On an existence theory for a fluid-beam problem encompassing possible contacts

Jean-Jérôme Casanova (1) , Céline Grandmont (2, 3, 4) , Matthieu Hillairet (5)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 SU - Sorbonne Université
3 COMMEDIA - COmputational Mathematics for bio-MEDIcal Applications
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
5 IMAG - Institut Montpelliérain Alexander Grothendieck

In this paper we consider a coupled system of pdes modelling the interaction between a two-dimensional incompressible viscous fluid and a one-dimensional elastic beam located on the upper part of the fluid domain boundary. We design a functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity. We then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanishes.

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Jérôme Casanova, Céline Grandmont, Matthieu Hillairet. On an existence theory for a fluid-beam problem encompassing possible contacts. Journal de l'École polytechnique — Mathématiques, 2021, 8, pp.933-971. ⟨10.5802/jep.162⟩. ⟨hal-02396915⟩