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Smooth solutions for the motion of a ball in an incompressible perfect fluid. (English) Zbl 1173.35105
The authors consider the system of partial and ordinary differential equations which describes the motion of a rigid ball in the ideal fluid, i.e. the Euler incompressible equations, coupled with the system of ordinary differential equations describing the motion of the ball.
In \(N\) space dimensions, provided the inital condition is sufficiently smooth, the authors show the existence of a smooth solution (with arbitrary smoothness if the data are so) to the studied system of differential equations. The proof is based on the abstract Kato-Lai theory from [T. Kato and C. Y. Lai, J. Funct. Anal. 56, 15–28 (1994; Zbl 0545.76007)]. The main difficulty is connected with the fact that a version of the Leray projection in domain changing with time has to be computed explicitely.
If \(N=2\), the solution is global in time. The proof is based on a priori estimates relating the velocity to the vorticity in an exterior domain. The regularity of the solution is expressed in Sobolev spaces, without any weights.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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