Large time behavior for the nonstationary Navier-Stokes flows in the half-space

Asymptotic behavior of higher-order spatial derivatives in Lr is established for incompressible Navier-Stokes flows in the half-space, which is a long-time unsolved problem. The main tools employed for solving this problem are the nonstationary Stokes solution formula, and the a priori estimates of the steady Stokes system in the half-space. Another main result is devoted to studying time L1-decay, and a partial answer to this open problem is given by means of a crucial and new estimate for the Stokes flow. Two Theorems 1.1, 1.2 in this article can be regarded as a great improvement and powerful extension of the work in [6,38].