caa a22 4500 606166246 CHVBK 20210128100657.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10440-014-9975-z doi (NATIONALLICENCE)springer-10.1007/s10440-014-9975-z Liu Qiao Department of Mathematics, Hunan Normal University, 410081, Changsha, Hunan, People's Republic of China aut A Regularity Criterion for the Navier-Stokes Equations in Terms of One Directional Derivative of the Velocity [Elektronische Daten] [Qiao Liu] We consider the regularity criterion for the three dimensional incompressible Navier-Stokes equations in terms of one directional derivative of the velocity. The result shows that if weak solution u satisfies $$\begin{aligned} \partial_{3}u\in L^{\frac{2}{1-r}}\bigl(0,T;\dot{M}^{p,\frac{3}{r}}\bigl( \mathbb{R}^{3}\bigr)\bigr) \end{aligned}$$ with 0<r<1 and $2\leq p\leq\frac{3}{r}$ , then u is regular on $(0,T]\times\mathbb{R}^{3}$ . Here, $\dot{M}^{p,\frac{3}{r}}(\mathbb{R}^{3})$ is the homogeneous Morrey-Campanato space. Springer Science+Business Media Dordrecht, 2014 Navier-Stokes equations nationallicence Regularity nationallicence Weak solution nationallicence Morrey-Campanato space nationallicence Acta Applicandae Mathematicae Springer Netherlands 140/1(2015-12-01), 1-9 0167-8019 140:1<1 2015 140 10440 https://doi.org/10.1007/s10440-014-9975-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10440-014-9975-z text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Liu Qiao Department of Mathematics, Hunan Normal University, 410081, Changsha, Hunan, People's Republic of China aut NATIONALLICENCE 773 0- Acta Applicandae Mathematicae Springer Netherlands 140/1(2015-12-01), 1-9 0167-8019 140:1<1 2015 140 10440