A dynamical systems approach to the study of local phenomena in fluid flows

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Buza, Gergely

Advisor dc:contributor.advisor

• Kerswell, Rich

• dynamical systems

Licence

• https://www.repository.cam.ac.uk/bitstreams/3a3e1e55-d59b-4379-b676-d0698274dbd1/download
• https://creativecommons.org/licenses/by/4.0/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.117101

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/382116