An Investigation of the Forced Navier-Stokes Equations in Two and Three Dimensions

This dissertation is devoted to expanding the classical theory of the forced Navier-Stokes equations. First, we study the regularity of solutions to the two dimensional Navier-Stokes equations with a singular or ``fractal'' forcing term. The classical theory tells us that the two dimensional Navier-Stokes equations gain two derivatives on a sufficiently smooth force. Following these classical methods we extend this result to spaces with negative fractional derivatives. However, these methods break down at a critical value. In this case, we show that one can still gain two derivatives locally in time. Next, we investigate the long-term behavior of both the two dimensional and three dimensional Navier-Stokes equations with a time-dependent force. When the force is independent of time, it is known that the long-term behavior of the Navier-Stokes equations is encapsulated within a set called the global attractor. The global attractor has a nice characterization, even in the three dimensional case, where we still do not know if there exists unique solutions. We present a framework for studying the existence of an analogous object, the pullback attractor, when the force depends on time. We study the existence and structure of these pullback attractors as well as the relationship between the pullback attractor and other existing notions of attractors. Finally, we apply our framework to the two dimensional and three dimensional Navier-Stokes equations with an appropriate time-dependent force. We also study the effect that the size of the force has on the size of the pullback attractor. Finally, we show that if the force is sufficiently small and periodic, there must exist a unique, smooth, periodic solution to the three dimensional Navier-Stokes equations.