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 $${\mathbb {R}}^3$$ . 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 $$h_\infty \in {\mathbb {R}}^3$$ 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 fluid-structure 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).