--- APC_amount: 1352,08 EUR OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '21894' 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." 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. article_processing_charge: Yes (in subscription journal) article_type: original arxiv: 1 author: - first_name: Federico full_name: Cornalba, Federico last_name: Cornalba - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Jonas full_name: Ingmanns, Jonas id: 71523d30-15b2-11ec-abd3-f80aa909d6b0 last_name: Ingmanns orcid: 0009-0008-1310-7946 - first_name: Claudia full_name: Raithel, Claudia last_name: Raithel 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 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 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. 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. 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. short: F. Cornalba, J.L. Fischer, J. Ingmanns, C. Raithel, The Annals of Probability 54 (2026) 155–215. corr_author: '1' date_created: 2026-05-20T08:25:25Z date_published: 2026-01-01T00:00:00Z date_updated: 2026-05-21T07:21:25Z day: '01' ddc: - '510' department: - _id: JuFi doi: 10.1214/25-aop1763 ec_funded: 1 external_id: arxiv: - '2303.00429' file: - access_level: open_access checksum: 3e60c0e25a1c96342029a7d2b031505f content_type: application/pdf creator: dernst date_created: 2026-05-21T07:11:27Z date_updated: 2026-05-21T07:11:27Z file_id: '21906' file_name: 2026_AnnalsProbability_Cornalba.pdf file_size: 865745 relation: main_file success: 1 file_date_updated: 2026-05-21T07:11:27Z has_accepted_license: '1' intvolume: ' 54' issue: '1' keyword: - Weakly interacting particle systems - fluctuating hydrodynamics - Dean-Kawasaki equation - stochastic PDEs - numerical approximation language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '01' oa: 1 oa_version: Published Version page: 155-215 project: - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: The Annals of Probability publication_identifier: eissn: - 2168-894X issn: - 0091-1798 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' scopus_import: '1' status: public title: Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 54 year: '2026' ... --- OA_place: publisher OA_type: hybrid _id: '19027' abstract: - lang: eng 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.' acknowledgement: The work of the authors was supported by the Austrian Science Fund (FWF) projectF65. article_processing_charge: Yes (in subscription journal) article_type: original arxiv: 1 author: - first_name: Federico full_name: Cornalba, Federico id: 2CEB641C-A400-11E9-A717-D712E6697425 last_name: Cornalba orcid: 0000-0002-6269-5149 - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X citation: 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 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 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. 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. short: F. Cornalba, J.L. Fischer, SIAM Journal on Numerical Analysis 63 (2025) 262–287. corr_author: '1' date_created: 2025-02-16T23:02:34Z date_published: 2025-02-01T00:00:00Z date_updated: 2025-09-30T10:30:31Z day: '01' ddc: - '510' department: - _id: JuFi doi: 10.1137/23M1617345 external_id: arxiv: - '2311.08872' isi: - '001447583400011' file: - access_level: open_access checksum: 53505647e848ed50f7e0d00c369b14e7 content_type: application/pdf creator: dernst date_created: 2025-02-17T08:32:23Z date_updated: 2025-02-17T08:32:23Z file_id: '19029' file_name: 2025_SIAMNumerAnaly_Cornalba.pdf file_size: 2435019 relation: main_file success: 1 file_date_updated: 2025-02-17T08:32:23Z has_accepted_license: '1' intvolume: ' 63' isi: 1 issue: '1' language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: 262-287 project: - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: SIAM Journal on Numerical Analysis publication_identifier: eissn: - 1095-7170 issn: - 0036-1429 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 63 year: '2025' ... --- _id: '17887' 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. 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." article_number: '77' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Helmut full_name: Abels, Helmut last_name: Abels - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Maximilian full_name: Moser, Maximilian id: a60047a9-da77-11eb-85b4-c4dc385ebb8c last_name: Moser citation: ama: Abels H, Fischer JL, Moser M. Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. Archive for Rational Mechanics and Analysis. 2024;248(5). doi:10.1007/s00205-024-02020-9 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. 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. 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. short: H. Abels, J.L. Fischer, M. Moser, Archive for Rational Mechanics and Analysis 248 (2024). date_created: 2024-09-08T22:01:10Z date_published: 2024-09-03T00:00:00Z date_updated: 2025-09-08T09:11:41Z day: '03' ddc: - '510' department: - _id: JuFi doi: 10.1007/s00205-024-02020-9 ec_funded: 1 external_id: arxiv: - '2311.02997' isi: - '001305530600001' pmid: - '39239088' file: - access_level: open_access checksum: 98493a05b84e4513b6394dfad4851ddf content_type: application/pdf creator: dernst date_created: 2024-09-09T08:43:32Z date_updated: 2024-09-09T08:43:32Z file_id: '17938' file_name: 2024_ArchiveRatAnalysis_Abels.pdf file_size: 811131 relation: main_file success: 1 file_date_updated: 2024-09-09T08:43:32Z has_accepted_license: '1' intvolume: ' 248' isi: 1 issue: '5' language: - iso: eng month: '09' oa: 1 oa_version: Published Version pmid: 1 project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication: Archive for Rational Mechanics and Analysis publication_identifier: eissn: - 1432-0673 issn: - 0003-9527 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Approximation of classical two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 248 year: '2024' ... --- _id: '17481' 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." 