Decay of weak solutions and the singular set of the three-dimensional Navier-Stokes equations

We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ( parallel to u parallel to(2) = integral vertical bar u vertical bar(2)) but in. nite enstrophy ( parallel to Du parallel to(2) = integral vertical bar curl u vertical bar(2)). We show that this has to be reflected in the solution for small times, so that in particular parallel to Du( t)parallel to -> +infinity as t -> 0. We also give some limitations on this 'backwards blow-up', and give an elementary proof that the upper box-counting dimension of the set of singular times can be no larger than one half. Although similar in flavour, this final result neither implies, nor is implied by, Scheffer's result that the 1/2-dimensional Hausdorff measure of the singular set is zero.