Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting

TL;DR

This paper demonstrates that the uniqueness of Leray-Hopf weak solutions to the Navier-Stokes equations fails on hyperbolic spaces, indicating ill-posedness and a breakdown of classical theorems in negatively curved geometries.

Contribution

It proves non-uniqueness of Leray-Hopf solutions for Navier-Stokes on hyperbolic spaces and extends results to general negatively curved manifolds, challenging existing well-posedness assumptions.

Findings

Non-uniqueness of solutions on hyperbolic space

Breakdown of classical Liouville theorem in hyperbolic setting

Ill-posedness of Navier-Stokes in negatively curved geometries

Abstract

We consider the Navier-Stokes equation on $\mathbb{H}^{2}(-a^{2})$, the two dimensional hyperbolic space with constant sectional curvature $-a^{2}$. We prove an ill-posedness result in the sense that the uniqueness of the Leray-Hopf weak solutions to the Navier-Stokes equation breaks down on $\mathbb{H}^{2}(-a^{2})$. We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.