Euler and Navier-Stokes equations on the hyperbolic plane

摘要

We show that nonuniqueness of the Leray-Hopf solutions of the Navier-Stokes equation on the hyperbolic plane H~2 observed by Chan and Czubak is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on H~n whenever n ≥ 3. We also describe the corresponding general Hamiltonian framework of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.