Longtime Behavior of Solutions of Non-Autonomous Equations in Fluid Dynamics

Abstract

We will treat a non-autonomous dynamical system and investigate some long-time behavior of the solution. Our motivation is to study a dynamical system derived from a moving boundary problem. In this case the domain of the generator of the corresponding evolution equation is time-dependent. By this reason, the known theory of the process is not applicable to such a problem. The purpose of the present paper is to extend the notion of the process to be able to apply to above problems. Making use of the extended process, we can study the 2-D Navier-Stokes equation in a given time-dependent region.