Comptes Rendus. Mathématique (Feb 2023)

Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data

  • Ferriere, Guillaume,
  • Hillairet, Matthieu

DOI
https://doi.org/10.5802/crmath.357
Journal volume & issue
Vol. 361, no. G2
pp. 453 – 485

Abstract

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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].