The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on ℝ2. We show the uniqueness of the Leray-Hopf solutions breaks down on ℍ2(-a2), the two dimensional hyperbolic space with constant sectional curvature -a2. 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.
- Hyperbolic space
- Liouville theorem