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 $\varepsilon>0$ such that solutions remain regular on $[0,T]$ if $\|u_0\|_{L^3}^3\int_0^T\int_{\mathbb R^3}|\nabla v(s)|^2|v(s)|\,\rm d x\,\rm d t\leq \varepsilon$. 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 $\|u(T-t)\|_{L^p}\ge ct^{-(p-3)/2p}$ on any solution that blows up in $L^p$ at time $T$.

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