Non-existence of splash singularities for the two-fluid Euler--Navier-Stokes system

TL;DR

This paper proves that in a two-fluid system with one inviscid and one viscous fluid, splash singularities cannot form in finite time, even with complex vorticity effects, extending understanding of fluid interface stability.

Contribution

It establishes the non-existence of splash singularities in a two-fluid Euler--Navier-Stokes system with vorticity, a novel result in fluid dynamics.

Findings

Splash singularities are precluded in the system.

Vorticity effects do not lead to finite-time singularities.

First result addressing vorticity's role in preventing splash singularities.

Abstract

We consider a system of two incompressible fluids separated by a free interface. The first fluid is inviscid, governed by the Euler system, while the second fluid has positive viscosity and is governed by the Navier-Stokes system. We formulate a notion of splash-type self-intersection singularities, and show that this system cannot form such a singularity in finite time. The main obstacle in our analysis is to handle the effect of bulk vorticity induced by the Navier-Stokes flow. To the best of our knowledge this is the first result on preclusion of splash-type singularities in the presence of non-trivial vorticity interior to one of the fluids.