Global regularity of the Navier-Stokes equation on thin three dimensional domains with periodic boundary conditions

TL;DR

This paper establishes the global regularity of Navier-Stokes solutions on thin 3D domains with periodic boundary conditions, under certain initial data and forcing controls, extending previous results with a perturbation approach.

Contribution

It provides a new version of global regularity results for Navier-Stokes on thin domains, using a perturbation method based on reduced dependence on the thin coordinate.

Findings

Solutions remain globally regular under specified initial and forcing conditions.

The approach treats the 3D equation as a perturbation of a lower-dimensional problem.

Results extend previous work by Raugel, Sell, Moise, Temam, and Ziane.

Abstract

This paper gives another version of results due to Raugel and Sell, and similar results due to Moise, Temam and Ziane, that state the following: the solution of the Navier-Stokes equation on a thin 3 dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and the forcing term, where the control is larger than that obtainable via ``small data'' estimates. The approach taken is to consider the three dimensional equation as a perturbation of the equation when the vector field does not depend upon the coordinate in the thin direction.