Multi-vortices and lower bounds for the attractor dimension of 2d Navier-Stokes equations

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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.

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 Novgorod

A. 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, Guildford

S. 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 Novgorod

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