Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity

TL;DR

This paper establishes local well-posedness and instant analyticity for the two-phase Navier-Stokes equations with surface tension and gravity, addressing cases including Rayleigh-Taylor instability.

Contribution

It provides the first rigorous proof of well-posedness and analyticity for the two-phase Navier-Stokes problem with gravity and free interface.

Findings

Proved local well-posedness for the two-phase Navier-Stokes with gravity.

Showed solutions become real analytic instantaneously.

Addressed the case of Rayleigh-Taylor instability.

Abstract

We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously.