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By Mathias Getzlaff

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Hence we can model the Lie extensions and formulate the controllability results in terms of indices k ∈ Z2 of controlled modes. Define iteratively a sequence of sets Kj ⊂ Z2 as follows: j = 2, . . , Kj = Kj−1 {m + n|m, n ∈ Kj−1 m = n m ∧ n = 0}. 1. 3). 2 (controllability in finite-dimensional projection). Let K1 be a saturating set of controlled forcing modes, and let L be any finitedimensional subspace of H 2 (T2 ). Then for any T > 0 the Navier–Stokes / Euler equations on T2 is time-T solidly controllable in finite-dimensional projections and is time-T L2 -approximately controllable.

277 Special local solutions . . . . . . . . . . . . . . . . . . . . . . 279 The asymptotic behavior of the family {uε }ε . . . . . . . . . . 281 Conclusion and remarks . . . . . . . . . . . . . . . . . . . . . 282 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Efim Dinaburg and Yakov Sinai Existence Theorems for the 3D–Navier–Stokes System Having as Initial Conditions Sums of Plane Waves .

1. 2. 3. 4. 5. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Special oscillating initial data . . . . . . . . . . . . . . . . . . 277 Special local solutions . . . . . . . . . . . . . . . . . . . . . . 279 The asymptotic behavior of the family {uε }ε . . . . . . . . . . 281 Conclusion and remarks . . . . . . . . . . . . . . . . . . . . . 282 References . . . . . . . . .

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