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We examine the Navier-Stokes equations (NS) on a thin      3 3      -dimensional domain      Ω  ε  = Q 2 ×  ( 0 , ε  ) {\Omega _\varepsilon } = {Q_2} \times (0,\varepsilon )      , where      Q 2 {Q_2}       is a suitable bounded domain in      R 2 {\mathbb {R}^2}       and      ε  \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      R ( ε  ) \mathcal {R}(\varepsilon )       in      H 1 ( Ω  ε  ) {H^1}({\Omega _\varepsilon })       and      S ( ε  ) \mathcal {S}(\varepsilon )       in      W 1 , ∞  ( ( 0 , ∞  ) , L 2 ( Ω  ε  ) ) {W^{1,\infty }}((0,\infty ),{L^2}({\Omega _\varepsilon }))       such that if      U 0 ∈  R ( ε  ) {U_0} \in \mathcal {R}(\varepsilon )       and      F ∈  S ( ε  ) F \in \mathcal {S}(\varepsilon )      , then (NS) has a strong solution      U ( t ) U(t)       that remains in      H 1 ( Ω  ε  ) {H^1}({\Omega _\varepsilon })       for all      t ≥  0 t \geq 0       and in      H 2 ( Ω  ε  ) {H^2}({\Omega _\varepsilon })       for all      t &gt; 0 t &gt; 0      . We show that the set of strong solutions of (NS) has a local attractor      A ε  {\mathfrak {A}_\varepsilon }       in      H 1 ( Ω  ε  ) {H^1}({\Omega _\varepsilon })      , which is compact in      H 2 ( Ω  ε  ) {H^2}({\Omega _\varepsilon })      . Furthermore, this local attractor      A ε  {\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,      A ε  {\mathfrak {A}_\varepsilon }       is upper semicontinuous at      ε  = 0 \varepsilon = 0      .

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