# Weak-strong uniqueness for the Navier-Stokes equation for two fluids with surface tension

## TL;DR

This paper establishes a weak-strong uniqueness principle for two-fluid Navier-Stokes equations with surface tension, showing varifold solutions coincide with strong solutions when they exist, thus ensuring uniqueness in the absence of singularities.

## Contribution

It introduces a relative entropy functional for free boundary problems, proving uniqueness of varifold solutions in two-fluid Navier-Stokes systems with surface tension.

## Key findings

- Varifold solutions coincide with strong solutions when they exist.
- The relative entropy functional effectively controls interface errors.
- Discontinuous velocity gradients at the interface require adapted entropy measures.

## Abstract

In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension - like, for example, the evolution of oil bubbles in water. Our main result is a weak-strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier-Stokes equation: As long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities the concept of varifold solutions - whose global in time existence has been shown by Abels [2] for general initial data - does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.