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 T𝜖 = (0,l1) × (0,l2) × (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 𝜖0 > 0 such that the Navier–Stokes equations are globally well-posed for a class of large initial data satisfying

hu0L2(T𝜖) C 𝜖1 2 |ln𝜖|3 2 ,3u0L2(T𝜖) 1 C𝜖1 2 ,

where h = (1,2) and 0 < 𝜖 𝜖0. This improves the result of Kukavica and Ziane (Journal of Differential Equations 234:(2) (2007), 485–506), where the initial data u0 is required to satisfy

u0L2(T𝜖) 1 C𝜖1 2 |ln𝜖|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
Authors
Na Zhao
Institute of Applied Physics and Computational Mathematics
Beijing
China