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March  2006, 16(1): 67-86. doi: 10.3934/dcds.2006.16.67

Regularity of the Navier-Stokes equation in a thin periodic domain with large data

Igor Kukavica 1, and  Mohammed Ziane 1,

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089

Received  September 2005 Revised  February 2006 Published  June 2006

Let $\Omega=[0,L_1]\times[0,L_2]\times[0,\epsilon]$ where $L_1,L_2>0$ and $\epsilon\in(0,1)$. We consider the Navier-Stokes equations with periodic boundary conditions and prove that if

$ \|\| \nabla u_0\|\|_{L^2(\Omega)} \le \frac{1}{C(L_1,L_2)\epsilon^{1/6}} $

then there exists a unique global smooth solution with the initial datum $u_0$.

Keywords: thin domains, weak solutions, strong solutions, Navier-Stokes equations, regularity..
Mathematics Subject Classification: Primary: 35Q30, 35K1.
Citation: Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
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