Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space

Asymptotic pro les are deduced for weak and strong solutions of the incompressible Navier Stokes equations in the whole space. It is shown that if the initial velocity satisfies a specific moment condition, the corresponding solution behaves like the first-order spatial derivatives of the heat kernel. Higher-order asymptotics are also deduced in case the initial data admit vector potentials with spatial decay of order -n. We further note that the results are not optimal and suggest by means of an example that there exist solutions with a more rapid ( space-time) decay property if we require certain symmetry conditions to the initial data.