Integrated Research on Besov Spaces and Koopman Operators for Data-Driven Fluid Dynamics Analysis

Reference No.	2025a042
Type/Category	Grant for General Research-Short-term Joint Research
Title of Research Project	Integrated Research on Besov Spaces and Koopman Operators for Data-Driven Fluid Dynamics Analysis
Principal Investigator	Isao Ishikawa（Center for Science Adventure and Collaborative Research Advancement, Kyoto University・Program-Specific Associate Professor）
Research Period	June 18,2025. - June 22,2025.
Keyword(s) of Research Fields	Besov space, Koopman operator, Dynamic Mode Decomposition, Navier-Stokes Equation
Abstract for Research Report	The objective of this study is to advance data-driven analysis of dynamical systems through the integration of Koopman operator theory within Besov spaces, with a focus on applications in fluid dynamics. The aim is to establish a new methodology called BesovDMD (Dynamic Mode Decomposition in Besov Spaces). Besov spaces play a essential role in the analysis of partial differential equations (PDEs), providing precise descriptions of function smoothness and decay. By linking the functional analysis in Besov spaces with Koopman operator theory and data-driven analysis of systems geverned by a PDE such as fluids, the study aims to enable more detailed structural understanding of complex dynamical systems. This research will develop matrix approximation techniques for Koopman operators using wavelet expansions as well as their generalizations—namely atom and molecule decompositions. It will also investigate the relationship between the approximated matrices and the properties of the original dynamical systems. Furthermore, the project aims to foster interdisciplinary collaboration by inviting experts in both theoretical and applied fluid dynamics for discussions and information exchange. The expected outcomes are as follows: 1. Theoretical Advancement: Clarify the mathematical characteristics of Koopman operators in Besov spaces and establish the theoretical foundation of the BesovDMD algorithm. 2. Algorithm Development: Propose new estimation algorithms for Koopman operators using Besov space theory, particularly highlighting their potential for application in fluid dynamics and related fields. 3. Interdisciplinary Workshop: Host a workshop during the research period to bridge the gap between theory and application, aiming for new interdisciplinary developments in data-driven science.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Isao Ishikawa（Kyoto University・Program-Specific Associate Professor） Masahiro Ikeda（Osaka University・Associate Professor） Koichi Taniguchi（Shizuoka University・Associate Professor）