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

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 $\varepsilon = 0$.