Gaining two derivatives on a singular force in the 2D Navier-Stokes equations

It has long been known, for the autonomous 2D Navier-Stokes equations with singular forcing, that there exist unique solutions which gain one derivative, globally. On the other hand, if the forcing term smooth enough, it is known that the solution gains two derivatives globally. In this paper, we explore classical techniques to show that if the force is sufficiently smooth, then the solution gains two derivatives globally. These methods break down when the force becomes singular. In this scenario, we use a Littlewood-Paley decomposition in Fourier space to show that a solutions gain two derivatives locally in time and the interval of time depends only on the size of the initial data.