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. 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.

中文翻译：

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双曲平面上的斯托克斯悖论的对立

我们表明在双曲平面的2D外部域上存在稳定Stokes方程的非平凡（H ^ 1_0 \）解。因此，我们证明在双曲线环境中没有斯托克斯悖论。实际上，我们构造的解既满足了无滑移边界条件又满足了无穷大的要求。这意味着该解决方案在某种意义上实际上是自相矛盾的解决方案，因为流体在移动时没有任何物理原因在移动。我们还显示了在相同设置下稳定Navier–Stokes方程的非平凡解的存在，而在欧几里得情况下类似问题是开放的。