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Article

Trajectory attractors for non-autonomous dissipative 2d Euler equations

Discrete and Continuous Dynamical Systems - Series B. 2015. Vol. 20. No. 3. P. 811-832.

We construct the trajectory attractor AΣ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms −r(t)u such that 0<α≤r(t)≤β, for t≥0. External forces g(x,t),x∈T2,t≥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 ν>0 also have the trajectory attractor AνΣ. Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that AνΣ→AΣ as viscosity ν→0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit AminΣ⊆AΣ of the trajectory attractors AνΣ as ν→0+. Every set AνΣ is connected. We prove that AminΣ is a connected invariant subset of AΣ. The problem of the connectedness of the trajectory attractor AΣ itself remains open.