Difference estimate for weak solutions to the Navier-Stokes equations in a thin spherical shell and on the unit sphere

TL;DR

This paper derives estimates for the difference between weak solutions of the Navier-Stokes equations in a thin spherical shell and on the unit sphere, using a weak-strong uniqueness approach.

Contribution

It introduces a method to compare weak solutions in a thin shell and on a sphere by extending solutions and applying weak-strong uniqueness.

Findings

Difference estimates for weak solutions are established.

Extension method effectively compares solutions in different geometries.

Results contribute to understanding fluid behavior in thin spherical domains.

Abstract

We consider the Navier-Stokes equations in a three-dimensional thin spherical shell and on the two-dimensional unit sphere, and estimate the difference of weak solutions on the thin spherical shell and the unit sphere. Assuming that the weak solution on the thin spherical shell is a Leray-Hopf weak solution satisfying the energy inequality, we derive difference estimates for the two weak solutions by the weak-strong uniqueness argument. The main idea is to extend the weak solution on the unit sphere properly to an approximate solution on the thin spherical shell, and to use the extension as a strong solution in the weak-strong uniqueness argument.