Asymptotic analysis of the Navier-Stokes equations in the domains

We are interested in this article with the Navier–Stokes equations of viscous incompressible fluids in three dimensional thin domains. Let Ωe be the thin domain Ωe = ω × (0, e), where ω is a suitable domain in R and 0 < e < 1. Our aim is to derive an asymptotic expansion of the strong solution u of the Navier–Stokes equations in the thin domain Ωe when e is small, which is valid uniformly in time. This study should give a better understanding of the global existence results in thin domains obtained previously; see [15]–[17] and [23], [22]. We consider in this work two types of boundary conditions: the Dirichlet-periodic boundary condition and the purely periodic condition. For the first type of boundary condition we derive an asymptotic expansion of the solution u in terms of the solution of the associated Stokes problem. More precisely, we prove that the solution can be written, for e small, as