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Trajectory attractors for 3D damped Euler equations and their approximation

This work was supported by Moscow Center for Fundamental and Applied Mathematics, Agreement with the Ministry of Science and Higher Education of the Russian Federation, No. 075-15-2019-1623 and by the Russian Science Foundation grant No.19-71-30004 (sections 2-4). The second author was partially supported by the Leverhulme grant No. RPG-2021-072 (United Kingdom)

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

    Mathematics Subject Classification: 35B40, 35B45, 35L70.


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