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Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data
Guillaume Ferriere1 ; Matthieu Hillairet2 
1 Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, France
2 IMAG, Univ Montpellier, CNRS, Montpellier, France
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].

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Accepté le :
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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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Guillaume Ferriere and Matthieu Hillairet},
     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},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
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     language = {en},
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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/

[1] Marco Bravin On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 3, pp. 303-319 | Numdam | Zbl

[2] Maity Debayan; Sylvain Ervedoza; Marius Tucsnak Large time behaviour for the motion of a solid in a viscous incompressible fluid (2020)

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[9] Jiao He; Dragoş Iftimie small solid body with large density in a planar fluid is negligible, J. Dyn. Differ. Equations, Volume 31 (2019) no. 3, pp. 1671-1688 | Zbl

[10] Hideo Kozono; Takayoshi Ogawa Decay properties of strong solutions for the Navier–Stokes equations in two-dimensional unbounded domains, Arch. Ration. Mech. Anal., Volume 122 (1993) no. 1, pp. 1-17 | DOI | Zbl

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