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Critical regularity criteria for Navier-Stokes equations in terms of one directional derivative of the velocity (2007.10888v1)

Published 21 Jul 2020 in math.AP

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 $\partial_3 \mathbf{u} \in Lp((0,T); Lq(\mathbb{R}3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q\leq 6$. 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 u3 \in Lp((0,T); Lq(\mathbb{R}3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q< \infty$.

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