We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface \(\Sigma \) without boundary and flows along \(\Sigma \). Local-in-time well-posedness is established in the framework of \(L_p\)-\(L_q\)-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on \(\Sigma \), and we show that each equilibrium on \(\Sigma \) is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.

我们考虑了不可压缩的粘性流体的运动，该流体完全覆盖了无边界的光滑，紧凑和嵌入的超表面\（\ Sigma \），并沿\（\ Sigma \）流动。在\（L_p \） - \（L_q \）-最大规律性的框架中建立时间局部的适时性。我们将均衡集的特征描述为\（\ Sigma \）上所有Killing向量场的集合，并且我们证明\（\ Sigma \）上的每个平衡都是稳定的。而且，表明随着时间趋于无穷大，任何接近平衡的解都全局存在并且以指数速率收敛到（可能不同的）平衡。