Antithesis of the Stokes paradox on the hyperbolic plane

TL;DR

This paper demonstrates the absence of the Stokes paradox on the hyperbolic plane by proving the existence of nontrivial solutions to the steady Stokes and Navier-Stokes equations in that setting, contrasting with the Euclidean case.

Contribution

It establishes the nonexistence of the Stokes paradox on the hyperbolic plane and shows solutions to the Navier-Stokes equations, which remains open in Euclidean space.

Findings

Existence of nontrivial $H^1_0$ solutions to steady Stokes equations on hyperbolic exterior domains.

Existence of solutions to steady Navier-Stokes equations on the hyperbolic plane.

Contrasts with the Euclidean case where these problems are unresolved.

Abstract

We show there exists a nontrivial $H^1_0$ 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. 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.