Some results on the Navier-Stokes equations in thin 3D domains

We consider the Navier-Stokes equations on thin 3D domains, supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works. An important tool in the proofs are some Sobolev embeddings into anisotropic L^p-type spaces. Better results are proved in the purely periodic case, where the conservation of enstrophy property is used. We also give a new uniqueness criterium for weak Leray solutions.