  In this paper we study a 2D Oldroyd free-boundary model which describes the
evolution of a viscoelastic fluid. We prove existence of splash singularities,
namely points where the boundary remains smooth but self-intersects. This paper
extends the previous results obtained for infinite Weissenberg number to the
case of any finite Weissenberg number. The main difficulty of this paper is due
to the non linear balance law of the elastic tensor which cannot be reduced, as
in the case of infinite Weissenberg, to the transport equations for the
deformation gradient. Our strategy in accurate local existence result depending
on the Weissenberg number and the combination of Conformal and Lagrangian
transformations. The existence of splash singularities is guarantee by a
suitable choice of the initial data combined with stability estimates.

Analysis of PDEs