On the “viscous incompressible fluid + rigid body” system with Navier conditions

Abstract

In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole system “viscous incompressible fluid + rigid body” is assumed to occupy the full space ${\mathbb{R}}^3$. We start by proving the existence of global weak solutions to the Cauchy problem. Then, we exhibit several properties of these solutions. First, we show that the added-mass effect can be computed which yields better-than-expected regularity (in time) of the solid velocity-field. More precisely we prove that the solid translation and rotation velocities are in the Sobolev space $H^1$. Second, we show that the case with the body fixed can be thought as the limit of infinite inertia of this system, that is when the solid density is multiplied by a factor converging to $+\infty$. Finally we prove the convergence in the energy space of weak solutions “à la Leray” to smooth solutions of the system “inviscid incompressible fluid + rigid body” as the viscosity goes to zero, till the lifetime $T$ of the smooth solution of the inviscid system. Moreover we show that the rate of convergence is optimal with respect to the viscosity and that the solid translation and rotation velocities converge in $H^1(0,T)$.