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

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

2006-12-15

arxiv.org/abs/math/0612425v1

Nonlinearity 20 (2007) 1185-1191

Mathematics

Analysis of PDEs

Scientific paper

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 ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl} u|^2$). We show that this has to be reflected in the solution for small times, so that in particular $\|Du(t)\|\to+\infty$ as $t\to0$. We also give some limitations on this `backwards blowup', 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.

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