Separation points and pressure gradients for the nonstationary Stokes system in the half-space with localized boundary data

Abstract

We study separation points of the nonstationary Stokes system in a half-space with zero initial data and localized boundary data $ψ(x')ϕ(t)\mathbf e_j$, $1\le j\le n$, where $ψ$ is supported in the closed unit ball. A separation point is a boundary point outside $\operatorname{supp}ψ$ where a component takes both signs on positive-measure sets in every sufficiently small adjoining space-time neighborhood. By evaluating the spatial convolutions appearing in the solution formula of the Stokes system via confluent hypergeometric functions, we obtain explicit formulas for the first normal derivatives of tangential components, the second normal derivative of the normal component, and tangential pressure derivatives at the boundary. For normal data, suitable Hölder regularity and strict log-concavity of $ϕ$ yield a unique separation point for each tangential component at every point of its coordinate axis outside the ball, and exactly two for the normal component at each sufficiently distant boundary point. We prove monotonicity and large-distance asymptotics of their temporal coordinates and quantitative temporal vanishing estimates for the normal derivatives. For tangential data in $n\ge3$, we prove the existence and uniqueness of separation points of tangential and normal components, including at fixed spatial points while $ϕ$ is positive. We derive spatial asymptotics at large distance and time from data moments and, under a geometric condition, supporting faces of $\operatorname{supp}ψ$. We characterize tangential pressure-gradient signs for both tangential and normal data, locating favorable and adverse regions near the boundary. For radial $ψ$, additional assumptions yield existence, uniqueness, and asymptotics of pressure-gradient zeros and zero curves, including those governed by the zero of the Caputo half-derivative of $ϕ$.