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We study the global attractors for the damped 3D Euler--Bardina equationswith the regularization parameter $\alpha>0$ and Ekman damping coefficient$\gamma>0$ endowed with periodic boundary conditions as well as their dampedEuler limit $\alpha\to0$. We prove that despite the possible non-uniqueness ofsolutions of the limit Euler system and even the non-existence of suchsolutions in the distributional sense, the limit dynamics of the correspondingdissipative solutions introduced by P.\,Lions can be described in terms ofattractors of the properly constructed trajectory dynamical system. Moreover,the convergence of the attractors $\Cal A(\alpha)$ of the regularized system tothe limit trajectory attractor $\Cal A(0)$ as $\alpha\to0$ is also establishedin terms of the upper semicontinuity in the properly defined functional space.

Trajectory attractors for 3D damped Euler equations and their approximation