Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Navier-Stokes equations on thin $ 3$D domains. I. Global attractors and global regularity of solutions

Authors: Geneviève Raugel and George R. Sell
Journal: J. Amer. Math. Soc. 6 (1993), 503-568
MSC: Primary 35Q30; Secondary 34D45, 35B65, 58F39, 76D05
MathSciNet review: 1179539
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Abstract: We examine the Navier-Stokes equations (NS) on a thin $ 3$-dimensional domain $ {\Omega _\varepsilon } = {Q_2} \times (0,\varepsilon )$, where $ {Q_2}$ is a suitable bounded domain in $ {\mathbb{R}^2}$ and $ \varepsilon $ is a small, positive, real parameter. We consider these equations with various homogeneous boundary conditions, especially spatially periodic boundary conditions. We show that there are large sets $ \mathcal{R}(\varepsilon )$ in $ {H^1}({\Omega _\varepsilon })$ and $ \mathcal{S}(\varepsilon )$ in $ {W^{1,\infty }}((0,\infty ),{L^2}({\Omega _\varepsilon }))$ such that if $ {U_0} \in \mathcal{R}(\varepsilon )$ and $ F \in \mathcal{S}(\varepsilon )$, then (NS) has a strong solution $ U(t)$ that remains in $ {H^1}({\Omega _\varepsilon })$ for all $ t \geq 0$ and in $ {H^2}({\Omega _\varepsilon })$ for all $ t > 0$. We show that the set of strong solutions of (NS) has a local attractor $ {\mathfrak{A}_\varepsilon }$ in $ {H^1}({\Omega _\varepsilon })$, which is compact in $ {H^2}({\Omega _\varepsilon })$. Furthermore, this local attractor $ {\mathfrak{A}_\varepsilon }$ turns out to be the global attractor for all the weak solutions (in the sense of Leray) of (NS). We also show that, under reasonable assumptions, $ {\mathfrak{A}_\varepsilon }$ is upper semicontinuous at $ \varepsilon = 0$.

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Keywords: Attractor, global attractor, global regularity, Navier-Stokes equations, three-dimensional space
Article copyright: © Copyright 1993 American Mathematical Society