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 Uspekhi Mat. Nauk, 2014, Volume 69, Issue 2(416), Pages 177–200 (Mi umn9583)

A unified approach to determining forms for the 2D Navier–Stokes equations — the general interpolants case

C. Foiasa, M. S. Jollyb, R. Kravchenkoc, E. S. Titide

a Texas A&M University, College Station, USA
b Indiana University, Bloomington, USA
c University of Chicago, Chicago, USA
d Weizmann Institute of Science, Rehovot, Israel
e University of California, Irvine, USA

Abstract: It is shown that the long-time dynamics (the global attractor) of the 2D Navier–Stokes system is embedded in the long-time dynamics of an ordinary differential equation, called a determining form, in a space of trajectories which is isomorphic to $C^1_b(\mathbb{R};\mathbb{R}^N)$ for sufficiently large $N$ depending on the physical parameters of the Navier–Stokes equations. A unified approach is presented, based on interpolant operators constructed from various determining parameters for the Navier–Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, and so on. There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, and thus its solutions converge to the set of steady states of the determining form as the time goes to infinity. The second is that these steady states of the determining form can be uniquely identified with the trajectories in the global attractor of the Navier–Stokes system. It should be added that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems.
Bibliography: 23 titles.

Keywords: Navier–Stokes equation, inertial manifold, determining forms, determining modes, dissipative dynamical systems.

 Funding Agency Grant Number National Science Foundation DMS-1109784DMS-1008661DMS-1109638DMS-1009950DMS-1109640DMS-1109645 Minerva Stiftung The work of the first author was supported by NSF (grant no. DMS-1109784), that of the second author by NSF (grant nos. DMS-1008661 and DMS-1109638), and that of the fourth author by NSF (grant nos. DMS-1009950, DMS-1109640, and DMS-1109645), as well as the Minerva Stiftung/Foundation.

DOI: https://doi.org/10.4213/rm9583

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English version:
Russian Mathematical Surveys, 2014, 69:2, 359–381

Bibliographic databases:

UDC: 517.954+517.957
MSC: Primary 35Q30; Secondary 76D05

Citation: C. Foias, M. S. Jolly, R. Kravchenko, E. S. Titi, “A unified approach to determining forms for the 2D Navier–Stokes equations — the general interpolants case”, Uspekhi Mat. Nauk, 69:2(416) (2014), 177–200; Russian Math. Surveys, 69:2 (2014), 359–381

Citation in format AMSBIB
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• https://doi.org/10.4213/rm9583
• http://mi.mathnet.ru/eng/umn/v69/i2/p177

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Azouani, E. S. Titi, “Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm”, Evol. Equ. Control Theory, 3:4 (2014), 579–594
2. H. Bessaih, E. Olson, E. S. Titi, “Continuous data assimilation with stochastically noisy data”, Nonlinearity, 28:3 (2015), 729–753
3. M. S. Jolly, T. Sadigov, E. S. Titi, “A determining form for the damped driven nonlinear Schrödinger equation Fourier modes case”, J. Differential Equations, 258:8 (2015), 2711–2744
4. A. Farhat, M. S. Jolly, E. S. Titi, “Continuous data assimilation for the 2D Bénard convection through velocity measurements alone”, Phys. D, 303 (2015), 59–66
5. M. Abu Hamed, Ya. Guo, E. S. Titi, “Inertial manifolds for certain subgrid-scale $\alpha$-models of turbulence”, SIAM J. Appl. Dyn. Syst., 14:3 (2015), 1308–1325
6. A. Farhat, E. Lunasin, E. S. Titi, “Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field”, J. Math. Fluid Mech., 18:1 (2016), 1–23
7. D. A. F. Albanez, H. J. Nussenzveig Lopes, E. S. Titi, “Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model”, Asymptot. Anal., 97:1-2 (2016), 139–164
8. Lukaszewicz G., Kalita P., “Navier–Stokes Equations: An Introduction With Applications”, Navier-Stokes Equations: An Introduction With Applications, Advances in Mechanics and Mathematics, Springer, 2016, 1–390
9. M. S. Jolly, T. Sadigov, E. S. Titi, “Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation — Fourier modes case”, Nonlinear Anal. Real World Appl., 36 (2017), 287–317
10. Lunasin E., Titi E.S., “Finite Determining Parameters Feedback Control For Distributed Nonlinear Dissipative Systems - a Computational Study”, Evol. Equ. Control Theory, 6:4 (2017), 535–557
11. Foias C., Jolly M.S., Lithio D., Titi E.S., “One-Dimensional Parametric Determining Form For the Two-Dimensional Navier–Stokes Equations”, J. Nonlinear Sci., 27:5 (2017), 1513–1529
12. Bai L., Yang M., “A Determining Form For a Nonlocal System”, Adv. Nonlinear Stud., 17:4 (2017), 705–713
13. A. Biswas, J. Hudson, A. Larios, Yu. Pei, “Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields”, Asymptotic Anal., 108:1-2 (2018), 1–43
14. Ch. R. Doering, E. M. Lunasin, A. Mazzucato, “Introduction to special issue: nonlinear partial differential equations in mathematical fluid dynamics”, Physica D, 376:SI (2018), 1–4
15. M. Ozluk, M. Kaya, “On the weak solutions and determining modes of the g-Benard problem”, Hacet. J. Math. Stat., 47:6 (2018), 1453–1466
16. A. Larios, L. G. Rebholz, C. Zerfas, “Global in time stability and accuracy of imex-fem data assimilation schemes for Navier-Stokes equations”, Comput. Meth. Appl. Mech. Eng., 345 (2019), 1077–1093
17. A. Biswas, C. Foias, C. F. Mondaini, E. S. Titi, “Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 36:2 (2019), 295–326
18. A. Cheskidov, M. Dai, “Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier-Stokes equations”, Proc. R. Soc. Edinb. Sect. A-Math., 149:2 (2019), 429–446
19. M. S. Jolly, V. R. Martinez, T. Sadigov, E. S. Titi, “A determining form for the subcritical surface quasi-geostrophic equation”, J. Dyn. Differ. Equ., 31:3, SI (2019), 1457–1494
20. Larios A., Pei Yu., “Approximate Continuous Data Assimilation of the 2D Navier-Stokes Equations Via the Voigt-Regularization With Observable Data”, Evol. Equ. Control Theory, 9:3 (2020), 733–751
21. Cheskidov A., Dai M., “On the Determining Wavenumber For the Nonautonomous Subcritical Sqg Equation”, J. Dyn. Differ. Equ., 32:3 (2020), 1511–1525
22. Carlson E., Hudson J., Larios A., “Parameter Recovery For the 2 Dimensional Navier-Stokes Equations Via Continuous Data Assimilation”, SIAM J. Sci. Comput., 42:1 (2020), A250–A270
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