Large time behaviour for the motion of a solid in a viscous incompressible fluid

Abstract

In this article, we study the long-time behaviour of a system describing the coupled motion of a rigid body and of a viscous incompressible fluid in which the rigid body is contained. We assume that the system formed by the rigid body and the fluid fills the entire space R 3 . In the case in which the rigid body is a ball, we prove the local existence of mild solutions and, when the initial data are small, the global existence of solutions for this system with a precise description of their large time behavior. Our main result asserts, in particular, that if the initial datum is small enough in suitable norms then the position of the center of the rigid ball converges to some h ∞ ∈ R 3 as time goes to infinity. This result contrasts with those known for the analogues of our system in 2 or 1 space dimensions, where it has been proved that the body quits any bounded set, provided that we wait long enough. To achieve this result, we use a "monolithic" type approach, which means that we consider a linearized problem in which the equations of the solid and of the fluid are still coupled. An essential role is played by the properties of the semigroup, called fluid-structure semigroup, associated to this coupled linearized problem. The generator of this semigroup is called the fluidstructure operator. Our main tools are new L p − L q estimates for the fluid-structure semigroup. Note that these estimates are proved for bodies of arbitrary shape. The main ingredients used to study the fluid-structure semigroup and its generator are resolvent estimates which provide both the analyticity of the fluid-structure semigroup (in the spirit of a classical work of Borchers and Sohr) and L p − L q decay estimates (by adapting a strategy due to Iwashita).

Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation

“…Let us emphasize that using the L p − L q setting allows us to weaken the regularity on the initial conditions (see for instance [33]). Moreover, this "L p − L q " framework is interesting even for studies in fluid-structure interaction problems done in the "L 2 − L 2 " framework: let us quote for instance the uniqueness of weak solutions ( [21,8]), the asymptotic behavior for large time ( [17,18]), and the asymptotic behavior for small structures ( [28]).…”

Rigorous analysis of the interaction problem of sea ice with a rigid body

“…It is a classical problem in fluid mechanics to study the movement of rigid or elastic bodies immersed in a fluid, see e.g. the works of Galdi [16], Hoffmann and Starovoitov [26], Desjardins and Esteban [11,12], Gunzberger, Lee and Seregin [22], Feireisl, Hillairet and Nečasová [14], Cumsille, Takahashi and Tucsnak [7,8], Geissert, Götze and Hieber [19], Maity, Raymond, Roy and Vanninathan [35,43], and the recent work of Ervedoza, Maity and Tucsnak [13]. We also refer to the survey article [17] by Galdi and Neustupa for the stationary case.…”

“…We also refer to the survey article [17] by Galdi and Neustupa for the stationary case. The very recent article [13] by Ervedoza, Maity and Tucsnak discusses the long-time behaviour of a system accounting for the motion of a rigid body enclosed in a viscous incompressible fluid.…”

Rigorous analysis of the interaction problem of sea ice with a rigid body

“…It is a classical problem in fluid mechanics to study the movement of rigid or elastic bodies immersed in a fluid, see e.g. the works of Galdi [15], Hoffmann and Starovoitov [24], Desjardins and Esteban [10,11], Gunzberger, Lee and Seregin [21], Feireisl, Hillairet and Necasova [13], Tucsnak, Cumsille and Takahashi [7,8], Geissert, Götze and Hieber [18], Raymond, Maity, Roy and Vanninathan [32,39], and the recent work of Ervedoza, Maity and Tucsnak [12]. We also refer to the survey article [16] by Galdi and Neustupa for the stationary case.…”

“…We also refer to the survey article [16] by Galdi and Neustupa for the stationary case. The very recent article [12] by Ervedoza, Maity and Tucsnak discusses the longtime behaviour of a system accounting for the motion of a rigid body enclosed in a viscous incompressible fluid. In this article, we investigate the problem of interaction of a rigid body in sea ice.…”