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On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data. (English) Zbl 1366.35144
Summary: 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 M. C. Lopes Filho et al. [J. Differ. Equ. 229, No. 1, 154–171 (2006; Zbl 1102.76009)], 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. [loc. cit.].

35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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