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].

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
     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},
     doi = {10.5802/crmath.357},
     language = {en},
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JO  - Comptes Rendus. Mathématique
PY  - 2023
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VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.357
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%0 Journal Article
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%A Matthieu Hillairet
%T Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data
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%D 2023
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%I Académie des sciences, Paris
%R 10.5802/crmath.357
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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.

[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)

[3] Manuel Del Pino; Jean Dolbeault Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., Volume 81 (2002) no. 9, pp. 847-875 | DOI | Zbl

[4] Sylvain Ervedoza; Matthieu Hillairet; Christophe Lacave Long-time behavior for the two-dimensional motion of a disk in a viscous fluid, Commun. Math. Phys., Volume 329 (2014) no. 1, pp. 325-382 | DOI | Zbl

[5] Hiroshi Fujita; Tosio Kato On the Navier–Stokes initial value problem. I, Arch. Ration. Mech. Anal., Volume 16 (1964), pp. 269-315 | DOI | Zbl

[6] Giovanni P. Galdi An introduction to the mathematical theory of the Navier–Stokes equations. Steady-state problems, Springer Monographs in Mathematics, Springer, 2011 | Zbl

[7] Thierry Gallay; Yasunori Maekawa Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity, Anal. PDE, Volume 6 (2013) no. 4, pp. 973-991 | DOI | Zbl

[8] Thierry Gallay; C. Eugene Wayne Global stability of vortex solutions of the two-dimensional Navier–Stokes equation, Commun. Math. Phys., Volume 255 (2005) no. 1, pp. 97-129 | DOI | Zbl

[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

[11] Christophe Lacave; Takéo Takahashi Small moving rigid body into a viscous incompressible fluid, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 3, pp. 1307-1335 | DOI | Zbl

[12] Frank W. J. Olver Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press Inc., 1974 | Zbl

[13] Amnon Pazy Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1983 | DOI | Zbl

[14] Hermann Sohr The Navier–Stokes equations. An elementary functional analytic approach, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2001 | DOI | Zbl

[15] Takéo Takahashi; Marius Tucsnak Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., Volume 6 (2004) no. 1, pp. 53-77 | DOI | Zbl

[16] Roger Temam Navier–Stokes Equations. Theory and numerical analysis, Studies in Mathematics and its Applications, 2, North-Holland, 1977 | Zbl

[17] Yun Wang; Zhouping Xin Analyticity of the semigroup associated with the fluid-rigid body problem and local existence of strong solutions, J. Funct. Anal., Volume 261 (2011) no. 9, pp. 2587-2616 | DOI | Zbl

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