Singular limit problem for the Navier–Stokes equations in a curved thin domain

Abstract

We consider the incompressible Navier–Stokes equations with slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions on given data, we establish the convergence on the limit surface of the average in the thin direction of a strong solution to the bulk equations as the width of the curved thin domain tends to zero. Moreover, we characterize the limit as a unique weak solution to limit equations, which are the damped and weighted Navier–Stokes equations on the limit surface.