weizsäcker Model.” Nonlinearity. IOP Publishing, 2020. https://doi.org/10.1088/1361-6544/ab9728.","apa":"Fischer, J. L., & Kniely, M. (2020). Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model. Nonlinearity. IOP Publishing. https://doi.org/10.1088/1361-6544/ab9728","ama":"Fischer JL, Kniely M. Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model. Nonlinearity. 2020;33(11):5733-5772. doi:10.1088/1361-6544/ab9728"},"publication_identifier":{"issn":["0951-7715"],"eissn":["1361-6544"]},"abstract":[{"lang":"eng","text":"In the computation of the material properties of random alloys, the method of 'special quasirandom structures' attempts to approximate the properties of the alloy on a finite volume with higher accuracy by replicating certain statistics of the random atomic lattice in the finite volume as accurately as possible. In the present work, we provide a rigorous justification for a variant of this method in the framework of the Thomas–Fermi–von Weizsäcker (TFW) model. Our approach is based on a recent analysis of a related variance reduction method in stochastic homogenization of linear elliptic PDEs and the locality properties of the TFW model. Concerning the latter, we extend an exponential locality result by Nazar and Ortner to include point charges, a result that may be of independent interest."}],"publication_status":"published","article_type":"original","publication":"Nonlinearity","volume":33,"date_created":"2020-10-25T23:01:16Z","type":"journal_article","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 3.0 Unported (CC BY 3.0)","short":"CC BY (3.0)","legal_code_url":"https://creativecommons.org/licenses/by/3.0/legalcode"},"file":[{"checksum":"ed90bc6eb5f32ee6157fef7f3aabc057","file_name":"2020_Nonlinearity_Fischer.pdf","file_id":"8710","access_level":"open_access","file_size":1223899,"success":1,"relation":"main_file","date_updated":"2020-10-27T12:09:57Z","date_created":"2020-10-27T12:09:57Z","content_type":"application/pdf","creator":"cziletti"}],"ddc":["510"],"file_date_updated":"2020-10-27T12:09:57Z","month":"11","doi":"10.1088/1361-6544/ab9728","has_accepted_license":"1","issue":"11","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","day":"01","status":"public","quality_controlled":"1","department":[{"_id":"JuFi"}],"oa_version":"Published Version","date_published":"2020-11-01T00:00:00Z","external_id":{"isi":["000576492700001"],"arxiv":["1906.12245"]},"oa":1,"intvolume":" 33","isi":1,"corr_author":"1","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer","first_name":"Julian L"},{"id":"2CA2C08C-F248-11E8-B48F-1D18A9856A87","first_name":"Michael","orcid":"0000-0001-5645-4333","last_name":"Kniely","full_name":"Kniely, Michael"}],"arxiv":1,"page":"5733-5772"},{"doi":"10.1137/20M1322182","month":"12","file_date_updated":"2021-01-25T07:48:39Z","day":"15","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","issue":"6","has_accepted_license":"1","oa":1,"date_published":"2020-12-15T00:00:00Z","project":[{"grant_number":"665385","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"International IST Doctoral Program"}],"external_id":{"isi":["000600695200027"]},"quality_controlled":"1","status":"public","department":[{"_id":"JuFi"}],"oa_version":"Published Version","page":"6222-6233","acknowledgement":"This work was supported by the European Union's Horizon 2020 Research and Innovation\r\nProgramme under Marie Sklodowska-Curie grant agreement 665385 and by the Deutsche\r\nForschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy, EXC-2047/1--390685813.","corr_author":"1","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X","first_name":"Julian L"},{"last_name":"Laux","full_name":"Laux, Tim","first_name":"Tim"},{"last_name":"Simon","full_name":"Simon, Theresa M.","first_name":"Theresa M."}],"intvolume":" 52","isi":1,"year":"2020","title":"Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Society for Industrial and Applied Mathematics","publication_identifier":{"issn":["0036-1410"],"eissn":["1095-7154"]},"citation":{"ama":"Fischer JL, Laux T, Simon TM. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. 2020;52(6):6222-6233. doi:10.1137/20M1322182","apa":"Fischer, J. L., Laux, T., & Simon, T. M. (2020). Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/20M1322182","chicago":"Fischer, Julian L, Tim Laux, and Theresa M. Simon. “Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2020. https://doi.org/10.1137/20M1322182.","ista":"Fischer JL, Laux T, Simon TM. 2020. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. 52(6), 6222–6233.","mla":"Fischer, Julian L., et al. “Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies.” SIAM Journal on Mathematical Analysis, vol. 52, no. 6, Society for Industrial and Applied Mathematics, 2020, pp. 6222–33, doi:10.1137/20M1322182.","short":"J.L. Fischer, T. Laux, T.M. Simon, SIAM Journal on Mathematical Analysis 52 (2020) 6222–6233.","ieee":"J. L. Fischer, T. Laux, and T. M. Simon, “Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies,” SIAM Journal on Mathematical Analysis, vol. 52, no. 6. Society for Industrial and Applied Mathematics, pp. 6222–6233, 2020."