We consider the three-dimensional Navier–Stokes equations on the whole space and on the three-dimensional torus . We give a simple proof of the local existence of (finite energy) solutions in for initial data , based on energy estimates and regularisation of the initial data with the heat semigroup.
We also provide a lower bound on the existence time of a strong solution in terms of the solution of the heat equation with such initial data: there is an absolute constant such that solutions remain regular on if . This implies the regularity criterion due to von Wahl. We also derive simple a priori estimates in for that recover the well known lower bound on any solution that blows up in at time .
The key ingredients are a calculus inequality (valid on and for functions on bounded domains with zero average) and the bound on the pressure , valid both on the whole space and for periodic boundary conditions.
Cite this article
James C. Robinson, Witold Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations. Rend. Sem. Mat. Univ. Padova 131 (2014), pp. 159–178DOI 10.4171/RSMUP/131-9