Comptes Rendus
Partial differential equations
Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 453-485.

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].

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/crmath.357

Guillaume Ferriere 1; Matthieu Hillairet 2

1 Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, France
2 IMAG, Univ Montpellier, CNRS, Montpellier, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data},
     journal = {Comptes Rendus. Math\'ematique},
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Guillaume Ferriere; Matthieu Hillairet. Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 453-485. doi : 10.5802/crmath.357. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.357/

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