Asymptotic Profiles of Nonstationary Incompressible Navier--Stokes Flows in the Whole Space

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Asymptotic profiles 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.