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Navier-Stokes equations in three-dimensional thin domains with various boundary conditions

Abstract. In this work we develop new methods for studying the Navier-Stokes equations in thin domains. We consider various boundary conditions and establish the global existence of strong solutions when the initial data belong to “large sets”. Our work was inspired by the recent interesting results of G. Raugel and G. Sell [RS 1,2,3] which, in the periodic case, give global existence for smooth solutions of the 3D Navier-Stokes equations in thin domains for large sets of initial conditions. We extend their results in several ways: we consider numerous boundary conditions and as it will appear hereafter, the passage from one boundary condition to another one is not necessarily straightforward; also the set of initial data for which we obtain the global existence in time is larger in our case and its characterization is simpler. The proof of our improved results is based on precise estimates of the dependence of some classical constants on the thickness ǫ of the domain; e.g. Sobolev type constants and the regularity constant for the corresponding Stokes problem. As an application, we study the behavior of the average of the strong solution in the thin direction when the thickness of the domain goes to zero; we prove its convergence to the strong solution of a 2D Navier-Stokes system of equations.