--- _id: '14587' abstract: - lang: eng text: "This thesis concerns the application of variational methods to the study of evolution problems arising in fluid mechanics and in material sciences. The main focus is on weak-strong stability properties of some curvature driven interface evolution problems, such as the two-phase Navier–Stokes flow with surface tension and multiphase mean curvature flow, and on the phase-field approximation of the latter. Furthermore, we discuss a variational approach to the study of a class of doubly nonlinear wave equations.\r\nFirst, we consider the two-phase Navier–Stokes flow with surface tension within a bounded domain. The two fluids are immiscible and separated by a sharp interface, which intersects the boundary of the domain at a constant contact angle of ninety degree. We devise a suitable concept of varifolds solutions for the associated interface evolution problem and we establish a weak-strong uniqueness principle in case of a two dimensional ambient space. In order to focus on the boundary effects and on the singular geometry of the evolving domains, we work for simplicity in the regime of same viscosities for the two fluids.\r\nThe core of the thesis consists in the rigorous proof of the convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow for a suitable class of multi- well potentials and for well-prepared initial data. We even establish a rate of convergence. Our relative energy approach relies on the concept of gradient-flow calibration for branching singularities in multiphase mean curvature flow and thus enables us to overcome the limitations of other approaches. To the best of the author’s knowledge, our result is the first quantitative and unconditional one available in the literature for the vectorial/multiphase setting.\r\nThis thesis also contains a first study of weak-strong stability for planar multiphase mean curvature flow beyond the singularity resulting from a topology change. Previous weak-strong results are indeed limited to time horizons before the first topology change of the strong solution. We consider circular topology changes and we prove weak-strong stability for BV solutions to planar multiphase mean curvature flow beyond the associated singular times by dynamically adapting the strong solutions to the weak one by means of a space-time shift.\r\nIn the context of interface evolution problems, our proofs for the main results of this thesis are based on the relative energy technique, relying on novel suitable notions of relative energy functionals, which in particular measure the interface error. Our statements follow from the resulting stability estimates for the relative energy associated to the problem.\r\nAt last, we introduce a variational approach to the study of nonlinear evolution problems. This approach hinges on the minimization of a parameter dependent family of convex functionals over entire trajectories, known as Weighted Inertia-Dissipation-Energy (WIDE) functionals. We consider a class of doubly nonlinear wave equations and establish the convergence, up to subsequences, of the associated WIDE minimizers to a solution of the target problem as the parameter goes to zero." acknowledgement: The research projects contained in this thesis have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819). alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Alice full_name: Marveggio, Alice id: 25647992-AA84-11E9-9D75-8427E6697425 last_name: Marveggio citation: ama: Marveggio A. Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences. 2023. doi:10.15479/at:ista:14587 apa: Marveggio, A. (2023). Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:14587 chicago: Marveggio, Alice. “Weak-Strong Stability and Phase-Field Approximation of Interface Evolution Problems in Fluid Mechanics and in Material Sciences.” Institute of Science and Technology Austria, 2023. https://doi.org/10.15479/at:ista:14587. ieee: A. Marveggio, “Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences,” Institute of Science and Technology Austria, 2023. ista: Marveggio A. 2023. Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences. Institute of Science and Technology Austria. mla: Marveggio, Alice. Weak-Strong Stability and Phase-Field Approximation of Interface Evolution Problems in Fluid Mechanics and in Material Sciences. Institute of Science and Technology Austria, 2023, doi:10.15479/at:ista:14587. short: A. Marveggio, Weak-Strong Stability and Phase-Field Approximation of Interface Evolution Problems in Fluid Mechanics and in Material Sciences, Institute of Science and Technology Austria, 2023. date_created: 2023-11-21T11:41:05Z date_published: 2023-11-21T00:00:00Z date_updated: 2024-03-22T13:21:28Z day: '21' ddc: - '515' degree_awarded: PhD department: - _id: GradSch - _id: JuFi doi: 10.15479/at:ista:14587 ec_funded: 