Remarks on the weak formulation of the Navier-Stokes equations on the 2D hyperbolic space

TL;DR

This paper addresses the formulation of the Navier-Stokes equations on 2D hyperbolic space to restore the uniqueness of Leray-Hopf solutions, contrasting with known non-uniqueness results.

Contribution

It proposes a modified weak formulation of the Navier-Stokes equations on hyperbolic space that ensures solution uniqueness.

Findings

Uniqueness of solutions is restored with the new formulation

Contrasts with previous non-uniqueness results in hyperbolic settings

Provides a framework for well-posedness in hyperbolic geometry

Abstract

The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on $\R^{2}$. In our previous work we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored.