On the $(x,t)$ asymptotic properties of solutions of the Navier–Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial boundary value problem for the Navier–Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data assumed only continuous and decreasing at infinity as $|x|^{-\mu}$, $\mu\in(\frac12,n)$. We prove pointwise estimates in the space variable. Moreover, if $\mu\in[1,n)$ and the initial data is suitably small, the above solutions in global (in time) and we prove space-time pointwise estimates.