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 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.