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

We consider the Navier–Stokes equations on a 3D periodic thin domain $T_{𝜖} = (0, l_{1}) \times (0, l_{2}) \times (0, 𝜖)$ . We show that there exists an absolute (large) constant $C$ such that for any $C^{*} > 0$ 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

∥ \partial_{h} u_{0} ∥_{L^{2} (T_{𝜖})} \leq \frac{C^{*}}{𝜖^{\frac{1}{2}} | ln 𝜖 |^{\frac{3}{2}}}, ∥ \partial_{3} u_{0} ∥_{L^{2} (T_{𝜖})} \leq \frac{1}{C 𝜖^{\frac{1}{2}}},

where $\partial_{h} = (\partial_{1}, \partial_{2})$ and $0 < 𝜖 \leq 𝜖_{0}$ . 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

∥ \nabla u_{0} ∥_{L^{2} (T_{𝜖})} \leq \frac{1}{C 𝜖^{\frac{1}{2}} | ln 𝜖 |^{\frac{3}{2}}} .

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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

Authors

Na Zhao
Institute of Applied Physics and Computational Mathematics Beijing China

Vol. 309, No. 1, 2020

Na Zhao

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors