Gergely Buza

Rigorization of model reduction about equilibria in fluid dynamics

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 talk will provide existence and uniqueness results for spectral submanifolds, smooth invariant foliations and Koopman eigenfunctions in the full, infinite-dimensional phase space of the Navier-Stokes system, alongside avenues to make the approximation procedure rigorous.