Euler and Navier-Stokes equations on the hyperbolic plane

Abstract

We show that nonuniqueness of the Leray-Hopf solutions of the Navier-Stokes equation on the hyperbolic plane ℍ2 observed by Chan and Czubak is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on ℍn whenever n ≥ 3. We also describe the corresponding general Hamiltonian framework of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.