1 file: - access_level: open_access checksum: 6c7db4cc86da6cdc79f7f358dc7755d4 content_type: application/pdf creator: amarvegg date_created: 2023-11-29T09:09:31Z date_updated: 2023-11-29T09:09:31Z file_id: '14626' file_name: thesis_Marveggio.pdf file_size: 2881100 relation: main_file success: 1 - access_level: closed checksum: 52f28bdf95ec82cff39f3685f9c48e7d content_type: application/zip creator: amarvegg date_created: 2023-11-29T09:10:19Z date_updated: 2024-03-20T12:28:32Z file_id: '14627' file_name: Thesis_Marveggio.zip file_size: 10189696 relation: source_file file_date_updated: 2024-03-20T12:28:32Z has_accepted_license: '1' language: - iso: eng license: https://creativecommons.org/licenses/by-nc-sa/4.0/ month: '11' oa: 1 oa_version: Published Version page: '228' project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication_identifier: issn: - 2663 - 337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '11842' relation: part_of_dissertation status: public - id: '14597' relation: part_of_dissertation status: public status: public supervisor: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X title: Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences tmp: image: /images/cc_by_nc_sa.png legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) short: CC BY-NC-SA (4.0) type: dissertation user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2023' ... --- _id: '11842' abstract: - lang: eng text: We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier–Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With respect to the motion of the interface, we consider pure transport by the fluid flow. Along the boundary of the domain, a complete slip boundary condition for the fluid velocities and a constant ninety degree contact angle condition for the interface are assumed. In the present work, we devise for the resulting evolution problem a suitable weak solution concept based on the framework of varifolds and establish as the main result a weak-strong uniqueness principle in 2D. The proof is based on a relative entropy argument and requires a non-trivial further development of ideas from the recent work of Fischer and the first author (Arch. Ration. Mech. Anal. 236, 2020) to incorporate the contact angle condition. To focus on the effects of the necessarily singular geometry of the evolving fluid domains, we work for simplicity in the regime of same viscosities for the two fluids. acknowledgement: The authors warmly thank their former resp. current PhD advisor Julian Fischer for the suggestion of this problem and for valuable initial discussions on the subjects of this paper. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819) , and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. article_number: '93' article_processing_charge: No article_type: original author: - first_name: Sebastian full_name: Hensel, Sebastian id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87 last_name: Hensel orcid: 0000-0001-7252-8072 - first_name: Alice full_name: Marveggio, Alice id: 25647992-AA84-11E9-9D75-8427E6697425 last_name: Marveggio citation: ama: Hensel S, Marveggio A. Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety degree contact angle and same viscosities. Journal of Mathematical Fluid Mechanics. 2022;24(3). doi:10.1007/s00021-022-00722-2 apa: Hensel, S., & Marveggio, A. (2022). Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety degree contact angle and same viscosities. Journal of Mathematical Fluid Mechanics. Springer Nature. https://doi.org/10.1007/s00021-022-00722-2 chicago: Hensel, Sebastian, and Alice Marveggio. “Weak-Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities.” Journal of Mathematical Fluid Mechanics. Springer Nature, 2022. https://doi.org/10.1007/s00021-022-00722-2. ieee: S. Hensel and A. Marveggio, “Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety degree contact angle and same viscosities,” Journal of Mathematical Fluid Mechanics, vol. 24, no. 3. Springer Nature, 2022. ista: Hensel S, Marveggio A. 2022. Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety degree contact angle and same viscosities. Journal of Mathematical Fluid Mechanics. 