Suppose to have the unit square D = [0, 1] x [0, 1] and two-dim Navier-Stokes equations + continuity (e.g. dimensionless form uux+vuy = -px +1/Rey (uxx+uyy)). Are there a Reynolds number value Rey, continuous (or smooth) Boundary Conditions BC for velocity field (u, v) and pressure p, an initial (t = 0) C1 field (u0, v0), a C1 solution (u, v) for NS + BC + (u0, v0) for that Rey number, an internal point P of D and a finite time T such that in the time interval [0, T] the point P describes a traiectory (or curve) dense (in the topological sense) in D, that is the image of P under the flux t --> (u(t), v(t)) is a set Q such that Closure(Q) = D?

Gianluca