We show that so-called splash singularities cannot develop in the case of locally smooth solutions of the two-fluid interfaces in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro, Córdoba, Fefferman, Gancedo, and Gómez-Serrano in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.
"On the absence of splash singularities in the case of two-fluid interfaces." Duke Math. J. 165 (3) 417 - 462, 15 February 2016. https://doi.org/10.1215/00127094-3166629