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 ${\mathbb{R}}^3$ and on the three-dimensional torus ${\mathbb{T}}^3$. We give a simple proof of the local existence of (finite energy) solutions in $L^3$ for initial data $u_0\in L^2\cap L^3$, 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 $\epsilon >0$ such that solutions remain regular on $[0,T]$ if $\Vert u_0{\Vert }_{L^3}^3{\int }_0^T{\int }_{{\mathbb{R}}^3}|\nabla v(s){|}^2|v(s)|\mathrm{dx}\mathrm{dt}\le \epsilon$. This implies the $u\in C^0([0,T];L^3)$ 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 $\Vert u(T-t){\Vert }_{L^p}\ge c{t}^{-(p-3)/2p}$ on any solution that blows up in $L^p$ at time $T$.

The key ingredients are a calculus inequality $\Vert u{\Vert }_{L^{3p}}^p\le c\int |u{|}^{p-2}|\nabla u{|}^2$ (valid on ${\mathbb{R}}^3$ and for functions on bounded domains with zero average) and the bound on the pressure $\Vert p{\Vert }_{L^r}\le c_r\Vert u{\Vert }_{L^{2r}}^2$, valid both on the whole space and for periodic boundary conditions.