Strong solutions to the Navier-Stokes equations on thin 3D domains
  We prove the existence of strong solutions to Navier-Stokes equations in three dimensional thin domains. Our proof is based on the energy and the Poincar\'e inequalities as well as contraction principle argument and is free of the mean value operator. The price we pay for the simplicity of the proof are stronger assumptions on the initial velocity and the forcing term. We need to assume that their derivatives with respect to time belong to certain Lebesgue space. 
B. Nowakowski, W. Zaj\k{a}czkowski
math.AP