Multi-valued dynamical systems on time-dependent metric spaces with applications to Navier–Stokes equations

In this paper, we develop an extension of the theoretical framework of multi-valued dynamical systems for families of time-dependent phase spaces, where special attention was paid to the relationship between the pullback attractors of homeomorphically equivalent dynamical systems. We apply this theory to show that the 3D Navier–Stokes equations defined on a non-cylindrical domain, satisfying certain hypotheses about the energy inequality, generate an upper-semicontinuous multi-valued dynamical system, and then, by means of the energy method, we show that this system is asymptotically compact and has a pullback attractor on a tempered universe. Using current techniques we also prove that pullback attractors associated with the single-valued dynamical systems that satisfy the smoothing property have finite fractal dimension. This latter result is applied to show that the 2D Navier–Stokes equations on a non-cylindrical domain has a pullback attractor with finite fractal dimension.