In this paper, we analyse the long-time behavior of solutions to a coupled system describing the motion of a rigid disk in a 2D viscous incompressible fluid. Following previous approaches in [4, 15, 17] we look at the problem in the system of coordinates associated with the center of mass of the disk. Doing so, we introduce a further nonlinearity to the classical Navier Stokes equations. In comparison with the classical nonlinearities, this new term lacks time and space integrability, thus complicating strongly the analysis of the long-time behavior of solutions.
We provide herein two refined tools: a refined analysis of the Gagliardo–Nirenberg inequalities and a thorough description of fractional powers of the so-called fluid-structure operator [2]. On the basis of these two tools we extend decay estimates obtained in [4] to arbitrary initial data and show local stability of the Lamb-Oseen vortex in the spirit of [7, 8].
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Ferriere, Guillaume 1 ; Hillairet, Matthieu 2
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@article{CRMATH_2023__361_G2_453_0,
author = {Ferriere, Guillaume and Hillairet, Matthieu},
title = {Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data},
journal = {Comptes Rendus. Math\'ematique},
pages = {453--485},
year = {2023},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
number = {G2},
doi = {10.5802/crmath.357},
language = {en},
url = {https://www.numdam.org/articles/10.5802/crmath.357/}
}
TY - JOUR AU - Ferriere, Guillaume AU - Hillairet, Matthieu TI - Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data JO - Comptes Rendus. Mathématique PY - 2023 SP - 453 EP - 485 VL - 361 IS - G2 PB - Académie des sciences, Paris UR - https://www.numdam.org/articles/10.5802/crmath.357/ DO - 10.5802/crmath.357 LA - en ID - CRMATH_2023__361_G2_453_0 ER -
%0 Journal Article %A Ferriere, Guillaume %A Hillairet, Matthieu %T Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data %J Comptes Rendus. Mathématique %D 2023 %P 453-485 %V 361 %N G2 %I Académie des sciences, Paris %U https://www.numdam.org/articles/10.5802/crmath.357/ %R 10.5802/crmath.357 %G en %F CRMATH_2023__361_G2_453_0
Ferriere, Guillaume; Hillairet, Matthieu. Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data. Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 453-485. doi: 10.5802/crmath.357
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