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

Abstract

The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on R-2. We show the uniqueness of the Leray-Hopf solutions breaks down on H-2(-a(2)), the two dimensional hyperbolic space with constant sectional curvature -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.