We consider the Stokes system in the half-space with localized boundary data. We prove that there exist boundary influxes for which the induced flow exhibits flow reversal and boundary-layer separation. When boundary-layer separation occurs, we further investigate the dynamics and asymptotics of the separation point. Moreover, by a perturbation argument, we construct solutions to the Navier–Stokes equations in the half-space that exhibit the same qualitative behavior as in the Stokes case. This is a joint work with T.-K. Chang and C.-H. Min.