Abstract
We study the long-time dynamics of a coupled system consisting of the 2D Navier-Stokes equations and full von Karman elasticity equations. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.
Highlights
We assume that the motion of the fluid and the elastic cylindrical shell is radially symmetric
We prove the well-posedness of the system considered and investigate the longtime dynamics of solutions to the coupled problem in (1)-(11)
To conclude the proof of the existence of weak solutions we only need to show that any function ψ in L0T can be approximated by a sequence of functions of the form (52)
Summary
While in the first paper a nonlinear system describing the interaction of a viscous incompressible fluid in a bounded vessel with a flat elastic part of the boundary moving in the in-plane directions only is considered, the second one deals with the transversal displacement on a flexible flat part of the boundary All these sources deal with the case of bounded reservoirs Ω and a flat elastic shallow shell or plate. Regarding infinite reservoirs Ω we can mention works [8, 12] which establish the existence of a compact global attractors to linearized around a Poiseuille type flow Navier-Stokes systems in unbounded domains coupled with a nonlinear equation on the boundary accounting for the transverse displacements.
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