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On the effect of external forces on incompressible fluid motions at large distances

We study incompressible Navier--Stokes flows in~$\R^d$ with small and well localized data and external force~$f$. We establish pointwise estimates for large~$|x|$ of the form \hbox{$c_t|x|^{-d}\le |u(x,t)|\le c'_t|x|^{-d}$}, where $c_t>0$ whenever $\int_0^t\!\!\int f(x,s)\,dx\,ds\not=\vec 0$. This sharply contrasts with the case of the Navier--Stokes equations without force, studied in [Brandolese, Vigneron, J. Math. Pures Appl. 88, 64--86 (2007)], where the spatial asymptotic behavior was $|u(x,t)|\simeq C_t|x|^{-d-1}$. In particular, this shows that external forces with non-zero mean, no matter how small and well localized (say, compactly supported in space-time), increase the velocity of fluid particles at {\it all times~$t$} and at \emph{at all points~$x$} in the far-field. As an application of our analysis on the pointwise behavior, we deduce sharp upper and lower bounds of weighted $L^p$-norms for strong solutions, extending the results obtained in [Bae, Brandolese, Jin, Asymptotic behavior for the Navier--Stokes equations with nonzero external forces, Nonlinear analysis, doi:10.1016/j.na.2008.10.074] for weak solutions, by considering here a larger (and in fact, optimal) class of weight functions.