Abstract
A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations is presented, which does not use Kolmogorov flows. By applying this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions.
Similar content being viewed by others
REFERENCES
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (North-Holland, Amsterdam, 1992).
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. (Springer-Verlag, New York, 1997).
A. A. Ilyin, A. Miranville, and E. S. Titi, “Small viscosity sharp estimates for the global attractor of the 2-D damped/driven Navier–Stokes equations,” Commun. Math. Sci. 2, 403–426 (2004).
A. A. Ilyin, K. Patni, and S. V. Zelik, “Upper bounds for the attractor dimension of damped Navier–Stokes equations in \({{\mathbb{R}}^{2}}\),” Discrete Contin. Dyn. Syst. 36 (4), 2085–2102 (2016).
S. V. Zelik, “Attractors: Then and now,” Russ. Math. Surv. 78 (4), 635–777 (2023).
V. X. Liu, “A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier–Stokes equations,” Commun. Math. Phys. 158, 327–339 (1993).
L. D. Meshalkin and Ya. G. Sinai, “Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid,” J. Appl. Math. Mech. 25, 1700–1705 (1961).
M. M. Vishik, “Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid,” Part I, arXiv:1805.09426 (2018);
Part 2, arXiv:1805.09440 (2018).
D. Albritton, E. Brué, and M. Colombo, “Non-uniqueness of Leray solutions of the forced Navier–Stokes equations,” Ann. Math. (2) 196 (1), 415–455 (2022).
A. Mielke and S. V. Zelik, Multi-Pulse Evolution and Space-Time Chaos in Dissipative Systems (Am. Math. Soc., Providence, R.I., 2009).
Funding
This work was supported by the Russian Science Foundation, project no. 23-71-30008.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kostianko, A.G., Ilyin, A.A., Stone, D. et al. Multi-vortices and Lower Bounds for the Attractor Dimension of 2D Navier–Stokes Equations. Dokl. Math. 109, 179–182 (2024). https://doi.org/10.1134/S1064562424702016
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562424702016