# On singular limit equations for incompressible fluids in moving thin domains

## TL;DR

This paper derives and analyzes the limiting equations for incompressible Euler and Navier-Stokes flows in moving thin domains as they degenerate into two-dimensional surfaces, highlighting their energy structures and relations to stationary manifold equations.

## Contribution

It provides a heuristic derivation of singular limit equations for fluid flows in moving thin domains and compares them with equations on stationary manifolds.

## Key findings

- Derivation of limit equations on degenerate moving surfaces
- Analysis of energy structure relations between original and limit equations
- Comparison with stationary manifold equations using Levi-Civita connection

## Abstract

We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are described in terms of the Levi-Civita connection.