Russian Journal of Mathematical Physics

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

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

  • V. V. Chepyzhov
  • M. I. Vishik


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.


Mathematical Physic Weak Solution Periodic Boundary Condition Planetary Boundary Layer Global Attractor 
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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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