The steady problem of the motion of a rigid ball in a Stokes–Poiseuille flow: differentiability of the solution with respect to the ball position

This paper deals with the steady problem of the motion of a rigid body in a viscous incompressible fluid that fills a cylindrical domain. The fluid flow is governed by the Stokes equation and tends to Poiseuille flow at infinity. The body is a ball that moves according to the laws of classical mechanics. The unique solvability of this problem was proved in an earlier work of the authors. Here, the differentiability of the solution in the function space $L^2$ with respect to the position of the ball is established.