On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data

Abstract

A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weak solutions when the initial vorticity is a bounded Radon measure with distinguished sign and lies in the Sobolev space $H^{-1}$. In this paper we are interested in the case where there is a rigid body immersed in the fluid moving under the action of the fluid pressure. We succeed to prove the existence of solutions à la Delort in a particular case with a mirror symmetry assumption already considered by Lopes Filho et al. (2006) [11], where it was assumed in addition that the rigid body is a fixed obstacle. The solutions built here satisfy the energy inequality and the body acceleration is bounded. When the mass of the body becomes infinite, the body does not move anymore and one recovers a solution in the sense of Lopes Filho et al. (2006) [11].