Flow reversal of the Stokes system with localized boundary data in the half space

We consider the unsteady Stokes system in the half-space with zero initial data and nonzero, space-time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign in the half-space. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half-space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.