},"_id":"9039","date_updated":"2025-07-10T12:01:32Z","volume":52,"publication":"SIAM Journal on Mathematical Analysis","article_type":"original","publication_status":"published","abstract":[{"lang":"eng","text":"We give a short and self-contained proof for rates of convergence of the Allen--Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen--Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems."}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"creator":"dernst","content_type":"application/pdf","date_updated":"2021-01-25T07:48:39Z","date_created":"2021-01-25T07:48:39Z","file_size":310655,"relation":"main_file","success":1,"access_level":"open_access","file_id":"9041","checksum":"21aa1cf4c30a86a00cae15a984819b5d","file_name":"2020_SIAM_Fischer.pdf"}],"ddc":["510"],"type":"journal_article","date_created":"2021-01-24T23:01:09Z"},{"corr_author":"1","author":[{"first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0001-7252-8072","last_name":"Hensel","full_name":"Hensel, Sebastian","first_name":"Sebastian","id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87"}],"page":"967-1087","intvolume":" 236","isi":1,"date_published":"2020-05-01T00:00:00Z","external_id":{"isi":["000511060200001"]},"project":[{"grant_number":"665385","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","name":"International IST Doctoral Program","call_identifier":"H2020"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"oa":1,"quality_controlled":"1","status":"public","department":[{"_id":"JuFi"}],"oa_version":"Published Version","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","day":"01","has_accepted_license":"1","doi":"10.1007/s00205-019-01486-2","file_date_updated":"2020-11-20T09:14:22Z","month":"05","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"10007"}]},"ddc":["530","532"],"file":[{"checksum":"f107e21b58f5930876f47144be37cf6c","file_name":"2020_ArchRatMechAn_Fischer.pdf","file_id":"8779","access_level":"open_access","file_size":1897571,"relation":"main_file","success":1,"date_created":"2020-11-20T09:14:22Z","date_updated":"2020-11-20T09:14:22Z","content_type":"application/pdf","creator":"dernst"}],"date_created":"2020-02-16T23:00:50Z","type":"journal_article","publication":"Archive for Rational Mechanics and Analysis","volume":236,"abstract":[{"lang":"eng","text":"In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities, the concept of varifold solutions—whose global in time existence has been shown by Abels (Interfaces Free Bound 9(1):31–65, 2007) for general initial data—does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy."}],"publication_status":"published","article_type":"original","citation":{"ista":"Fischer JL, Hensel S. 2020. Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. 236, 967–1087.","mla":"Fischer, Julian L., and Sebastian Hensel. “Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension.” Archive for Rational Mechanics and Analysis, vol. 236, Springer Nature, 2020, pp. 967–1087, doi:10.1007/s00205-019-01486-2.","ieee":"J. L. Fischer and S. Hensel, “Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension,” Archive for Rational Mechanics and Analysis, vol. 236. Springer Nature, pp. 967–1087, 2020.","short":"J.L. Fischer, S. Hensel, Archive for Rational Mechanics and Analysis 236 (2020) 967–1087.","apa":"Fischer, J. L., & Hensel, S. (2020). Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. Springer Nature. https://doi.org/10.1007/s00205-019-01486-2","chicago":"Fischer, Julian L, and Sebastian Hensel. “Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension.” Archive for Rational Mechanics and Analysis. Springer Nature, 2020. https://doi.org/10.1007/s00205-019-01486-2.","ama":"Fischer JL, Hensel S. Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension. Archive for Rational Mechanics and Analysis. 2020;236:967-1087. doi:10.1007/s00205-019-01486-2"},"publication_identifier":{"issn":["0003-9527"],"eissn":["1432-0673"]},"date_updated":"2026-04-08T07:01:01Z","_id":"7489","year":"2020","title":"Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Springer Nature"},{"department":[{"_id":"JuFi"}],"oa_version":"Preprint","quality_controlled":"1","status":"public","publist_id":"7770","external_id":{"arxiv":["1408.1587"],"isi":["000449108500010"]},"date_published":"2019-01-05T00:00:00Z","oa":1,"isi":1,"intvolume":" 266","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer"},{"last_name":"Kneuss","full_name":"Kneuss, Olivier","first_name":"Olivier"}],"arxiv":1,"page":"257 - 311","month":"01","doi":"10.1016/j.jde.2018.07.045","issue":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1408.1587"}],"day":"05","abstract":[{"lang":"eng","text":"We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide."