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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.

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids