# Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

### Matthieu Hillairet

Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France### Julien Lequeurre

Université de Lorraine, CNRS, IECL, Inria, F-57000 Metz, France### Céline Grandmont

Inria de Paris, 75012 Paris, France, Sorbonne Université, Lab. Jacques-Louis Lions, 75252 Paris, France

## Abstract

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled *via* interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

## Cite this article

Matthieu Hillairet, Julien Lequeurre, Céline Grandmont, Existence of local strong solutions to fluid–beam and fluid–rod interaction systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 4, pp. 1105–1149

DOI 10.1016/J.ANIHPC.2018.10.006