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

Abstract

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 > 0 t > 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 .