Navier--Stokes equations in a curved thin domain

We consider the three-dimensional incompressible Navier--Stokes equations in a curved thin domain with Navier's slip boundary conditions. The curved thin domain is defined as a region between two closed surfaces which are very close to each other and degenerates into a given closed surface as its width tends to zero. We establish the global-in-time existence and uniform estimates of a strong solution for large data when the width of the thin domain is very small. Moreover, we study a singular limit problem as the thickness of the thin domain tends to zero and rigorously derive limit equations on the limit surface, which are the damped and weighted Navier--Stokes equations on a surface with viscous term involving the Gaussian curvature of the surface. We prove the weak convergence of the average in the thin direction of a strong solution to the bulk Navier--Stokes equations and characterize the weak limit as a weak solution to the limit equations as well as provide estimates for the difference between solutions to the bulk and limit equations. To deal with the weighted surface divergence-free condition of the limit equations we also derive the weighted Helmholtz--Leray decomposition of a tangential vector field on a closed surface.