Fast Decaying Solutions of the Navier-Stokes Equation and Asymptotic Properties

While the basic global existence problem for the Navier-Stokes equations seems to remain open, there are related questions of some interest which are amenable to discussion: find large initial data giving rise to global solutions. Such initial data are known in the literature. A study shows that they have a peculiar property: they give rise to solutions which decay fast in very short time. A major result to be proved states that the set of trajectories induced by such initial data is dense in every open set (with respect to some fractional power norm). A further result states that if the exterior force f is zero, then such rapid decays cannot occur infinitely often along trajectories. This follows from some inequalities, connecting parallel to A(1/2) w(t)parallel to and parallel to A(1/2) w(t + delta)parallel to, with A the Stokes operator.