Long-time behavior of the tangential surface Navier–Stokes equation

We investigate the initial-value problem for the incompressible tangential Navier-Stokes equation with variable viscosity on a given two-dimensional surface without boundary. Existence of global weak and strong solutions under inhomogeneous forcing is proved by a fixed-point and continuation argument. Continuous dependence on data, backward uniqueness, and instantaneous regularization are also discussed. Depending on the effect of the inhomogeneous forcing on the dissipative and the nondissipative components of the system, we investigate the long-time behavior of solutions. We prove the existence and properties of the $σ$-global attractor, in the case of bounded trajectories, and of the so-called unbounded attractor, for unbounded trajectories.