Spatial Decay of Time-Dependent Incompressible Navier-Stokes Flows with Nonzero Velocity at Infinity

We consider the time-dependent Navier--Stokes system in a three-dimensional exterior domain with nonzero velocity at infinity. Under suitable assumptions on the data, it is shown that the velocity part of strong solutions, after subtraction of the far-field velocity, decays as $\bigl( |x| \cdot (1+|x|-x_1) \bigr) ^{-1}$, and its spatial gradient as $\bigl( |x| \cdot (1+|x|-x_1) ,\bigr) ^{-3/2}$, for $|x|\to \infty $. The solution class in question includes solutions obtained by $L^2$-variational methods and characterized by the fact that the velocity $u$ and its spatial gradient $ \nabla _xu$ are $L ^{ \infty } (L^2)$-functions, and $\nabla _xu$ is additionally $L^2$-integrable in time and in space.