Splash singularities for a 2D Oldroyd-B model with nonlinear Piola-Kirchhoff stress

In this paper consider a 2-D free boundary Oldroyd-B model at infinite Weissenberg number, under the assumption that the Piola-Kirchoff tensor, entering in the description of the extra-stress tensor, is given by a quadratic, convex energy functional. Our main goal is to investigate the existence of splash type singularities, namely points of self-intersection of the free boundary. The analysis of this problem requires to map the equations via a conformal transformation, in order to separate the singular points, and then to fix the free boundary via a Lagrangian change of coordinates. The investigation starts by proving local existence and stability results for a family of smooth initial configurations which, by considering a special class of initial data, allow us to show the existence of solutions having a self-intersecting configuration. As a consequence of this fact, we can conclude there exists a configuration, which has a singularity of splash type.