The initial value problem for the incompressible Navier–Stokes system on the whole space \(\mathbb {R}^n\) ( \(n \ge 2\) ) is considered. The initial datum \(\varvec{a}\) is assumed to be small in the norm of scaling invariant homogeneous Besov spaces \(\dot{B}_{r,\infty }^{-1+n/r}(\mathbb {R}^n)\) with \(n<r<\infty \) . We then investigate the decay rates of the scaling invariant norms of the corresponding global solution \(\varvec{u}\) by imposing an additional condition on the low-frequency part of \(\varvec{a}\) . In particular, we reveal logarithmic decay rates by changing the condition for \(\varvec{a}\) slightly. We show that while the decay rate of \(\Vert \varvec{u}(t,\cdot )\Vert _{\dot{B}_{r,\infty }^{-1+n/r}(\mathbb {R}^n)}\) improves up to \(O(t^{-n/r})\) , our decay rates of \(\varvec{u}\) are indeed optimal under specific restricted conditions for \(\varvec{a}\) .
