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Russian Journal of Mathematical Physics

, Volume 15, Issue 2, pp 156–170 | Cite as

Trajectory attractors for dissipative 2D Euler and Navier-Stokes equations

Article

Abstract

A trajectory attractor \( \mathfrak{A} \) is constructed for the 2D Euler system containing an additional dissipation term −ru, r > 0, with periodic boundary conditions. The corresponding dissipative 2D Navier-Stokes system with the same term −ru and with viscosity v > 0 also has a trajectory attractor, \( \mathfrak{A}_\nu \). Such systems model large-scale geophysical processes in atmosphere and ocean (see [1]). We prove that \( \mathfrak{A}_\nu \)\( \mathfrak{A} \) as v → 0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit \( \mathfrak{A}_{min} \) of the trajectory attractors \( \mathfrak{A}_\nu \) as v → 0+. We prove that \( \mathfrak{A}_{min} \) is a connected invariant subset of \( \mathfrak{A} \). The connectedness problem for the trajectory attractor \( \mathfrak{A} \) by itself remains open.

Keywords

Mathematical Physic Weak Solution Periodic Boundary Condition Planetary Boundary Layer Global Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRAS (Kharkevich Institute)MoscowRussia

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