Trajectory attractors for non-autonomous dissipative 2d Eulerequations

Vladimir V. Chepyzhov

doi:10.3934/dcdsb.2015.20.811

Article Contents

2015, Volume 20, Issue 3: 811-832. Doi: 10.3934/dcdsb.2015.20.811

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Trajectory attractors for non-autonomous dissipative 2d Euler equations

Vladimir V. Chepyzhov^1,

1.
Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4

Received: November 2013

Revised: May 2014

Published: May 2015

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Abstract

We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms $-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$. External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$ and with viscosity $\nu >0$ also have the trajectory attractor $\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that $\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$ as viscosity $\nu \rightarrow 0+$ in the corresponding metric space. Moreover, we establish the existence of the minimal limit $\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu \rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem of the connectedness of the trajectory attractor $\mathfrak{A}_{\Sigma }$ itself remains open.

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Mathematics Subject Classification: 35B40, 37C70, 37L30, 35Q35, 35Q31.

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