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." article_processing_charge: Yes article_type: original author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Alice full_name: Marveggio, Alice id: 25647992-AA84-11E9-9D75-8427E6697425 last_name: Marveggio 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. 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. 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. 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. short: J.L. Fischer, A. Marveggio, Annales de l’Institut Henri Poincaré C 41 (2024) 1117–1178. corr_author: '1' date_created: 2024-09-01T22:01:09Z date_published: 2024-01-24T00:00:00Z date_updated: 2026-08-12T09:15:49Z day: '24' ddc: - '510' department: - _id: JuFi doi: 10.4171/AIHPC/109 ec_funded: 1 external_id: isi: - '001293853900003' file: - access_level: open_access checksum: b5ad02d9abd5b4701269cd1ad0a1cc8f content_type: application/pdf creator: dernst date_created: 2024-09-09T07:46:42Z date_updated: 2024-09-09T07:46:42Z file_id: '17923' file_name: 2024_AnnInstHPoincare_Fischer.pdf file_size: 1348896 relation: main_file success: 1 file_date_updated: 2024-09-09T07:46:42Z has_accepted_license: '1' intvolume: ' 41' isi: 1 issue: '5' language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 1117-1178 project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication: Annales de l'Institut Henri Poincaré C publication_identifier: eissn: - 1873-1430 issn: - 0294-1449 publication_status: published publisher: EMS Press quality_controlled: '1' related_material: record: - id: '14597' relation: earlier_version status: public scopus_import: '1' status: public title: Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 41 year: '2024' ... --- _id: '14661' abstract: - lang: eng 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.' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Marcello full_name: Carioni, Marcello last_name: Carioni - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Anja full_name: Schlömerkemper, Anja last_name: Schlömerkemper citation: 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.' 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.' 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.' 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.' 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.' short: M. Carioni, J.L. Fischer, A. Schlömerkemper, Journal of Convex Analysis 30 (2023) 217–247. corr_author: '1' date_created: 2023-12-10T23:00:59Z date_published: 2023-01-01T00:00:00Z date_updated: 2024-10-09T21:07:35Z day: '01' department: - _id: JuFi external_id: arxiv: - '1811.09857' isi: - '001115503400013' intvolume: ' 30' isi: 1 issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1811.09857 month: '01' oa: 1 oa_version: Preprint page: 217-247 publication: Journal of Convex Analysis publication_identifier: eissn: - 2363-6394 issn: - 0944-6532 publication_status: published publisher: Heldermann Verlag quality_controlled: '1' scopus_import: '1' status: public title: 'External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 30 year: '2023' ... --- _id: '10550' abstract: - lang: eng 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. 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." article_number: '66' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Klemens full_name: Fellner, Klemens last_name: Fellner - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Michael full_name: Kniely, Michael id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87 last_name: Kniely orcid: 0000-0001-5645-4333 - first_name: Bao Quoc full_name: Tang, Bao Quoc last_name: Tang 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. 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. 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. 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. short: K. Fellner, J.L. Fischer, M. Kniely, B.Q. Tang, Journal of Nonlinear Science 33 (2023). date_created: 2021-12-16T12:15:35Z date_published: 2023-06-07T00:00:00Z date_updated: 2023-08-01T14:40:33Z day: '07' ddc: - '510' department: - _id: JuFi doi: 10.1007/s00332-023-09926-w external_id: arxiv: - '2109.12019' isi: - '001002343400002' file: - access_level: open_access checksum: f3f0f0886098e31c81116cff8183750b content_type: application/pdf creator: dernst date_created: 2023-06-19T07:33:53Z date_updated: 2023-06-19T07:33:53Z file_id: '13149' file_name: 2023_JourNonlinearScience_Fellner.pdf file_size: 742315 relation: main_file success: 1 file_date_updated: 2023-06-19T07:33:53Z has_accepted_license: '1' intvolume: ' 33' isi: 1 language: - iso: eng month: '06' oa: 1 oa_version: Published Version publication: Journal of Nonlinear Science publication_identifier: eissn: - 1432-1467 issn: - 0938-8974 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 33 year: '2023' ... --- _id: '10551' abstract: - lang: eng 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.' 