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