Spectral Submanifolds of the Navier-Stokes Equations
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 nonli...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
19.01.2023
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Subjects | |
Online Access | Get full text |
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Summary: | 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
behaviour 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, that computes the
desired manifolds as power series expansions. Results are shown within the
context of 2D channel flows. |
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DOI: | 10.48550/arxiv.2301.07898 |