We study the unsteady incompressible Navier–Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. This leads to a coupled system of nonlinear PDEs where the moving part of the boundary is an unknown of the problem. The known existence theory for weak solutions is extended to non-linear Koiter shell models. We introduce a priori estimates that reveal higher regularity of the shell displacement beyond energy estimates. These are essential for non-linear Koiter shell models, since such shell models are non-convex (with respect to terms of highest order). The estimates are obtained by introducing new analytical tools that allow dissipative effects of the fluid to be exploited for the (non-dissipative) solid. The regularity result depends on the geometric constitution alone and is independent of the approximation procedure; hence it holds for arbitrary weak solutions. The developed tools are further used to introduce a generalized Aubin–Lions-type compactness result suitable for fluid–structure interactions.
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Keywords: Fluid–structure interaction, weak solutions, regularity, compactness
@article{AIHPC_2022__39_6_1369_0,
author = {Muha, Boris and Schwarzacher, Sebastian},
title = {Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1369--1412},
year = {2022},
volume = {39},
number = {6},
doi = {10.4171/aihpc/33},
language = {en},
url = {https://www.numdam.org/articles/10.4171/aihpc/33/}
}
TY - JOUR AU - Muha, Boris AU - Schwarzacher, Sebastian TI - Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions JO - Annales de l'I.H.P. Analyse non linéaire PY - 2022 SP - 1369 EP - 1412 VL - 39 IS - 6 UR - https://www.numdam.org/articles/10.4171/aihpc/33/ DO - 10.4171/aihpc/33 LA - en ID - AIHPC_2022__39_6_1369_0 ER -
%0 Journal Article %A Muha, Boris %A Schwarzacher, Sebastian %T Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions %J Annales de l'I.H.P. Analyse non linéaire %D 2022 %P 1369-1412 %V 39 %N 6 %U https://www.numdam.org/articles/10.4171/aihpc/33/ %R 10.4171/aihpc/33 %G en %F AIHPC_2022__39_6_1369_0
Muha, Boris; Schwarzacher, Sebastian. Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1369-1412. doi: 10.4171/aihpc/33
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