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

We study incompressible Navier–Stokes flows in~${\mathbb{R}}^d$ with small and well localized data and external force~$f$. We establish pointwise estimates for large~$\mathrm{\mid }x\mathrm{\mid }$ of the form \hbox{$c_t\mathrm{\mid }x{\mathrm{\mid }}^{-d}\le \mathrm{\mid }u(x,t)\mathrm{\mid }\le c_t^{\mathrm{\prime }}\mathrm{\mid }x{\mathrm{\mid }}^{-d}$}, where $c_t>0$ whenever ${\int }_0^t\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int f(x,s)dxds≠\vec{0}$. 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 $\mathrm{\mid }u(x,t)\mathrm{\mid }\simeq C_t\mathrm{\mid }x{\mathrm{\mid }}^{-d-1}$. 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.