On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

Abstract In this paper, we study a nonlinear fluid-structure interaction problem between a ‘square-root’ viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions in both directions, while both the beam and the fluid are under the effect of time-periodic forces. By using a decoupling approach, at least one time-periodic weak solution to this problem is constructed which has a bounded energy and a fixed prescribed mass. The lack of a priori energy bounds is overcome by a series of estimates based on a careful choice of parameters. The most challenging one is the pressure estimate, which is obtained by utilizing the specific periodic geometry and the Bogovskiǐ operator on a fixed domain that has a uniform constant. With uniform estimates and improved regularity of the beam as in (Muha and Schwarzacher 2023 Ann. Inst. Henri Poin. Anal. Non Lineaire 39 1369–412), the time-periodic solution is constructed by a series of limit procedures, following the finite-dimensional time-space construction from (Feireisl et al 2012 Arch. Rational Mech. Anal. 204 74586).