Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems

Abstract

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image of a torus. Moreover the global attractor attracts all the solutions of the NS system with exponential rate, that is, the attractor is exponential. We also consider the 2D Navier–Stokes system with rapidly oscillating external force g(x,t,t/ε), which has the average as ε → 0+. We assume that the function g(x,t,z) has a bounded primitive with respect to z and the averaged NS system has a small Grashof number that provides a simple structure of the averaged global attractor. Then we prove that the distance from the global attractor of the original NS system to the attractor of the averaged NS system is less than a small power of ε.