On the existence of stationary splash singularities for the Euler equations

TL;DR

This paper demonstrates the existence of stationary solutions to the Euler equations featuring splash singularities, where fluid interfaces or water waves come arbitrarily close to touching, highlighting complex behaviors in fluid dynamics.

Contribution

It establishes the existence of stationary Euler solutions with splash singularities, a novel result in the study of fluid interfaces and water waves.

Findings

Stationary solutions with splash singularities exist for two-fluid Euler equations.

Stationary water waves with splash singularities are constructed.

Interfaces can be arbitrarily close to a splash in stationary configurations.

Abstract

In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.