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A regularity criterion for the Navier–Stokes equations via one diagonal entry of the velocity gradient
Communications in Mathematical Sciences ( IF 1.2 ) Pub Date : 2021-01-01 , DOI: 10.4310/cms.2021.v19.n4.a10
Zdeněk Skalák 1
Affiliation  

We study the conditional regularity of solutions to the Navier-Stokes equations in the three dimensional space. Let $u=(u_1,u_2,u_3)$ denote the velocity. We impose an additional condition only on one diagonal entry of the velocity gradient, namely $\partial_3 u_3$, and show, using a technique based on the mixed multiplier theorem and an anisotropic version of the Troisi inequality, that if $\partial_3 u_3$ lies in the space $L^\beta (0,T; L^q)$ with suitable $\beta,q$, then $u$ is regular on $(0,T]$. Our result improves and extends the analogous results known from the literature.

中文翻译:

Navier-Stokes 方程的规律性准则,通过速度梯度的一个对角输入

我们研究了 Navier-Stokes 方程在三维空间中解的条件规律性。让 $u=(u_1,u_2,u_3)$ 表示速度。我们仅对速度梯度的一个对角线条目施加附加条件,即 $\partial_3 u_3$,并使用基于混合乘数定理和 Troisi 不等式的各向异性版本的技术表明,如果 $\partial_3 u_3$位于空间 $L^\beta (0,T; L^q)$ 和合适的 $\beta,q$,那么 $u$ 在 $(0,T]$ 上是正则的。我们的结果改进并扩展了类似的文献中已知的结果。
更新日期:2021-01-01
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