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 $T_{𝜖}=\left(0,l_1\right)\times \left(0,l_2\right)\times \left(0,𝜖\right)$. 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

$$\parallel {\partial }_h{u}_0{\parallel }_{L^2\left(T_{𝜖}\right)}\le \frac{C^{\ast }}{{𝜖}^{\frac{1}{2}}|ln𝜖{|}^{\frac{3}{2}}},\parallel {\partial }_3{u}_0{\parallel }_{L^2\left(T_{𝜖}\right)}\le \frac{1}{C{𝜖}^{\frac{1}{2}}},$$

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

Milestones

Received: 9 June 2018

Revised: 23 February 2020

Accepted: 10 September 2020

Published: 26 December 2020