A regularity criterion for the Navier-Stokes equations based on the gradient of one velocity component

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• 作者：Zdeněk Skalá ; k ^{skalak@mat.fsv.cvut.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Navier&ndash ; Stokes equations ; Regularity of solutions ; Regularity criteria
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：437
• 期：1
• 页码：474-484
• 全文大小：344 K

文摘

We show that if u   is a Leray solution to the Navier–Stokes equations in the full three-dimensional space with an initial condition from ontainer hidden">$W_{0,\sigma }^{1,2}$, T>0ontainer hidden">$T>0$ and 136f768d0d9a5c1aa1c9ab4eb12a" title="Click to view the MathML source">u∈L^t(0,T;L^s)ontainer hidden">$u\in L^t(0,T;L^s)$, where 2/t+3/s=59/30ontainer hidden">$2/t+3/s=59/30$ for s∈(2,30/13]ontainer hidden">$s\in (2,30/13]$ and 2/t+3/s=7/4+1/(2s)ontainer hidden">$2/t+3/s=7/4+1/(2s)$ for s∈(30/13,3)ontainer hidden">$s\in (30/13,3)$ then u   is regular on (0,T)ontainer hidden">$(0,T)$. We prove our result as a special case of a more general method which might possibly bring a further improvement.