Interface boundary value problem for the Navier–Stokes equations in thin two-layer domains

We study a system of 3D Navier–Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.