Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2407-2443, September 2024.
Abstract. In this paper we studied the forward dynamics of nonautonomous dynamical systems in terms of forward attractors. We first reviewed the well-known uniform attractor theory, and then by weakening the uniformity of attraction we introduced semiuniform forward attractors and minimal (nonuniform) forward attractors. With these semiuniform attractors, a characterization of the structure of uniform attractors was given: a uniform attractor is composed of two semiuniform attractors and bounded complete trajectories connecting them. As a consequence, the nature of the forward attraction of a dissipative nonautonomous dynamical system was then revealed: the vector field in the distant future of the system determines the (nonuniform) forward asymptotic behavior. A criterion for certain semiuniform attractors to have finite fractal dimension was given and the finite dimensionality of uniform attractors was discussed. Forward attracting time-dependent sets were studied also. A sufficient condition and a necessary condition for a time-dependent set to be forward attracting were given with illustrative counterexamples. Forward attractors of a Navier–Stokes equation with asymptotically vanishing viscosity (with an Euler equation as the limit equation) and with time-dependent forcing were studied as applications.