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On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system

2007; American Institute of Mathematical Sciences; Volume: 17; Issue: 3 Linguagem: Inglês

10.3934/dcds.2007.17.481

1553-5231

Vladimir V. Chepyzhov, Edriss S. Titi, Mark Vishik,

Fluid Dynamics and Turbulent Flows

We study the relations between the global dynamics of the 3DLeray-$\alpha $ model and the 3D Navier-Stokes system. We provethat time shifts of bounded sets of solutions of the Leray-$\alpha$ model converges to the trajectoryattractor of the 3D Navier-Stokes system as time tends to infinity and $\alpha $ approaches zero. In particular, we show that thetrajectory attractor of the Leray-$\alpha $ model converges to thetrajectory attractor of the 3D Navier-Stokes system when $\alpha\rightarrow 0\+.$