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)." article_number: '76' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Federico full_name: Cornalba, Federico id: 2CEB641C-A400-11E9-A717-D712E6697425 last_name: Cornalba orcid: 0000-0002-6269-5149 - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X citation: 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 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 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. 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. 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. short: F. Cornalba, J.L. Fischer, Archive for Rational Mechanics and Analysis 247 (2023). corr_author: '1' date_created: 2021-12-16T12:16:03Z date_published: 2023-08-04T00:00:00Z date_updated: 2025-04-23T13:06:01Z day: '04' ddc: - '510' department: - _id: JuFi doi: 10.1007/s00205-023-01903-7 ec_funded: 1 external_id: arxiv: - '2109.06500' isi: - '001043086800001' pmid: - '37547904' file: - access_level: open_access checksum: 4529eeff170b6745a461d397ee611b5a content_type: application/pdf creator: dernst date_created: 2024-01-30T12:09:34Z date_updated: 2024-01-30T12:09:34Z file_id: '14904' file_name: 2023_ArchiveRationalMech_Cornalba.pdf file_size: 1851185 relation: main_file success: 1 file_date_updated: 2024-01-30T12:09:34Z has_accepted_license: '1' intvolume: ' 247' isi: 1 issue: '5' language: - iso: eng month: '08' oa: 1 oa_version: Published Version pmid: 1 project: - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: Archive for Rational Mechanics and Analysis publication_identifier: eissn: - 1432-0673 issn: - 0003-9527 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 247 year: '2023' ... --- _id: '12304' abstract: - lang: eng 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.' 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. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Nicola full_name: De Nitti, Nicola last_name: De Nitti - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X citation: 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 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. 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. 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. short: N. De Nitti, J.L. Fischer, Communications in Partial Differential Equations 47 (2022) 1394–1434. corr_author: '1' date_created: 2023-01-16T10:06:50Z date_published: 2022-07-01T00:00:00Z date_updated: 2024-10-09T21:03:57Z day: '01' department: - _id: JuFi doi: 10.1080/03605302.2022.2056702 external_id: arxiv: - '1907.05342' isi: - '000805689800001' intvolume: ' 47' isi: 1 issue: '7' keyword: - Applied Mathematics - Analysis language: - iso: eng main_file_link: - open_access: '1' url: ' https://doi.org/10.48550/arXiv.1907.05342' month: '07' oa: 1 oa_version: Preprint page: 1394-1434 publication: Communications in Partial Differential Equations publication_identifier: eissn: - 1532-4133 issn: - 0360-5302 publication_status: published publisher: Taylor & Francis quality_controlled: '1' scopus_import: '1' status: public title: Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 47 year: '2022' ... --- _id: '10547' abstract: - lang: eng 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." 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). article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Katharina full_name: Hopf, Katharina last_name: Hopf - first_name: Michael full_name: Kniely, Michael id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87 last_name: Kniely orcid: 0000-0001-5645-4333 - first_name: Alexander full_name: Mielke, Alexander last_name: Mielke citation: 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 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 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. 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. 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. 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. short: J.L. Fischer, K. Hopf, M. Kniely, A. Mielke, SIAM Journal on Mathematical Analysis 54 (2022) 220–267. date_created: 2021-12-16T12:08:56Z date_published: 2022-01-04T00:00:00Z date_updated: 2023-08-02T13:37:03Z day: '04' department: - _id: JuFi doi: 10.1137/20M1387237 external_id: arxiv: - '2012.03792 ' isi: - '000762768000006' intvolume: ' 54' isi: 1 issue: '1' keyword: - Energy-Reaction-Diffusion Systems - Cross Diffusion - Global-In-Time Existence of Weak/Renormalised Solutions - Entropy Method - Onsager System - Soret/Dufour Effect language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2012.03792 month: '01' oa: 1 oa_version: Preprint page: 220-267 publication: SIAM Journal on Mathematical Analysis publication_identifier: issn: - 0036-1410 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: Global existence analysis of energy-reaction-diffusion systems type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 54 year: '2022' ... --- _id: '10548' abstract: - lang: eng 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." 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). article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Mitia full_name: Duerinckx, Mitia last_name: Duerinckx - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Antoine full_name: Gloria, Antoine last_name: Gloria citation: 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 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 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. 