Self-intersecting Interfaces for Stationary Solutions of the Two-Fluid Euler Equations

Abstract

We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a \(\mathcal {C}^{2,\alpha }\) smooth curve that intersects itself at one point, and the vorticity density on the interface is of class \(\mathcal {C}^\alpha \). The proof consists in perturbing Crapper’s family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper.