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

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^3^3∫_0^T∫R^3|∇ v(s)|^2|v(s)|\, d x\, d t≤ 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\|_{L³ᵖ}^p≤ 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\|_{L²ʳ}^2 , valid both on the whole space and for periodic boundary conditions.