Advances in Differential Equations

Navier-Stokes equations in three-dimensional thin domains with various boundary conditions

R. Temam and M. Ziane

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In this work we develop methods for studying the Navier-Stokes equations in thin domains. We consider various boundary conditions and establish the global existence of strong solutions when the initial data belong to "large sets." Our work was inspired by the recent interesting results of G. Raugel and G. Sell [22, 23, 24] which, in the periodic case, give global existence for smooth solutions of the 3D Navier-Stokes equations in thin domains for large sets of initial conditions. We extend their results in several ways, we consider numerous boundary conditions and as it will appear hereafter, the passage from one boundary condition to another one is not necessarily straightforward. The proof of our improved results is based on precise estimates of the dependence of some classical constants on the thickness $\epsilon$ of the domain, e.g. Sobolev-type constants and the regularity constant for the corresponding Stokes problem.

Article information

Adv. Differential Equations Volume 1, Number 4 (1996), 499-546.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 76D05: Navier-Stokes equations [See also 35Q30]


Temam, R.; Ziane, M. Navier-Stokes equations in three-dimensional thin domains with various boundary conditions. Adv. Differential Equations 1 (1996), no. 4, 499--546.

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