On the weak uniqueness of “viscous incompressible fluid + rigid body” system with Navier slip-with-friction conditions in a 2D bounded domain

The existence of weak solutions to the “viscous incompressible fluid + rigid body” system with Navier slipwith-friction conditions in a 3D bounded domain has been recently proved by Gérard-Varet and Hillairet in [8]. In 2D for a fluid alone (without any rigid body) it is well-known since Leray that weak solutions are unique, continuous in time with L regularity in space and satisfy the energy equality. In this paper we prove that these properties also hold for the 2D “viscous incompressible fluid + rigid body” system. Introduction The problem of a rigid body immersed in an incompressible viscous fluid with different boundary conditions has been studied a lot in the past years. At a mathematical level we have a bounded domain Ω, independent in time, which is the union of two time-dependent domains F(t) and S(t), i.e. Ω = F(t)∪S(t) as in Figure 1, where F(t) is the part of the domain fulfilled by an incompressible viscous fluid, which satisfies Navier-Stokes equations and S(t) the part of the domain which is occupied by the body which rigidly moves following Newton’s laws. The problem is to study the evolution of the motion of the fluid and of the rigid body. Until the body does not touch the boundary, there are two separate boundaries: ∂Ω and ∂S(t), where we can impose different boundary conditions. The most classical setting for this problem is to prescribe no-slip boundary condition on both ∂S(t) and ∂Ω. In this case a wide literature is available regarding the existence of both weak and strong solutions, see [11], [17], [7], [15]. Moreover in the 2D case weak solutions are also continuous in time with values in Lσ and unique [10]. Another option is to prescribe Navier slip-with-friction boundary condition on both ∂Ω and ∂S(t). This condition naturally appears in the rugosity limit, see [3], and allows collision between the body and the boundary, see for example [9], in contrast with the lack of collision in the no-slip case [12]. In [16], the authors prove a first result of existence of weak solutions in the case where Ω = R. In the case of a bounded domain Ω of R the existence of weak solutions has been proven by Gérard-Varet and Hillairet in [8]. Their result can be easily adapted to the 2D case, see Theorem 1 below, which is the 2D counterpart of Theorem 1 in [8]. Indeed Theorem 1 involves a slightly wider set of test functions for which the density property mentioned in Lemma 1 is guaranteed. In this paper we prove that the weak solutions are continuous in time with values in Lσ and satisfy an energy equality, see Theorem 2. Finally we prove that the weak solutions are unique, which is the counterpart of [10] for Navier slip-with-friction boundary conditions, see Theorem 3. To establish the two first properties we rely on a change of variables due to [13], see Claim 1, and some regularization processes adapted to the body motion, see (24)-(25) (where Lemma 1 is used) and Claim 2. On the other hand to establish uniqueness we use some maximal regularity for an auxiliary system, see Theorem 4 below, thanks to the R-boundedness for the Stokes operator with Robin (i.e. Navier slip-with-friction) boundary conditions presented in [18]. Another interesting result is the work [1], where the authors study the imaginary ∗Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 351 cours de la Libération, F33405 Talence Cedex, France.