We discuss a 2D free-boundary Oldroyd model which describes the evolution of a viscoelastic fluid. We prove [3] the existence of splash singularities, namely points where the free-boundary remains smooth but self-intersects. This paper extends previous result obtained for infinite Weissenberg number by the authors in [1], [2] to the more realistic physical case of any finite Weissenberg number. The main difficulty faced here is due to the non-linear balance law of the elastic tensor, which cannot be reduced, as in the case of infinite Weissenberg, to a transport equation for the deformation gradient. Overcoming this difficulty requires a very accurate local existence theorem in terms of dependence on the Weissenberg number. The method in this case also is based on the combined use of conformal transformations and Lagrangian coordinates, whose formulation must however take into account the general balance law of the elastic tensor and its dependence on the Weissenberg number. The existence of the splash singularities is therefore guaranteed by an adequate choice of initial data, depending also on the elastic tensor, combined with stability estimates. Joint with Elena di Iorio and Stefano Spirito.
