Long-time behavior for 2D non-autonomous $g$-Navier–Stokes equations
Volume 103 / 2012
Annales Polonici Mathematici 103 (2012), 277-302
MSC: Primary 35B41; Secondary 35Q30, 37L30, 35D05.
DOI: 10.4064/ap103-3-5
Abstract
We study the first initial boundary value problem for the 2D non-autonomous $g$-Navier–Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite-dimensional pullback $\mathcal D_\sigma$-attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Furthermore, when the force is time-independent and “small”, the existence, uniqueness and global stability of a stationary solution are also studied.
