Critical regularity criteria for Navier-Stokes equations in terms of one directional derivative of the velocity

Abstract

In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely ${\partial }_3\mathbf{u}\in L^p((0,T);L^q({\mathbb{R}}^3)),\frac{2}{p}+\frac{3}{q}=2,\frac{3}{2}<q\le 6$. Moreover, we extend the range of $q$ while the solution is axisymmetric, i.e. the axisymmetric solution $\mathbf{m}u$ is regular in $(0,T]$, if ${\partial }_3{u}^3\in L^p((0,T);L^q({\mathbb{R}}^3)),\frac{2}{p}+\frac{3}{q}=2,\frac{3}{2}<q<\infty$.

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• An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range