Abstract: In this talk I will go through some results concerning the long time behavior of fluid-structure interaction problems forced by an inflow. In these problems, the fluid domain depends on time in an unknown fashion. As a result, the solution operator associated to the system acts on a time dependent phase space: this makes it impossible to describe the system in terms of a semigroup nor of a process. Also, convergence to equilibrium is not immediate as comparing different solutions to time-dependent and stationary problems involve working with different solid and fluid domains.
Nonetheless, I will show that it is possible to extend the notion of global attractor to this particular setting, and prove its existence and regularity. Moreover, if the inflow is sufficiently small, we can prove a return-to-rest result through a novel approach based on a solid-fattening parameter.
This talk is based on some joint works with D. Bonheure, F. Gazzola, M. Hillairet, V. Pata and G. Sperone.