24(3), 93. mla: Hensel, Sebastian, and Alice Marveggio. “Weak-Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities.” Journal of Mathematical Fluid Mechanics, vol. 24, no. 3, 93, Springer Nature, 2022, doi:10.1007/s00021-022-00722-2. short: S. Hensel, A. Marveggio, Journal of Mathematical Fluid Mechanics 24 (2022). date_created: 2022-08-14T22:01:45Z date_published: 2022-08-01T00:00:00Z date_updated: 2024-03-22T13:21:27Z day: '01' ddc: - '510' department: - _id: JuFi doi: 10.1007/s00021-022-00722-2 ec_funded: 1 external_id: arxiv: - '2112.11154' isi: - '000834834300001' file: - access_level: open_access checksum: 75c5f286300e6f0539cf57b4dba108d5 content_type: application/pdf creator: cchlebak date_created: 2022-08-16T06:55:22Z date_updated: 2022-08-16T06:55:22Z file_id: '11848' file_name: 2022_JMathFluidMech_Hensel.pdf file_size: 2045570 relation: main_file success: 1 file_date_updated: 2022-08-16T06:55:22Z has_accepted_license: '1' intvolume: ' 24' isi: 1 issue: '3' language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '08' oa: 1 oa_version: Published Version project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication: Journal of Mathematical Fluid Mechanics publication_identifier: eissn: - 1422-6952 issn: - 1422-6928 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '14587' relation: dissertation_contains status: public scopus_import: '1' status: public title: Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety degree contact angle and same viscosities 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: 24 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_processing_charge: No 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, 10.48550/ARXIV.2203.17143. mla: Fischer, Julian L., and Alice Marveggio. “Quantitative Convergence of the Vectorial Allen-Cahn Equation towards Multiphase Mean Curvature Flow.” ArXiv, doi:10.48550/ARXIV.2203.17143. short: J.L. Fischer, A. Marveggio, ArXiv (n.d.). date_created: 2023-11-23T09:30:02Z date_published: 2022-03-31T00:00:00Z date_updated: 2024-03-22T13:21:27Z 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: submitted related_material: record: - id: '14587' relation: dissertation_contains status: public status: public title: Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2022' ... --- _id: '8792' abstract: - lang: eng text: This paper is concerned with a non-isothermal Cahn-Hilliard model based on a microforce balance. The model was derived by A. Miranville and G. Schimperna starting from the two fundamental laws of Thermodynamics, following M. Gurtin's two-scale approach. The main working assumptions are made on the behaviour of the heat flux as the absolute temperature tends to zero and to infinity. A suitable Ginzburg-Landau free energy is considered. Global-in-time existence for the initial-boundary value problem associated to the entropy formulation and, in a subcase, also to the weak formulation of the model is proved by deriving suitable a priori estimates and by showing weak sequential stability of families of approximating solutions. At last, some highlights are given regarding a possible approximation scheme compatible with the a-priori estimates available for the system. acknowledgement: G. Schimperna has been partially supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). article_processing_charge: No article_type: original author: - first_name: Alice full_name: Marveggio, Alice id: 25647992-AA84-11E9-9D75-8427E6697425 last_name: Marveggio - first_name: Giulio full_name: Schimperna, Giulio last_name: Schimperna citation: ama: Marveggio A, Schimperna G. On a non-isothermal Cahn-Hilliard model based on a microforce balance. Journal of Differential Equations. 2021;274(2):924-970. doi:10.1016/j.jde.2020.10.030 apa: Marveggio, A., & Schimperna, G. (2021). On a non-isothermal Cahn-Hilliard model based on a microforce balance. Journal of Differential Equations. Elsevier. https://doi.org/10.1016/j.jde.2020.10.030 chicago: Marveggio, Alice, and Giulio Schimperna. “On a Non-Isothermal Cahn-Hilliard Model Based on a Microforce Balance.” Journal of Differential Equations. Elsevier, 2021. https://doi.org/10.1016/j.jde.2020.10.030. ieee: A. Marveggio and G. Schimperna, “On a non-isothermal Cahn-Hilliard model based on a microforce balance,” Journal of Differential Equations, vol. 274, no. 2. Elsevier, pp. 924–970, 2021. ista: Marveggio A, Schimperna G. 2021. On a non-isothermal Cahn-Hilliard model based on a microforce balance. Journal of Differential Equations. 274(2), 924–970. mla: Marveggio, Alice, and Giulio Schimperna. “On a Non-Isothermal Cahn-Hilliard Model Based on a Microforce Balance.” Journal of Differential Equations, vol. 274, no. 2, Elsevier, 2021, pp. 924–70, doi:10.1016/j.jde.2020.10.030. short: A. Marveggio, G. Schimperna, Journal of Differential Equations 274 (2021) 924–970. date_created: 2020-11-22T23:01:26Z date_published: 2021-02-15T00:00:00Z date_updated: 2023-08-04T11:12:16Z day: '15' department: - _id: JuFi doi: 10.1016/j.jde.2020.10.030 external_id: arxiv: - '2004.02618' isi: - '000600845300023' intvolume: ' 274' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2004.02618 month: '02' oa: 1 oa_version: Preprint page: 924-970 publication: Journal of Differential Equations publication_identifier: eissn: - '10902732' issn: - '00220396' publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: On a non-isothermal Cahn-Hilliard model based on a microforce balance type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 274 year: '2021' ...