Splash singularities for a general Oldroyd model with finite Weissenberg number

Abstract

In this paper we study a 2D free-boundary Oldroyd-B model which describes the evolution of a viscoelastic fluid. We prove the existence of splash singularities, namely points where the free-boundary remains smooth but self-intersects. This paper extends the previous results obtained for the infinite Weissenberg number by the authors in Di Iorio et al. (Splash singularity for a free boundary incompressible viscoelastic fluid model, 2018. arXiv:1806.11089; Splash singularity for a 2D Oldroyd-B model with nonlinear Piola-Kirchhoff stress, Nonlinear Differ Equ Appl 24:60, 2017) to the more realistic physical case of any finite Weissenberg number. The main difficulty faced in 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 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 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.