[{"alternative_title":["ISTA Thesis"],"tmp":{"short":"CC BY-NC-SA (4.0)","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode"},"related_material":{"record":[{"relation":"part_of_dissertation","id":"11842","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"14597"}]},"ddc":["515"],"year":"2023","doi":"10.15479/at:ista:14587","ec_funded":1,"title":"Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences","oa":1,"date_updated":"2023-11-30T13:25:03Z","article_processing_charge":"No","_id":"14587","publication_identifier":{"issn":["2663 - 337X"]},"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).","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","project":[{"grant_number":"948819","name":"Bridging Scales in Random Materials","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020"}],"oa_version":"Published Version","publication_status":"published","citation":{"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. <a href=\"https://doi.org/10.15479/at:ista:14587\">https://doi.org/10.15479/at:ista:14587</a>.","apa":"Marveggio, A. (2023). <i>Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:14587\">https://doi.org/10.15479/at:ista:14587</a>","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.","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.","ama":"Marveggio A. Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences. 2023. doi:<a href=\"https://doi.org/10.15479/at:ista:14587\">10.15479/at:ista:14587</a>","mla":"Marveggio, Alice. <i>Weak-Strong Stability and Phase-Field Approximation of Interface Evolution Problems in Fluid Mechanics and in Material Sciences</i>. Institute of Science and Technology Austria, 2023, doi:<a href=\"https://doi.org/10.15479/at:ista:14587\">10.15479/at:ista:14587</a>."},"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."}],"author":[{"first_name":"Alice","last_name":"Marveggio","full_name":"Marveggio, Alice","id":"25647992-AA84-11E9-9D75-8427E6697425"}],"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","degree_awarded":"PhD","has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"JuFi"}],"date_created":"2023-11-21T11:41:05Z","file":[{"success":1,"content_type":"application/pdf","relation":"main_file","creator":"amarvegg","file_id":"14626","file_name":"thesis_Marveggio.pdf","file_size":2881100,"date_created":"2023-11-29T09:09:31Z","checksum":"6c7db4cc86da6cdc79f7f358dc7755d4","date_updated":"2023-11-29T09:09:31Z","access_level":"open_access"},{"checksum":"52f28bdf95ec82cff39f3685f9c48e7d","date_created":"2023-11-29T09:10:19Z","file_size":10189696,"file_name":"Thesis_Marveggio.zip","access_level":"open_access","date_updated":"2023-11-29T09:28:30Z","file_id":"14627","creator":"amarvegg","relation":"source_file","content_type":"application/zip"}],"month":"11","date_published":"2023-11-21T00:00:00Z","publisher":"Institute of Science and Technology Austria","language":[{"iso":"eng"}],"file_date_updated":"2023-11-29T09:28:30Z","page":"228","day":"21","type":"dissertation","supervisor":[{"full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"}],"status":"public"},{"status":"public","supervisor":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X"}],"type":"dissertation","day":"14","file_date_updated":"2021-09-15T14:37:30Z","page":"300","language":[{"iso":"eng"}],"publisher":"Institute of Science and Technology Austria","date_published":"2021-09-14T00:00:00Z","month":"09","date_created":"2021-09-13T11:12:34Z","file":[{"checksum":"c8475faaf0b680b4971f638f1db16347","date_created":"2021-09-13T11:03:24Z","file_size":15022154,"file_name":"thesis_final_Hensel.zip","access_level":"closed","date_updated":"2021-09-15T14:37:30Z","file_id":"10008","creator":"shensel","relation":"source_file","content_type":"application/x-zip-compressed"},{"date_updated":"2021-09-14T09:52:47Z","access_level":"open_access","date_created":"2021-09-13T14:18:56Z","checksum":"1a609937aa5275452822f45f2da17f07","file_name":"thesis_final_Hensel.pdf","file_size":6583638,"file_id":"10014","creator":"shensel","content_type":"application/pdf","relation":"main_file"}],"department":[{"_id":"GradSch"},{"_id":"JuFi"}],"has_accepted_license":"1","degree_awarded":"PhD","author":[{"id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastian","last_name":"Hensel","full_name":"Hensel, Sebastian","orcid":"0000-0001-7252-8072"}],"abstract":[{"lang":"eng","text":"The present thesis is concerned with the derivation of weak-strong uniqueness principles for curvature driven interface evolution problems not satisfying a comparison principle. The specific examples being treated are two-phase Navier-Stokes flow with surface tension, modeling the evolution of two incompressible, viscous and immiscible fluids separated by a sharp interface, and multiphase mean curvature flow, which serves as an idealized model for the motion of grain boundaries in an annealing polycrystalline material. Our main results - obtained in joint works with Julian Fischer, Tim Laux and Theresa M. Simon - state that prior to the formation of geometric singularities due to topology changes, the weak solution concept of Abels (Interfaces Free Bound. 