Spectral Submanifolds of the Navier–Stokes Equations

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1052-1089, June 2024.
Abstract.Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of spectral submanifolds—manifolds invariant under the full nonlinear dynamics that are determined by their tangency to spectral subspaces of the linearized system. In light of the recently emerged interest in their use as tools in model reduction, we propose an extension of the relevant theory to the realm of fluid dynamics. We show the existence of a large (and the most pertinent) subclass of spectral submanifolds and foliations—describing the behavior of nearby trajectories—about fixed points and periodic orbits of the Navier–Stokes equations. Their uniqueness and smoothness properties are discussed in detail, due to their significance from the perspective of model reduction. The machinery is then put to work via a numerical algorithm developed along the lines of the parameterization method, which computes the desired manifolds as power series expansions. Results are shown within the context of two-dimensional channel flows.