  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.
