Approximation of a solution to the stationary Navier-Stokes equations in a curved thin domain by a solution to thin-film limit equations

TL;DR

This paper demonstrates that solutions to the stationary Navier-Stokes equations in a curved thin domain can be effectively approximated by solutions to surface limit equations as the domain's thickness approaches zero, using averaging and difference estimates.

Contribution

It provides a rigorous approximation framework connecting bulk solutions to surface solutions in curved thin domains for stationary Navier-Stokes equations.

Findings

Bulk solutions are well-approximated by surface solutions in thin domains.

The approximation improves as the domain thickness decreases.

A difference estimate quantifies the approximation accuracy.

Abstract

We consider the stationary Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under the slip boundary conditions. Our aim is to show that a solution to the bulk equations is approximated by a solution to limit equations on the surface appearing in the thin-film limit of the bulk equations. To this end, we take the average of the bulk solution in the thin direction and estimate the difference of the averaged bulk solution and the surface solution. Then we combine an obtained difference estimate on the surface with an estimate for the difference of the bulk solution and its average to get a difference estimate for the bulk and surface solutions in the thin domain, which shows that the bulk solution is approximated by the surface one when the thickness of the thin domain is sufficiently small.