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

Abstract

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 ($\Vert u{\Vert }^2=\int |u{|}^2$) but infinite enstrophy ($\Vert Du{\Vert }^2=\int |\mathrm{curl}u{|}^2$). We show that this has to be reflected in the solution for small times, so that in particular $\Vert Du(t)\Vert \to +\infty$ as $t\to 0$. 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.

Cited by in corpus (5)

• Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations
• Partial regularity for a surface growth model
• On the box-counting dimension of potential singular set for suitable weak solutions to the 3D Navier-Stokes equations
• An anisotropic partial regularity criterion for the Navier-Stokes equations
• Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation