The Euler and Navier-Stokes equations on the hyperbolic plane

TL;DR

This paper investigates the uniqueness of solutions to the Navier-Stokes equations on hyperbolic spaces, revealing non-uniqueness in 2D due to Hodge decomposition and its absence in higher dimensions, within a broader geometric framework.

Contribution

It demonstrates the role of Hodge decomposition in solution non-uniqueness on hyperbolic planes and extends the analysis to general Riemannian manifolds.

Findings

Non-uniqueness of Leray-Hopf solutions on hyperbolic plane due to Hodge decomposition

No non-uniqueness phenomenon on higher-dimensional hyperbolic spaces

Framework for hydrodynamics on complete Riemannian manifolds

Abstract

We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.