Abstract
The evolution of a determining form for the 2D Navier–Stokes equations (NSE) which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and a chosen, fixed steady state, with a dynamical convexity parameter \(\theta \), which will be called the characteristic determining parameter. That is, we show a separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter \(\theta \) which is governed by a one-dimensional ODE. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than \({\mathcal {O}}(\tau ^{-1/2})\), otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than \({\mathcal {O}}(\tau ^{-1})\), as \(\tau \rightarrow \infty \), where \(\tau \) is the evolutionary variable in determining form. The one-dimensional ODE is also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. It is shown that the zeros of the scalar function that governs the dynamics of \(\theta \), which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE
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Acknowledgements
The work of C.F. was partially supported by National Science Foundation (NSF) Grant DMS-1109784 and Office of Naval Research (ONR) Grant N00014-15-1-2333, that of M.S.J. by NSF Grant DMS-1418911 and the Leverhulme Trust Grant VP1-2015-036, and D.L. by NSF Grant DMS-1418911. The work of E.S.T. was supported in part by the ONR Grant N00014-15-1-2333 and the NSF Grants DMS-1109640 and DMS-1109645.
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Communicated by Peter Constantin.
This paper is dedicated to the memory of Professor George Sell, an initiator of the modern approach to ODEs and infinite-dimensional dynamical systems theory, a great friend, collaborator and teacher.
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Foias, C., Jolly, M.S., Lithio, D. et al. One-Dimensional Parametric Determining form for the Two-Dimensional Navier–Stokes Equations. J Nonlinear Sci 27, 1513–1529 (2017). https://doi.org/10.1007/s00332-017-9375-4
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DOI: https://doi.org/10.1007/s00332-017-9375-4
Keywords
- Navier–Stokes equations
- Global attractors
- Determining nodes
- Determining form
- Parametric determining form
- Determining parameter