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

Abstract

We consider the Navier-Stokes equation on H(−a), the two dimensional hyperbolic space with constant sectional curvature −a. 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 H(−a). 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.