9, 2007) to two-phase Navier-Stokes flow with surface tension and the weak solution concept of Laux and Otto (Calc. Var. Partial Differential Equations 55, 2016) to multiphase mean curvature flow (for networks in R^2 or double bubbles in R^3) represents the unique solution to these interface evolution problems within the class of classical solutions, respectively. To the best of the author's knowledge, for interface evolution problems not admitting a geometric comparison principle the derivation of a weak-strong uniqueness principle represented an open problem, so that the works contained in the present thesis constitute the first positive results in this direction. The key ingredient of our approach consists of the introduction of a novel concept of relative entropies for a class of curvature driven interface evolution problems, for which the associated energy contains an interfacial contribution being proportional to the surface area of the evolving (network of) interface(s). The interfacial part of the relative entropy gives sufficient control on the interface error between a weak and a classical solution, and its time evolution can be computed, at least in principle, for any energy dissipating weak solution concept. A resulting stability estimate for the relative entropy essentially entails the above mentioned weak-strong uniqueness principles. The present thesis contains a detailed introduction to our relative entropy approach, which in particular highlights potential applications to other problems in curvature driven interface evolution not treated in this thesis."}],"publication_status":"published","citation":{"mla":"Hensel, Sebastian. <i>Curvature Driven Interface Evolution: Uniqueness Properties of Weak Solution Concepts</i>. Institute of Science and Technology Austria, 2021, doi:<a href=\"https://doi.org/10.15479/at:ista:10007\">10.15479/at:ista:10007</a>.","ama":"Hensel S. Curvature driven interface evolution: Uniqueness properties of weak solution concepts. 2021. doi:<a href=\"https://doi.org/10.15479/at:ista:10007\">10.15479/at:ista:10007</a>","short":"S. Hensel, Curvature Driven Interface Evolution: Uniqueness Properties of Weak Solution Concepts, Institute of Science and Technology Austria, 2021.","ista":"Hensel S. 2021. Curvature driven interface evolution: Uniqueness properties of weak solution concepts. Institute of Science and Technology Austria.","ieee":"S. Hensel, “Curvature driven interface evolution: Uniqueness properties of weak solution concepts,” Institute of Science and Technology Austria, 2021.","apa":"Hensel, S. (2021). <i>Curvature driven interface evolution: Uniqueness properties of weak solution concepts</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:10007\">https://doi.org/10.15479/at:ista:10007</a>","chicago":"Hensel, Sebastian. “Curvature Driven Interface Evolution: Uniqueness Properties of Weak Solution Concepts.” Institute of Science and Technology Austria, 2021. <a href=\"https://doi.org/10.15479/at:ista:10007\">https://doi.org/10.15479/at:ista:10007</a>."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","project":[{"name":"International IST Doctoral Program","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"665385"},{"_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","name":"Bridging Scales in Random Materials","call_identifier":"H2020","grant_number":"948819"}],"oa_version":"Published Version","_id":"10007","publication_identifier":{"issn":["2663-337X"]},"oa":1,"date_updated":"2023-09-07T13:30:45Z","article_processing_charge":"No","title":"Curvature driven interface evolution: Uniqueness properties of weak solution concepts","ec_funded":1,"year":"2021","doi":"10.15479/at:ista:10007","ddc":["515"],"related_material":{"record":[{"relation":"part_of_dissertation","id":"10012","status":"public"},{"status":"public","id":"10013","relation":"part_of_dissertation"},{"id":"7489","relation":"part_of_dissertation","status":"public"}]},"alternative_title":["ISTA Thesis"]}]