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Multi-vortices and lower bounds for the attractor dimension of 2d Navier-Stokes equations
- Authors: Kostianko A.G.1,2, Ilyin A.A.3,2, Stone D.4, Zelik S.V.1,4,3,2
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Affiliations:
- Zhejiang Normal University
- HSE University
- Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
- University of Surrey
- Issue: Vol 516, No 1 (2024)
- Pages: 98-102
- Section: MATHEMATICS
- URL: https://journal-vniispk.ru/2686-9543/article/view/265339
- DOI: https://doi.org/10.31857/S2686954324020163
- EDN: https://elibrary.ru/XHPUVQ
- ID: 265339
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Abstract
A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations, which does not use Kolmogorov flows, is presented. Using 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.
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About the authors
A. G. Kostianko
Zhejiang Normal University; HSE University
Author for correspondence.
Email: a.kostianko@imperial.ac.uk
Department of Mathematics
Russian Federation, Zhejiang; Nizhny NovgorodA. A. Ilyin
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences; HSE University
Email: ilyin@keldysh.ru
Russian Federation, Moscow; Nizhny Novgorod
D. Stone
University of Surrey
Email: d.stonc@surrey.ac.uk
Department of Mathematics
United Kingdom, GuildfordS. V. Zelik
Zhejiang Normal University; University of Surrey; Keldysh Institute of Applied Mathematics, Russian Academy of Sciences;HSE University
Email: s.zelik@surrey.ac.uk
Department of Mathematics, Department of Mathematics
Russian Federation, Zhejiang, China; Guildford, UK; Moscow; Nizhny NovgorodReferences
- Бабин А.В., Вишик М.И. Аттракторы эволюционных уравнений. M.: Наука, 1989.
- Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. New York: Springer-Verlag, 1997.
- Ilyin A.A., Miranville A., Titi E.S. Small viscosity sharp estimates for the global attractor of the 2-D damped/driven Navier–Stokes equations // Commun. Math. Sci. 2004. V. 2. P. 403–426.
- Ilyin A.A., Patni K., Zelik S.V. Upper bounds for the attractor dimension of damped Navier–Stokes equations in R2 // Discrete Contin. Dyn. Syst. 2016 V. 36. N 4. P. 2085–2102.
- Zelik S.V. Attractors. Then and Now // Успехи математических наук. 2023. V. 78. N 4. P. 53–198.
- Liu V.X. A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier–Stokes equations // Comm. Math. Phys. 1993. V. 158. P. 327–339.
- Мешалкин Л.Д., Синай Я.Г. Исследование устойчивости стационарного решения одной системы уравнений плоского движения несжимаемой вязкой жидкости // Прикладная мат. мех. 1961. Т. 25. С. 1140–1143.
- Vishik M.M. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I arXiv:1805.09426 and Part 2 arXiv:1805.09440, 2018.
- Albritton D., Brué E., Colombo M. Non-uniqueness of Leray solutions of the forced Navier–Stokes equations // Ann. Math. (2) 2022. V. 196. N 1. P. 415–455.
- Mielke A., Zelik S.V. Multi-Pulse Evolution and Space-Time Chaos in Dissipative Systems // Mem. Amer. Math. Soc. 2009. V. 198. N 925. 97 p.
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