Degenerate pullback attractors for the 3D Navier-Stokes equations

As in our previous paper, the 3D Navier-Stokes equations with a translationally bounded force contain pullback attractors in a weak sense. Moreover, those attractors consist of complete bounded trajectories. In this paper, we present a sufficient condition under which the pullback attractors are degenerate. That is, if the Grashof constant is small enough, the pullback attractor will be a single point on a unique, complete, bounded, strong solution. We then apply our results to provide a new proof of the existence of a unique, strong, periodic solution to the 3D Navier-Stokes with a small, periodic forcing term

A Beckman-Quarles Type Theorem for Laguerre Transformations in the Dual Plane

In 1953, Beckman and Quarles proved a well-known result in Euclidean Geometry that any transformation preserving a distance r must be a rigid motion. In 1991, June Lester published an analogous result for circle-preserving transformations in the complex plane. In our paper, we introduce the notion of dual numbers and the geometry of the dual plan. We forcus on the set of vertical parabolas and non-vertical linear P with a distance between pairs of parabolas defined to be the difference of slopes at their point(s) of intersection. We then prove that any bijective transformation from P to itself which preserves our distance 1 induces a fractional linear or Laguerre transformation of the dual plane

Stability of Vortex Solutions to an Extended Navier-Stokes System

We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.Comment: 24 pages, 1 figure, updated to add authors' contact information and to address referee's comment

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