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. 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. 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. short: M. Duerinckx, J.L. Fischer, A. Gloria, Annals of Applied Probability 32 (2022) 1179–1209. corr_author: '1' date_created: 2021-12-16T12:10:16Z date_published: 2022-04-28T00:00:00Z date_updated: 2026-08-12T09:22:31Z day: '28' department: - _id: JuFi doi: 10.1214/21-AAP1705 external_id: arxiv: - '1910.04088' isi: - '000791003700011' intvolume: ' 32' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1910.04088 month: '04' oa: 1 oa_version: Preprint page: 1179-1209 publication: Annals of Applied Probability publication_identifier: issn: - 1050-5164 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' scopus_import: '1' status: public title: Scaling limit of the homogenization commutator for Gaussian coefficient fields type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 32 year: '2022' ... --- _id: '14597' 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." article_number: '2203.17143' article_processing_charge: No arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Alice full_name: Marveggio, Alice id: 25647992-AA84-11E9-9D75-8427E6697425 last_name: Marveggio citation: 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 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 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. ieee: J. L. Fischer and A. Marveggio, “Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow,” arXiv. . ista: Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow. arXiv, 2203.17143. 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. short: J.L. Fischer, A. Marveggio, ArXiv (n.d.). corr_author: '1' date_created: 2023-11-23T09:30:02Z date_published: 2022-03-31T00:00:00Z date_updated: 2026-08-12T09:15:49Z day: '31' department: - _id: JuFi doi: 10.48550/ARXIV.2203.17143 ec_funded: 1 external_id: arxiv: - '2203.17143' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2203.17143 month: '03' oa: 1 oa_version: Preprint project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication: arXiv publication_status: draft related_material: record: - id: '14587' relation: dissertation_contains status: public - id: '17481' relation: later_version status: public status: public title: Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2022' ... --- _id: '10549' 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. 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. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Stefan full_name: Neukamm, Stefan last_name: Neukamm 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 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 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. 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. 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. 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. short: J.L. Fischer, S. Neukamm, Archive for Rational Mechanics and Analysis 242 (2021) 343–452. date_created: 2021-12-16T12:12:33Z date_published: 2021-06-30T00:00:00Z date_updated: 2023-08-17T06:23:21Z day: '30' ddc: - '530' department: - _id: JuFi doi: 10.1007/s00205-021-01686-9 external_id: arxiv: - '1908.02273' isi: - '000668431200001' file: - access_level: open_access checksum: cc830b739aed83ca2e32c4e0ce266a4c content_type: application/pdf creator: cchlebak date_created: 2021-12-16T14:58:08Z date_updated: 2021-12-16T14:58:08Z file_id: '10558' file_name: 2021_ArchRatMechAnalysis_Fischer.pdf file_size: 1640121 relation: main_file success: 1 file_date_updated: 2021-12-16T14:58:08Z has_accepted_license: '1' intvolume: ' 242' isi: 1 issue: '1' keyword: - Mechanical Engineering - Mathematics (miscellaneous) - Analysis language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: 343-452 publication: Archive for Rational Mechanics and Analysis publication_identifier: eissn: - 1432-0673 issn: - 0003-9527 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 242 year: '2021' ... --- _id: '9335' abstract: - lang: eng 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.' acknowledgement: This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Daniel full_name: Matthes, Daniel last_name: Matthes citation: 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 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. 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. 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. date_created: 2021-04-18T22:01:42Z date_published: 2021-01-01T00:00:00Z date_updated: 2023-08-08T13:10:40Z day: '01' department: - _id: JuFi doi: 10.1137/19M1300017 external_id: arxiv: - '1911.04185' isi: - '000625044600003' intvolume: ' 59' isi: 1 issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1911.04185 month: '01' oa: 1 oa_version: Preprint page: 60-87 publication: SIAM Journal on Numerical Analysis publication_identifier: issn: - 0036-1429 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: The waiting time phenomenon in spatially discretized porous medium and thin film equations type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 59 year: '2021' ... --- _id: '9352' abstract: - lang: eng 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. 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.' