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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Navier-Stokes equations on thin $3$D domains. I. Global attractors and global regularity of solutions
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by Geneviève Raugel and George R. Sell PDF
J. Amer. Math. Soc. 6 (1993), 503-568 Request permission

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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • 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