Interfaces in incompressible flows

TL;DR

This paper reviews and extends models of gravity-capillary nonlinear waves in incompressible fluids, presenting new results on the well-posedness of the associated PDEs and covering both bidirectional and unidirectional wave equations.

Contribution

It introduces new models and provides rigorous analysis of their well-posedness, enhancing understanding of nonlinear wave dynamics in incompressible flows.

Findings

Derived new bidirectional and unidirectional wave equations

Established well-posedness results for the models

Unified previous works with new theoretical insights

Abstract

The motion of both internal and surface waves in incompressible fluids under capillary and gravity forces is a major research topic. In particular, we review the derivation of some new models describing the dynamics of gravity-capillary nonlinear waves in incompressible flows. These models take the form of both bidirectional and unidirectional nonlinear and nonlocal wave equations. More precisely, with the goal of telling a more complete story, in this paper we present the results in the works \cite{GS,aurther2019rigorous,granero2019models,granero2019well,granero2020well,granero2021motion, granero2021global} together with some new results regarding the well-posedness of the resulting PDEs.