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Dietmar full_name: Gallistl, Dietmar last_name: Gallistl - first_name: Dietmar full_name: Peterseim, Dietmar last_name: Peterseim citation: 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 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. 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. 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. short: J.L. Fischer, D. Gallistl, D. Peterseim, SIAM Journal on Numerical Analysis 59 (2021) 660–674. date_created: 2021-04-25T22:01:31Z date_published: 2021-03-09T00:00:00Z date_updated: 2023-08-08T13:13:37Z day: '09' department: - _id: JuFi doi: 10.1137/19M1308992 external_id: arxiv: - '1912.11646' isi: - '000646030400003' intvolume: ' 59' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1912.11646 month: '03' oa: 1 oa_version: Preprint page: 660-674 publication: SIAM Journal on Numerical Analysis publication_identifier: issn: - 0036-1429 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: A priori error analysis of a numerical stochastic homogenization method type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 59 year: '2021' ... --- _id: '10012' 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. 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. article_number: '2003.05478' article_processing_charge: No arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Sebastian full_name: Hensel, Sebastian id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87 last_name: Hensel orcid: 0000-0001-7252-8072 - first_name: Tim full_name: Laux, Tim last_name: Laux - first_name: Thilo full_name: Simon, Thilo last_name: Simon citation: 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' 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.' 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. .' 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.' 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.' short: J.L. Fischer, S. Hensel, T. Laux, T. Simon, ArXiv (n.d.). date_created: 2021-09-13T12:17:11Z date_published: 2020-03-11T00:00:00Z date_updated: 2026-04-08T07:01:01Z day: '11' department: - _id: JuFi doi: 10.48550/arXiv.2003.05478 ec_funded: 1 external_id: arxiv: - '2003.05478' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2003.05478 month: '03' oa: 1 oa_version: Preprint project: - _id: 2564DBCA-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '665385' name: International IST Doctoral Program publication: arXiv publication_status: draft related_material: record: - id: '10007' relation: dissertation_contains status: public status: public title: 'The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions' type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2020' ... --- _id: '8697' 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. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Michael full_name: Kniely, Michael id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87 last_name: Kniely orcid: 0000-0001-5645-4333 citation: 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 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 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. 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. 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. 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. short: J.L. Fischer, M. Kniely, Nonlinearity 33 (2020) 5733–5772. corr_author: '1' date_created: 2020-10-25T23:01:16Z date_published: 2020-11-01T00:00:00Z date_updated: 2026-04-02T14:31:34Z day: '01' ddc: - '510' department: - _id: JuFi doi: 10.1088/1361-6544/ab9728 external_id: arxiv: - '1906.12245' isi: - '000576492700001' file: - access_level: open_access checksum: ed90bc6eb5f32ee6157fef7f3aabc057 content_type: application/pdf creator: cziletti date_created: 2020-10-27T12:09:57Z date_updated: 2020-10-27T12:09:57Z file_id: '8710' file_name: 2020_Nonlinearity_Fischer.pdf file_size: 1223899 relation: main_file success: 1 file_date_updated: 2020-10-27T12:09:57Z has_accepted_license: '1' intvolume: ' 33' isi: 1 issue: '11' language: - iso: eng license: https://creativecommons.org/licenses/by/3.0/ month: '11' oa: 1 oa_version: Published Version page: 5733-5772 publication: Nonlinearity publication_identifier: eissn: - 1361-6544 issn: - 0951-7715 publication_status: published publisher: IOP Publishing quality_controlled: '1' scopus_import: '1' status: public title: Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode name: Creative Commons Attribution 3.0 Unported (CC BY 3.0) short: CC BY (3.0) type: journal_article user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd volume: 33 year: '2020' ... --- _id: '9039' 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. 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." article_processing_charge: No article_type: original author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Tim full_name: Laux, Tim last_name: Laux - first_name: Theresa M. full_name: Simon, Theresa M. last_name: Simon 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.' 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.' 