In the first part of the paper we study decays of solutions of the Navier-Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier-Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R 3 and β ∈ [1/2, 1), then there exist C 0 > 1 and δ 0 ∈ (0, 1) such thatwhere ||| · ||| β = ||A β · || + || · || is the graph norm and A is the Stokes operator. In the second part of the paper we characterize approximately the subspace of the phase space through which the trajectory of w(t)/||w(t)|| moves as t → ∞. To this purpose we use the resolution of identity of A, denoted as {E λ ; λ ≥ 0}. We prove, for example, that if Ω is such a domain that the Poincaré inequality holds, then there exists a ≥ 0 such that if µ > a then lim t→∞ Eµ(w(t)/||w(t)||) = 1.As a consequence of the presented results we get that no global strong solution of homogeneous Navier-Stokes equations can decay more quickly than exponentially for t → ∞ in the norm · .In the paper we also formulate some open problems.

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