Antithesis of the Stokes Paradox on the Hyperbolic Plane

Abstract

We show there exists a nontrivial H-0(1) solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. In fact, the solution we construct satisfies both the no-slip boundary condition and vanishing at infinity. This means that the solution is in some sense actually a paradoxical solution since the fluid is moving without having any physical cause to move. We also show the existence of a nontrivial solution to the steady Navier-Stokes equation in the same setting, whereas the analogous problem is open in the Euclidean case.