$L^p$ theory for the interaction between the incompressible Navier-Stokes system and a damped plate

Debayan Maity (1) , Takéo Takahashi (2, 3)

1 TIFR-CAM - Center for Applicable Mathematics [Bangalore]
2 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
3 IECL - Institut Élie Cartan de Lorraine

We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.

Mots clés: Incompressible Navier-Stokes System; Fluid-structure interaction; Strong solutions; Maximal Lp regularity

Domaines: Equations aux dérivées partielles [math.AP]

Debayan Maity, Takéo Takahashi. $L^p$ theory for the interaction between the incompressible Navier-Stokes system and a damped plate. Journal of Mathematical Fluid Mechanics, 2021, ⟨10.1007/s00021-021-00628-5⟩. ⟨hal-02294097v2⟩