Stochastic Navier--Stokes equations on a 3D thin domain

TL;DR

This paper proves that as a 3D thin domain's thickness approaches zero, the stochastic Navier--Stokes equations' solutions converge to the 2D case, justifying simplified models in applications.

Contribution

It establishes the convergence of solutions from 3D to 2D stochastic Navier--Stokes equations in thin domains, providing rigorous justification for using 2D models.

Findings

Martingale solutions in 3D converge to 2D solutions as thickness vanishes

Justifies 2D stochastic Navier--Stokes as an approximation for thin 3D domains

Provides mathematical foundation for simplified 2D models in applications

Abstract

Stochastic Navier--Stokes equations in a thin three-dimensional domain are considered, driven by additive noise. The convergence of martingale solution of the stochastic Navier--Stokes equations in a thin three-dimensional domain to the unique martingale solution of the 2D stochastic Navier--Stokes equations, as the thickness of the film vanishes, is established. Hence, we justify the approximation of 3D Navier--Stokes equations driven by random forcing by its corresponding two-dimensional setting in applications.