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

Abstract

We consider the three-dimensional Navier–Stokes equations on the whole space R3 and on the three-dimensional torus T3. We give a simple proof of the local existence of (finite energy) solutions in L3 for initial data u0∈L2∩L3, 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 e>0 such that solutions remain regular on [0,T] if ∥u0∥3L3∫T0∫R3|∇v(s)|2|v(s)|dxdt≤e. This implies the u∈C0([0,T];L3) regularity criterion due to von Wahl. We also derive simple a priori estimates in Lp for p>3 that recover the well known lower bound ∥u(T−t)∥Lp≥ct−(p−3)/2p on any solution that blows up in Lp at time T. The key ingredients are a calculus inequality ∥u∥pL3p≤c∫|u|p−2|∇u|2 (valid on R3 and for functions on bounded domains with zero average) and the bound on the pressure ∥p∥Lr≤cr∥u∥2L2r, valid both on the whole space and for periodic boundary conditions. Keywords: Navier-Stokes equations, critical spaces, calculus inequalities