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On upper and lower bounds of rates of decay for nonstationary Navier-Stokes flows in the whole space. (English) Zbl 1048.35063
The title problem is studied in $$\mathbb{R}^n$$ in homogeneous Besov spaces modulo polynomials. The main result states that the solution $$u(t)$$ for large time $$t$$ satisfies the estimate $0< c_q'\leq t^{{n\over 2}(1+{1\over n}-{1\over q})}\| u(t)\|_{L^q}\leq c_q$ with appropriate constants $$c_q$$, $$c_q'$$, if and only if the initial velocity satisfies some moment conditions (here $${n\over n+1}\leq q\leq\infty$$ for strong solution, and $${n\over n+1}\leq q\leq 2$$ or weak solution). This result is then applied to the flows with cyclic symmetry.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
##### Keywords:
homogeneous Besov spaces; flows with cyclic symmetry