#### Vol. 309, No. 1, 2020

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Global regularity of the Navier–Stokes equations on 3D periodic thin domain with large data

### Na Zhao

Vol. 309 (2020), No. 1, 223–256
DOI: 10.2140/pjm.2020.309.223
##### Abstract

We consider the Navier–Stokes equations on a 3D periodic thin domain ${T}_{𝜖}=\left(0,{l}_{1}\right)×\left(0,{l}_{2}\right)×\left(0,𝜖\right)$. We show that there exists an absolute (large) constant $C$ such that for any ${C}^{\ast }>0$ which can be arbitrarily large, there exists an ${𝜖}_{0}>0$ such that the Navier–Stokes equations are globally well-posed for a class of large initial data satisfying

 $\parallel {\partial }_{h}{u}_{0}{\parallel }_{{L}^{2}\left({T}_{𝜖}\right)}\le \frac{{C}^{\ast }}{{𝜖}^{\frac{1}{2}}|ln𝜖{|}^{\frac{3}{2}}},\phantom{\rule{1em}{0ex}}\parallel {\partial }_{3}{u}_{0}{\parallel }_{{L}^{2}\left({T}_{𝜖}\right)}\le \frac{1}{C{𝜖}^{\frac{1}{2}}},$

where ${\partial }_{h}=\left({\partial }_{1},{\partial }_{2}\right)$ and $0<𝜖\le {𝜖}_{0}$. This improves the result of Kukavica and Ziane (Journal of Differential Equations 234:(2) (2007), 485–506), where the initial data ${u}_{0}$ is required to satisfy

 $\parallel \nabla {u}_{0}{\parallel }_{{L}^{2}\left({T}_{𝜖}\right)}\le \frac{1}{C{𝜖}^{\frac{1}{2}}|ln𝜖{|}^{\frac{3}{2}}}.$
##### Keywords
Navier–Stokes equations, thin domain
##### Mathematical Subject Classification 2010
Primary: 35Q30, 76D05, 76N10