# Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations

## TL;DR

This paper constructs stationary solutions to the 2D two-fluid Euler equations with a self-intersecting interface, demonstrating splash singularities using perturbations of known solutions and novel weighted estimates.

## Contribution

It introduces a method to create stationary two-fluid solutions with self-intersecting interfaces, extending the understanding of splash singularities in fluid dynamics.

## Key findings

- Existence of stationary solutions with splash singularities.
- Introduction of weighted estimates for self-intersecting interfaces.
- Perturbation of Crapper's solutions to include a second fluid.

## 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 $C^{2,\alpha}$ smooth curve that intersects itself at one point, and the vorticity density on the interface is of class $C^\alpha$. 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.