}],"publication_status":"published","publication":"Journal of Differential Equations","volume":266,"date_created":"2018-12-11T11:44:54Z","type":"journal_article","publisher":"Elsevier","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","title":"Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity","year":"2019","date_updated":"2023-09-08T13:25:35Z","_id":"151","citation":{"short":"J.L. Fischer, O. Kneuss, Journal of Differential Equations 266 (2019) 257–311.","ieee":"J. L. Fischer and O. Kneuss, “Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity,” Journal of Differential Equations, vol. 266, no. 1. Elsevier, pp. 257–311, 2019.","ista":"Fischer JL, Kneuss O. 2019. Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations. 266(1), 257–311.","mla":"Fischer, Julian L., and Olivier Kneuss. “Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with L p Data and Applications to Nonlinear Elasticity.” Journal of Differential Equations, vol. 266, no. 1, Elsevier, 2019, pp. 257–311, doi:10.1016/j.jde.2018.07.045.","chicago":"Fischer, Julian L, and Olivier Kneuss. “Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with L p Data and Applications to Nonlinear Elasticity.” Journal of Differential Equations. Elsevier, 2019. https://doi.org/10.1016/j.jde.2018.07.045.","apa":"Fischer, J. L., & Kneuss, O. (2019). Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations. Elsevier. https://doi.org/10.1016/j.jde.2018.07.045","ama":"Fischer JL, Kneuss O. Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations. 2019;266(1):257-311. doi:10.1016/j.jde.2018.07.045"}},{"has_accepted_license":"1","issue":"2","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","day":"01","file_date_updated":"2020-07-14T12:47:34Z","month":"11","doi":"10.1007/s00205-019-01400-w","intvolume":" 234","isi":1,"corr_author":"1","author":[{"first_name":"Julian L","orcid":"0000-0002-0479-558X","full_name":"Fischer, Julian L","last_name":"Fischer","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"}],"page":"635–726","arxiv":1,"status":"public","quality_controlled":"1","oa_version":"Published Version","department":[{"_id":"JuFi"}],"date_published":"2019-11-01T00:00:00Z","external_id":{"isi":["000482386000006"],"arxiv":["1807.00834"]},"project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"oa":1,"date_updated":"2025-04-15T06:53:15Z","_id":"6617","citation":{"short":"J.L. Fischer, Archive for Rational Mechanics and Analysis 234 (2019) 635–726.","ieee":"J. L. Fischer, “The choice of representative volumes in the approximation of effective properties of random materials,” Archive for Rational Mechanics and Analysis, vol. 234, no. 2. Springer, pp. 635–726, 2019.","ista":"Fischer JL. 2019. The choice of representative volumes in the approximation of effective properties of random materials. Archive for Rational Mechanics and Analysis. 234(2), 635–726.","mla":"Fischer, Julian L. “The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials.” Archive for Rational Mechanics and Analysis, vol. 234, no. 2, Springer, 2019, pp. 635–726, doi:10.1007/s00205-019-01400-w.","chicago":"Fischer, Julian L. “The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials.” Archive for Rational Mechanics and Analysis. Springer, 2019. https://doi.org/10.1007/s00205-019-01400-w.","apa":"Fischer, J. L. (2019). The choice of representative volumes in the approximation of effective properties of random materials. Archive for Rational Mechanics and Analysis. Springer. https://doi.org/10.1007/s00205-019-01400-w","ama":"Fischer JL. The choice of representative volumes in the approximation of effective properties of random materials. Archive for Rational Mechanics and Analysis. 2019;234(2):635–726. doi:10.1007/s00205-019-01400-w"},"publication_identifier":{"issn":["0003-9527"],"eissn":["1432-0673"]},"title":"The choice of representative volumes in the approximation of effective properties of random materials","publisher":"Springer","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","year":"2019","date_created":"2019-07-07T21:59:23Z","type":"journal_article","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"file_size":1377659,"relation":"main_file","access_level":"open_access","file_id":"6626","checksum":"4cff75fa6addb0770991ad9c474ab404","file_name":"Springer_2019_Fischer.pdf","creator":"kschuh","content_type":"application/pdf","date_updated":"2020-07-14T12:47:34Z","date_created":"2019-07-08T15:56:47Z"}],"ddc":["500"],"abstract":[{"lang":"eng","text":"The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: a sample of the random material is chosen—the representative volume—and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well; for example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in materials science, Le Bris, Legoll and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably."}],"publication_status":"published","article_type":"original","publication":"Archive for Rational Mechanics and Analysis","volume":234}]