APPLICATIONS OF THE GREEN TENSOR ESTIMATES OF THE NONSTATIONARY STOKES SYSTEM IN THE HALF SPACE

In this paper, we present a series of applications of the pointwise estimates of the (unrestricted) Green tensor of the nonstationary Stokes system in the half space, established in our previous work [Comm. Math. Phys., 399 (2023), pp. 1291-1372]. First, we show the L¹-L^q estimates for the Stokes flow with possibly nonsolenoidal L¹ initial data, generalizing the results of Giga, Matsui, and Shimizu [Math. Z., 231 (1999), pp. 383-396] and Desch, Hieber, and Pr\" uss [J. Evol. Equ., 1 (2001), pp. 115-142]. Second, we construct mild solutions of the Navier-Stokes equations in the half space with mixed-type pointwise decay or with pointwise decay alongside boundary vanishing. Finally, we explore various coupled fluid systems in the half space including viscous resistive magnetohydrodynamic (MHD) equations, a coupled system for the flow and the magnetic field of MHD type, and the nematic liquid crystal flow. For each of these systems, we construct mild solutions in L^q, pointwise decay, and uniformly local L^q spaces.