Thin-film limit of the Navier-Stokes equations in a curved thin domains

Thin-film limit of the Navier-Stokes equations in a curved thin domains

We consider the incompressible Navier-Stokes equations with Navier's slip boundary conditions in a 3D curved thin domain around a given closed surface. Under suitable assumptions, we show that the average in the thin direction of a strong solution to the thin domain problem weakly converges on the closed surface as the thickness of the thin domain tends to zero. Moreover, we derive limit equations on the closed surface by characterizing the weak limit of the average as a unique weak solution to the limit equations. Our limit equations are the Navier-Stokes equations on the closed surface described in terms of a fixed coordinate system of the ambient Euclidean space. However, it turns out that they are intrinsic and agree with the Navier-Stokes equations on an abstract Riemannian manifold. We also compare our limit equations with limit equations derived by Temam and Ziane (1997) under different boundary conditions when the surface is a sphere and show that the difference in two limit equations is due to the curvature of the boundary of the curved thin domain.

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