{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/382116"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/382116","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"A dynamical systems approach to the study of local phenomena in fluid flows","abstract":"The emergence of data-driven methods has fueled a newfound interest in the utilization of nonlinear tools from dynamical systems theory. In fluid dynamics, prominent examples are Koopman eigenfunctions (through dynamic mode decomposition) and spectral submanifolds. Due to their immense popularity, both of these techniques have been studied extensively, to the point that most aspects regarding their implementation are now fully fleshed out. However, there is one issue that has remained mostly untouched, and it is perhaps the most pressing one -- the mathematical foundation of these tools. While the theory is well understood in the case of finite-dimensional systems, fluid dynamics is inherently infinite-dimensional, which calls for a more careful assessment. The goal of this thesis is to address precisely this concern, and provide a rigorous backbone to the routinely conducted numerical studies. To do so, we give existence and uniqueness results for spectral submanifolds, invariant foliations and Koopman eigenfunctions in the full, infinite-dimensional phase space of certain partial differential equations, notably the Navier-Stokes system on a bounded domain.","abstract_html":"The emergence of data-driven methods has fueled a newfound interest in the utilization of nonlinear tools from dynamical systems theory. In fluid dynamics, prominent examples are Koopman eigenfunctions (through dynamic mode decomposition) and spectral submanifolds. Due to their immense popularity, both of these techniques have been studied extensively, to the point that most aspects regarding their implementation are now fully fleshed out. However, there is one issue that has remained mostly untouched, and it is perhaps the most pressing one -- the mathematical foundation of these tools. While the theory is well understood in the case of finite-dimensional systems, fluid dynamics is inherently infinite-dimensional, which calls for a more careful assessment. The goal of this thesis is to address precisely this concern, and provide a rigorous backbone to the routinely conducted numerical studies. To do so, we give existence and uniqueness results for spectral submanifolds, invariant foliations and Koopman eigenfunctions in the full, infinite-dimensional phase space of certain partial differential equations, notably the Navier-Stokes system on a bounded domain.","abstract_has_math":false,"creators":["Buza, Gergely"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kerswell, Rich"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-09-25","date_published":"2024-09-25","updated_at":"2026-07-24T01:33:28Z","subjects":["dynamical systems"],"languages":["eng"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/3a3e1e55-d59b-4379-b676-d0698274dbd1/download","https://creativecommons.org/licenses/by/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.117101","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kerswell, Rich"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Harding Distinguished Postgraduate Scholars Programme"]},{"key":"dc:creator","label":"Author","values":["Buza, Gergely"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2024-09-25"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/382116"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["dynamical systems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/3a3e1e55-d59b-4379-b676-d0698274dbd1/download","https://creativecommons.org/licenses/by/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.117101"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/9d27546b-f60a-4d1d-aef8-b5d09ea99d41/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The emergence of data-driven methods has fueled a newfound interest in the utilization of nonlinear tools from dynamical systems theory. In fluid dynamics, prominent examples are Koopman eigenfunctions (through dynamic mode decomposition) and spectral submanifolds. Due to their immense popularity, both of these techniques have been studied extensively, to the point that most aspects regarding their implementation are now fully fleshed out. However, there is one issue that has remained mostly untouched, and it is perhaps the most pressing one -- the mathematical foundation of these tools. While the theory is well understood in the case of finite-dimensional systems, fluid dynamics is inherently infinite-dimensional, which calls for a more careful assessment. The goal of this thesis is to address precisely this concern, and provide a rigorous backbone to the routinely conducted numerical studies. To do so, we give existence and uniqueness results for spectral submanifolds, invariant foliations and Koopman eigenfunctions in the full, infinite-dimensional phase space of certain partial differential equations, notably the Navier-Stokes system on a bounded domain."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["0aafdc572a0f1cf87529cf9c7eb782ba","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["A dynamical systems approach to the study of local phenomena in fluid flows"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kerswell, Rich"],"dc:contributor.sponsor":["Harding Distinguished Postgraduate Scholars Programme"],"dc:creator":["Buza, Gergely"],"dc:date.issued":["2024-09-25"],"dc:description.abstract":["The emergence of data-driven methods has fueled a newfound interest in the utilization of nonlinear tools from dynamical systems theory. In fluid dynamics, prominent examples are Koopman eigenfunctions (through dynamic mode decomposition) and spectral submanifolds. Due to their immense popularity, both of these techniques have been studied extensively, to the point that most aspects regarding their implementation are now fully fleshed out. However, there is one issue that has remained mostly untouched, and it is perhaps the most pressing one -- the mathematical foundation of these tools. While the theory is well understood in the case of finite-dimensional systems, fluid dynamics is inherently infinite-dimensional, which calls for a more careful assessment. The goal of this thesis is to address precisely this concern, and provide a rigorous backbone to the routinely conducted numerical studies. To do so, we give existence and uniqueness results for spectral submanifolds, invariant foliations and Koopman eigenfunctions in the full, infinite-dimensional phase space of certain partial differential equations, notably the Navier-Stokes system on a bounded domain."],"dc:format.checksum.md5":["0aafdc572a0f1cf87529cf9c7eb782ba","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.117101"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/9d27546b-f60a-4d1d-aef8-b5d09ea99d41/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/382116"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/3a3e1e55-d59b-4379-b676-d0698274dbd1/download","https://creativecommons.org/licenses/by/4.0/"],"dc:subject":["dynamical systems"],"dc:title":["A dynamical systems approach to the study of local phenomena in fluid flows"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:33:28Z"}