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

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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

Abstract
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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 & Continuous Dynamical Systems - A, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67

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