Strong trajectory and global  $\mathbf{W^{1,p}}$ -attractors for the damped-driven Euler system in  $\mathbb R^2$

Vladimir Chepyzhov; Alexei Ilyin; Sergey Zelik

doi:10.3934/dcdsb.2017109

Article Contents

2017, Volume 22, Issue 5: 1835-1855. Doi: 10.3934/dcdsb.2017109

This issue Previous Article Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic Next Article Smooth attractors for weak solutions of the SQG equation with critical dissipation

Strong trajectory and global $\mathbf{W^{1,p}}$ -attractors for the damped-driven Euler system in $\mathbb R^2$

1.
Institute for Information Transmission Problems, Moscow 127994, Russia
2.
National Research University Higher School of Economics, Moscow 101000, Russia
3.
Keldysh Institute of Applied Mathematics, Moscow 125047, Russia
4.
University of Surrey, Department of Mathematics, Guildford, GU2 7XH, UK

* Corresponding author: V. Chepyzhov
* Corresponding author: V. Chepyzhov

Received: December 31, 2015

Revised: February 29, 2016

Published: August 2017

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Abstract

We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space $H^1$ . A similar result on the strong attraction holds in the spaces $H^1\cap\{u:\ \|\text{curl}\, u\|_{L^p}<∞\}$ for $p≥2$ .

Keywords:

Damped and driven Euler equations,
global and trajectory attractors.

Mathematics Subject Classification: Primary:35B40, 35B41;Secondary:35Q35.

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