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

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.