Existence, uniqueness, and asymptotic stability results for the 3-D Navier-Stokes equations on multi-connected domains with time-dependent inhomogeneous boundary conditions

We consider solutions of the 3-D Navier-Stokes equations (NSE) on a multi-connected bounded domain $ \Omega\subset\mathbb{R^{\mathrm{3}}} $ with time-dependent inhomogeneous boundary values $ \beta = \beta(x, t) $ on $ \partial\Omega = \varGamma $ where $ \varGamma $ is as in [3] a union of Lipschitz surfaces $ \varGamma_{0}, \varGamma_{1}, ..., \varGamma_{l} $ whose interiors $ \mathring{\varGamma_{i}} $ are disjoint. The general flux condition $ \sum\nolimits_{j = 0}^{l} \int_{\varGamma_{j}}\beta\cdot{\mathbf n}dS = 0 $ must be satisfied due to compatibility considerations. Here we generalize the results in [3] to allow the time-dependent case $ \beta = \beta(x, t) $ with $ \beta\in C_{B}([0, T_{0});H^{1/2}(\varGamma) $ where $ T_{0}\in(0, \infty] $.