Abstract
This chapter presents the study of a particular problem concerning the motion of a two-dimensional incompressible nonviscous fluid. The behavior of a fluid for which the initial vorticity is sharphy concentrated around a finite number of points in the plane is reasonably described by a system called the point vortex system with finite degrees of freedom. The chapter reviews a mixed problem in which the singular part (vortex) and the bounded part (wave) are both present. The problem is not trivial because the proof of the existence of the time evolution for the bounded part is based on a quasi-Lipschitz condition for the velocity field, while the field produced by the point vortices is infinite in some points. To overcome this difficulty, a priori estimate is used on the possible approach of a trajectory to the point vortices. The possibility of a rigorous derivation of the model from the usual Euler equation is also discussed in the chapter.
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