# A local smoothness criterion for solutions of the 3D Navier-Stokes equations

### James C. Robinson

University of Warwick, Coventry, UK### Witold Sadowski

University of Warsaw, Poland

## Abstract

We consider the three-dimensional Navier–Stokes equations on the whole space $R_{3}$ and on the three-dimensional torus $T_{3}$. We give a simple proof of the local existence of (finite energy) solutions in $L_{3}$ for initial data $u_{0}∈L_{2}∩L_{3}$, 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 $v(t)$ of the heat equation with such initial data: there is an absolute constant $ε>0$ such that solutions remain regular on $[0,T]$ if $∥u_{0}∥_{L_{3}}∫_{0}∫_{R_{3}}∣∇v(s)∣_{2}∣v(s)∣dxdt≤ε$. This implies the $u∈C_{0}([0,T];L_{3})$ regularity criterion due to von Wahl. We also derive simple a priori estimates in $L_{p}$ for $p>3$ that recover the well known lower bound $∥u(T−t)∥_{L_{p}}≥ct_{−(p−3)/2p}$ on any solution that blows up in $L_{p}$ at time $T$.

The key ingredients are a calculus inequality $∥u∥_{L_{3p}}≤c∫∣u∣_{p−2}∣∇u∣_{2}$ (valid on $R_{3}$ and for functions on bounded domains with zero average) and the bound on the pressure $∥p∥_{L_{r}}≤c_{r}∥u∥_{L_{2r}}$, 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. 159–178

DOI 10.4171/RSMUP/131-9