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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.
Grandmont, Céline 1, 2 ; Hillairet, Matthieu 3 ; Lequeurre, Julien 4
@article{AIHPC_2019__36_4_1105_0,
author = {Grandmont, C\'eline and Hillairet, Matthieu and Lequeurre, Julien},
title = {Existence of local strong solutions to fluid{\textendash}beam and fluid{\textendash}rod interaction systems},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1105--1149},
publisher = {Elsevier},
volume = {36},
number = {4},
year = {2019},
doi = {10.1016/j.anihpc.2018.10.006},
mrnumber = {3955112},
zbl = {1428.35291},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.10.006/}
}
TY - JOUR AU - Grandmont, Céline AU - Hillairet, Matthieu AU - Lequeurre, Julien TI - Existence of local strong solutions to fluid–beam and fluid–rod interaction systems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2019 SP - 1105 EP - 1149 VL - 36 IS - 4 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.10.006/ DO - 10.1016/j.anihpc.2018.10.006 LA - en ID - AIHPC_2019__36_4_1105_0 ER -
%0 Journal Article %A Grandmont, Céline %A Hillairet, Matthieu %A Lequeurre, Julien %T Existence of local strong solutions to fluid–beam and fluid–rod interaction systems %J Annales de l'I.H.P. Analyse non linéaire %D 2019 %P 1105-1149 %V 36 %N 4 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.10.006/ %R 10.1016/j.anihpc.2018.10.006 %G en %F AIHPC_2019__36_4_1105_0
Grandmont, Céline; Hillairet, Matthieu; Lequeurre, Julien. Existence of local strong solutions to fluid–beam and fluid–rod interaction systems. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1105-1149. doi: 10.1016/j.anihpc.2018.10.006
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