Boundedness, almost periodicity and stability of certain Navier-Stokes flows in unbounded domains

We investigate the existence, uniqueness and stability of bounded and almost periodic mild solutions to several Navier-Stokes flow problems. In our strategy, we propose a general framework for studying the semi-linear evolution equations with certain smoothing properties of the linear part and with the local Lipschitz continuity of the nonlinear operator. Our method is based on interpolation functors combined with differential inequalities. Our abstract results are applied to Navier-Stokes-Oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to Navier-Stokes equations on aperture domains and/or in Besov spaces.