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. corr_author: '1' date_created: 2021-01-24T23:01:09Z date_published: 2020-12-15T00:00:00Z date_updated: 2025-07-10T12:01:32Z day: '15' ddc: - '510' department: - _id: JuFi doi: 10.1137/20M1322182 ec_funded: 1 external_id: isi: - '000600695200027' file: - access_level: open_access checksum: 21aa1cf4c30a86a00cae15a984819b5d content_type: application/pdf creator: dernst date_created: 2021-01-25T07:48:39Z date_updated: 2021-01-25T07:48:39Z file_id: '9041' file_name: 2020_SIAM_Fischer.pdf file_size: 310655 relation: main_file success: 1 file_date_updated: 2021-01-25T07:48:39Z has_accepted_license: '1' intvolume: ' 52' isi: 1 issue: '6' language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 6222-6233 project: - _id: 2564DBCA-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '665385' name: International IST Doctoral Program publication: SIAM Journal on Mathematical Analysis publication_identifier: eissn: - 1095-7154 issn: - 0036-1410 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: 'Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 52 year: '2020' ... --- _id: '7489' 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.' article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Sebastian full_name: Hensel, Sebastian id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87 last_name: Hensel orcid: 0000-0001-7252-8072 citation: 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 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. 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. 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. short: J.L. Fischer, S. Hensel, Archive for Rational Mechanics and Analysis 236 (2020) 967–1087. corr_author: '1' date_created: 2020-02-16T23:00:50Z date_published: 2020-05-01T00:00:00Z date_updated: 2026-04-08T07:01:01Z day: '01' ddc: - '530' - '532' department: - _id: JuFi doi: 10.1007/s00205-019-01486-2 ec_funded: 1 external_id: isi: - '000511060200001' file: - access_level: open_access checksum: f107e21b58f5930876f47144be37cf6c content_type: application/pdf creator: dernst date_created: 2020-11-20T09:14:22Z date_updated: 2020-11-20T09:14:22Z file_id: '8779' file_name: 2020_ArchRatMechAn_Fischer.pdf file_size: 1897571 relation: main_file success: 1 file_date_updated: 2020-11-20T09:14:22Z has_accepted_license: '1' intvolume: ' 236' isi: 1 language: - iso: eng month: '05' oa: 1 oa_version: Published Version page: 967-1087 project: - _id: 2564DBCA-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '665385' name: International IST Doctoral Program - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Archive for Rational Mechanics and Analysis publication_identifier: eissn: - 1432-0673 issn: - 0003-9527 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '10007' relation: dissertation_contains status: public scopus_import: '1' status: public title: Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 236 year: '2020' ... --- _id: '151' 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. article_processing_charge: No arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Olivier full_name: Kneuss, Olivier last_name: Kneuss citation: 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 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 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. 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. short: J.L. Fischer, O. Kneuss, Journal of Differential Equations 266 (2019) 257–311. date_created: 2018-12-11T11:44:54Z date_published: 2019-01-05T00:00:00Z date_updated: 2023-09-08T13:25:35Z day: '05' department: - _id: JuFi doi: 10.1016/j.jde.2018.07.045 external_id: arxiv: - '1408.1587' isi: - '000449108500010' intvolume: ' 266' isi: 1 issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1408.1587 month: '01' oa: 1 oa_version: Preprint page: 257 - 311 publication: Journal of Differential Equations publication_status: published publisher: Elsevier publist_id: '7770' quality_controlled: '1' scopus_import: '1' status: public title: Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 266 year: '2019' ... --- _id: '6617' 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.' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X citation: 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 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 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. 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. short: J.L. Fischer, Archive for Rational Mechanics and Analysis 234 (2019) 635–726. corr_author: '1' date_created: 2019-07-07T21:59:23Z date_published: 2019-11-01T00:00:00Z date_updated: 2025-04-15T06:53:15Z day: '01' ddc: - '500' department: - _id: JuFi doi: 10.1007/s00205-019-01400-w external_id: arxiv: - '1807.00834' isi: - '000482386000006' file: - access_level: open_access checksum: 4cff75fa6addb0770991ad9c474ab404 content_type: application/pdf creator: kschuh date_created: 2019-07-08T15:56:47Z date_updated: 2020-07-14T12:47:34Z file_id: '6626' file_name: Springer_2019_Fischer.pdf file_size: 1377659 relation: main_file file_date_updated: 2020-07-14T12:47:34Z has_accepted_license: '1' intvolume: ' 234' isi: 1 issue: '2' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: 635–726 project: - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Archive for Rational Mechanics and Analysis publication_identifier: eissn: - 1432-0673 issn: - 0003-9527 publication_status: published publisher: Springer quality_controlled: '1' scopus_import: '1' status: public title: The choice of representative volumes in the approximation of effective properties of random materials tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 234 year: '2019' ...