The vanishing limit of a rigid body in three-dimensional viscous incompressible fluid

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space ${\text{rline}}^3$. When the small rigid body shrinks to a “massless” point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier-Stokes equations in the full space. Based on some $L^p-L^q$ estimates of the fluid-structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit.