Sharp one component regularity for Navier-Stokes

Abstract

We consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e\in {\mathbb{S}}^2$ and $0<T^{\ast }<\infty$. J. Chemin and P. Zhang \cite{CP} proved the regularity of $v$ on $(0,T^{\ast }]$ if there exists $p\in (4,6)$ such that $${\int }_0^{T^{\ast }}\Vert v\cdot e{\Vert }_{{\dot{H}}^{\frac{1}{2}+\frac{2}{p}}}^{p}dt<\infty .$$ J. Chemin, P. Zhang and Z. F. Zhang \cite{CPZ} extended the range of $p$ to $(4,\infty )$. In this article we settle the case $p\in [2,4]$. Our proof also works for the case $p\in (4,\infty )$.