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

Guillaume Ferriere (1) , Matthieu Hillairet (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 IMAG - Institut Montpelliérain Alexander Grothendieck

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, 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. On the basis of these two tools we extend previous decay estimates to arbitrary initial data and show local stability of the Lamb-Oseen vortex.

Domaines: Equations aux dérivées partielles [math.AP]

Guillaume Ferriere, Matthieu Hillairet. Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data. 2022. ⟨hal-03511566v1⟩