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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
DOI: https://doi.org/10.1090/S0894-0347-1993-1179539-4
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